mydate mydate

# Inferring health conditions
from fMRI-graph data

P.G.L. Porta Mana 1, C. Bachmann 2,∗,A. Morrison 2,3,4

###### Resumo

## 1

Automated classification methods for disease diagnosis are currently in the
limelight, especially for imaging data. Classification does not fully meet a
clinician’s needs, however: in order to combine the results of multiple tests
and decide on a course of treatment, a clinician needs the likelihood of a
given health condition rather than binary classification yielded by such
methods. We illustrate how likelihoods can be derived step by step from first
principles and approximations, and how they can be assessed and selected,
using fMRI data from a publicly available data set containing schizophrenic
and healthy control subjects, as a working example. We start from the basic
assumption of partial exchangeability, and then the notion of sufficient
statistics and the “method of translation” (Edgeworth, 1898) combined with
conjugate priors. This method can be used to construct a likelihood that can
be used to compare different data-reduction algorithms. Despite the
simplifications and possibly unrealistic assumptions used to illustrate the
method, we obtain classification results comparable to previous, more
realistic studies about schizophrenia, whilst yielding likelihoods that can
naturally be combined with the results of other diagnostic tests.

##  2 Keywords:

disease diagnosis, decision theory, sufficient statistics, exchangeability,
parametric statistical model, schizophrenia, fMRI, Bayesian probability theory

mydate 30 January 2018; updated 3 May 2018

##  3 Introduction

A 29-year-old man seeks medical advice because he finds himself in a very
confused state. The clinician, after listening to the complaints of the
patient, identifies some diseases that would account for the symptoms.
However, the presentation is not clear cut, and treatment for some of the
potential conditions have significant side effects. To come to a decision on
the best course of action, the clinician decides to perform the differential
diagnosis in a mathematically sound manner (Sox et al., 2013), first assigning
an initial probability for the patient’s being healthy or having each of the
potential conditions, taking into account age, sex, familial factors,
symptoms, a psychological evaluation, the incidence of the disease, and
similar prior information:

$\mathrm{P}(\text{health condition}\mathpunct{\,|\ }\text{prior info}).$ (1)

Then she orders one or more diagnostic tests to make a better informed
assessment of the probabilities of the considered diseases. Among these tests
she orders a structural and functional magnetic-resonance imaging (MRI) scan.
The advantage of MRI lies in the non-invasive monitoring of brain structure
and activity; the structural image (sMRI) is used to exclude morphological
changes in the brain such as tumours, and the functional imaging (fMRI) can
provide information about changes in brain activity.

With the results of the tests and of the sMRI and fMRI, the clinician updates
her initial or prior probability to a “post-test” or posterior probability
based on the results, according to Bayes’s theorem:

$\overbrace{\vphantom{\bigg{\\{}}\mathrm{P}(\text{health
condition}\mathpunct{\,|\ }\text{results of all tests}\mathbin{\ \land\
}\text{prior info})}^{\makebox[0.0pt][c]{\scriptsize\emph{post-test
probability}}}\mathrel{\ \propto\
}\overbrace{\vphantom{\bigg{\\{}}\mathrm{P}(\text{health
condition}\mathpunct{\,|\ }\text{prior
info})}^{\makebox[0.0pt][c]{\scriptsize\emph{initial probability}}}\mathbin{\
\times\ }{}\\\\[3.0pt]
\text{\scriptsize\emph{likelihoods}}\left\\{\;\begin{aligned}
\mathrm{P}(\text{result of first test}\mathpunct{\,|\ }\text{health
condition}\mathbin{\ \land\ }\text{prior info})&\mathbin{\ \times\ }{}\\\
\mathrm{P}(\text{result of second test}\mathpunct{\,|\ }\text{health
condition}\mathbin{\ \land\ }\text{prior info})&\mathbin{\ \times\ }{}\\\
{}\dotsb&\mathbin{\ \times\ }{}\\\ \mathrm{P}(\text{result of
sMRI}\mathpunct{\,|\ }\text{health condition}\mathbin{\ \land\ }\text{prior
info})&\mathbin{\ \times\ }{}\\\ \mathrm{P}(\text{result of
fMRI}\mathpunct{\,|\ }\text{health condition}\mathbin{\ \land\ }\text{prior
info})&\end{aligned}\right.$ (2)

where “$\land$” denotes logical conjunction (“and”), and we have reasonably
assumed that the result of each test does not depend on those of the other
tests, i.e. that their likelihoods are independent Jaynes, 2003, § 4.2; Sox et
al., 2013, § 4.7.

In the update formula above, the initial probability is assessed by the
clinician. To calculate the post-test probability she needs the probabilities
for each test result conditional on the health condition, either “healthy” or
“presumptive disease”. These probabilities are called the _likelihoods for the
health condition_ in view of each test. The term “likelihood” has its standard
technical meaning in the present work: the probability of a proposition $A$
given $B$ is $\mathrm{P}(A\mathpunct{|}B)$, while the likelihood of $A$ in
view of $B$ is $\mathrm{P}(B\mathpunct{|}A)$, i.e., $A$ appears in the
conditional Jaynes, 2003, § 4.1; Good, 1950, § 6.1. A proposition can have
high probability but low likelihood and vice versa. Probabilities, not
likelihoods, are what we base our decisions upon.

The final, post-test probability is necessary to the clinician to decide upon
a course of action Sox et al., 2013, ch. 6; Goodman, 1999; Murphy, 2012, §
5.7; for example, to treat the patient according to one or another specific
treatment, to dismiss him, or to order more tests. To make such decision the
clinician will combine her post-test probabilities for the health conditions
with a utility table (a reminder of decision theory is given in § 6.3).

In the following we assume that one of the presumptive diseases the clinician
has in her mind is schizophrenia. Although currently MRI does not play a role
in a diagnosis of schizophrenia, there are substantial efforts to develop such
analyses for this purpose (Silva and Calhoun, 2014). In this work, we focus on
the diagnosis of this particular disease simply as a concrete worked example,
to demonstrate how results of a diagnostic test, in this case results from
function MRI imaging, can be incorporated in the diagnostic process in a
principled fashion.

In short, we address the question: how can we assign a numerical value to the
likelihood

$\mathrm{P}(\text{fMRI result}\mathpunct{\,|\ }\text{health
condition}\mathbin{\ \land\ }\text{prior info}).$ (3)

of each health condition (’healthy’ or ’schizophrenic’) in view of the fMRI
result? We will propose an answer that can be applied for any brain disease.

To this end it is useful to mark out some features of the approach presented
so far:

  1. I.

Modularity. The update formula (2) combines evidence from different tests, and
this combination does not need to be done at once. The clinician can multiply
her initial probability by the likelihood from the first test, normalize, and
thus obtain a “post-first-test” probability. Later she can multiply this
probability by the likelihood from the second test, normalize, and thus obtain
a post-second-test probability; and so on with any number of other tests, a
number that the clinician needs not fix in advance. She can therefore store
the value of the likelihood from the fMRI result, to later combine it with new
likelihoods from future tests to form a new, better-informed post-test
probability.

  2. II.

Decision-theoretic character. The clinician’s final goal is not simply a
healthy/schizophrenic classification, but a _decision_ upon a course of action
about the patient Sox et al., 2013, chs 6, 7; Jaynes, 2003, chs 13, 14; Raiffa
and Schlaifer, 2000. This distinction is important: for example, a treatment
without contraindications might be recommended even if there is only a 10%
probability that the disease is present; or a dangerous treatment might be
recommended only if there is a 90% probability that the disease is present.

The modularity of the present approach extends to the decision stage, because
the post-test probability can be used with different decisions and utilities,
which can also be updated later on. For example, after beginning a treatment
the clinician happens to read about a new kind of treatment, having new
benefits and contraindications. Using the post-test probabilities she already
has, she may re-evaluate her decision using an updated utility table that
includes the new treatment.

  3. III.

Incomplete knowledge. In general, we lack a complete biological understanding
of the relation between brain activity and the health condition under study.
In this case, the likelihoods can only be assessed by relying on examples of
known _health condition–fMRI data_ pairs, usually called a training or
calibrating dataset. Moreover, this training dataset is often very small.

  4. IV.

High dimensionality. The fMRI data are positive-valued vectors with
$10^{7}$–$10^{8}$ components or more (Lindquist, 2008). This high dimension
impacts the calculation of likelihoods and probabilities.

The first two points above are great advantages of the present approach, and
also the reasons why it cannot be based on machine learning algorithms for
deterministic classification; such methods give the clinician a dichotomous,
“healthy/schizophrenic” answer, with no associated uncertainty. This answer
cannot be used by the clinician to weigh the benefits and risks of different
courses of action, the assessment of which needs the probabilities of the
health conditions. Probabilistic algorithms, on the other hand, are not
flexible for combining evidence: they give a probability for the health
condition, not a likelihood; and only the latter can be combined with the
likelihoods from other tests, or stored for later reuse and combination.

We therefore approach the question of assigning the likelihoods (3) by means
of the probability calculus, the same calculus from which eq. (2) is derived.
What we will do is in essence no different from current Bayesian statistical
analyses and modelling; but we would like to emphasize some aspects of this
modelling that are usually left in the background. The probability calculus
can be regarded as the extension of formal logic (truth calculus) to plausible
inference (Jeffreys, 2003; Jaynes, 2003; Hailperin, 1996), a view also
supported in medicine (Greenland, 1998; Maclure, 1998; Goodman, 1999), which
has been proven with increasing rigour by Koopman (1940b; 1940a; 1941), Cox
(1946; 1961; 1979), Pólya (1949; 1968), and many others (Horvitz et al., 1986;
Paris, 2006; Halpern, 1999; Snow, 1998; Dupré and Tipler, 2009; Terenin and
Draper, 2017). The derivation of a probability proceeds much like an “axioms
$\rightarrow$ logic rules $\rightarrow$ theorem” derivation in formal logic:
one starts from the probabilities of some propositions, and by applying the
probability rules, arrives at the probability of the desired proposition, eq.
(3) in our case.

We will show this procedure step by step in the case of our problem, in order
to expose where assumptions and approximations enter the derivation. These may
be improved by other researchers, or replaced by different ones when the
method is applied to a different problem. Our discussion is inspired by
Mosteller & Wallace’s (Mosteller and Wallace, 1963) brilliant, thoughtful
analysis of a statistically similar problem in a very different context.

The approach we follow deals naturally with the four points listed above. The
small size of the training dataset, point III. above, is not an issue because
the probability calculus allows for training datasets of any size. In fact,
the calculus allows us to continuously update our inferences given new
training data, making our inferences more and more precise and less likely to
be affected by outliers.

The unmanageable size of our data space, point IV. above, will force us to
make auxiliary assumptions that will translate into the choice of a reduced
data space, discussed in § 4.3, and into the use of parametric statistical
models, discussed in § 4.4. Regarding the latter, we will emphasize that
assumptions about relevant and irrelevant information in our data may
translate into mathematical statistical models. It is often difficult to
relate biophysical considerations about quantities measured in the brain to
the shape of a probability distribution, especially in multidimensional
quantities. The notion of _sufficient statistics_ Dawid, 2013; Bernardo and
Smith, 2000, ch. 4; Lindley, 2008, § 5.5; Diaconis and Freedman, 1981;
Cifarelli and Regazzini, 1982; Lauritzen, 1988; Kallenberg, 2005, discussed in
§ 4.4.2, is a helpful bridge between biophysical considerations and
probability distributions. The idea is that it may be easier for us to
conceive a connection between biophysical considerations and some special
statistics of our measurements, than between biophysical considerations and an
abstract multidimensional distribution function. This “translation” is
powerful, because if one finds the assumptions about relevance or irrelevance
of some data unreasonable, one can then make different assumptions, resulting
in a different statistical model.

Models inspired by sufficient statistics – especially their comparison and
selection – can nevertheless be computationally demanding owing to the
multidimensional integrals in their formulae, even when these are addressed by
modern numerical methods such as Monte Carlo MacKay, 2003, ch. IV; Murphy,
2012, chs 23–24. In the present study we shall use analytically tractable
statistical models, but availing ourselves of Edgeworth’s “Method of
Translation” 1898; Johnson, 1949; Mead, 1965: the simple but potentially very
fertile idea of transforming a quantity into a normally distributed one,
discussed in § 4.4.3.

The possible combined choices of reduction of the data space, of sufficient
statistics, and of transformations into normal variable, lead to a variety of
possible models and likelihoods to be used by the clinician. Which is the
“best” one? We discuss several criteria for choice in § 4.5, settling on one
based on expected utility. We also briefly discuss the remarkable observation
that common Bayesian criteria based on weight of evidence and Bayes factors
Jeffreys, 2003, chs V, VI, A; Good, 1950; MacKay, 1992a; Kass and Raftery,
1995 for the fMRI data gives results opposite to those of the expected-utility
criterion.

In this article, we will calculate the likelihoods for the health conditions,
eq. (3), and assess the models for so doing, in the following steps. First, in
§ 4.1, we briefly discuss schizophrenia and the use of fMRI to diagnose it,
introducing a concrete dataset of fMRI data for schizophrenic and healthy
patients. We then show that a simple and natural assumption, called
_exchangeability_ , would lead to a unique value of the likelihoods (3) if the
training dataset were large enough (§ 4.2). However, with a small training
dataset we must face two problems: unmanageably large dimensions of the data
space, and the need to specify prior beliefs, also involving functions on
infinite-dimensional spaces.

To solve the first problem, in § 4.3 we assume that information adequate for
our health inference can be found in a reduced data space of the fMRI, which
we construct from time correlations between groups of voxels. To solve the
second problem we introduce parametric statistical models in § 4.4 using the
notions of _sufficient statistics_ and of transformation into normal
variables, mentioned above. We discuss how these models learn from the data
and select three models as possible candidates. We then consider several
criteria to select one of the three models against our data, as an example,
and discuss how a more realistic assessment could be made in a real
application (§ 4.5 ). We conclude with a discussion (§ 5) on how the choice of
sufficient statistics and prior probabilities could be improved, and on the
relation to machine-learning methods.

Our statistical terminology and notation follow ISO standards (iso, 2009,
2006).

##  4 Results

###  4.1 Selection of clinical use case and fMRI-data acquisition

Schizophrenia is a psychiatric disorder that comprises various symptoms that
are categorized into positive (e.g. hallucinations), negative (e.g. loss of
motivation) and cognitive (e.g. memory impairment) disease patterns. A common
disease cause for all these widespread symptoms is still unknown. Functional
magnetic resonance imaging (fMRI) has been used to gain insight into
modifications in functional connectivity in this disease. In the resting
state, functional connectivity is measured either by asking the subject to
fulfil a certain task or at rest, instructing the subject to think about
nothing specific but not fall asleep. In this condition, both increased and
decreased functional connectivity have been reported in the default mode
network, although the hyperactivity seems to be reported more often (Hu et
al., 2017). Moreover, widespread connectivity changes in the dorsal attention
network and the executive control network have been detected (Woodward et al.,
2011; Yu et al., 2016).

Beyond these individual sub-networks, many studies have found profound changes
in macroscopic brain structures, e.g. a shrinkage of whole brain and
ventricular volume, reduced gray matter in frontal, temporal cortex and
thalamus, and changes in white matter volume in frontal and temporal cortex
(Shenton et al., 2010; Ellison-Wright and Bullmore, 2009, 2010). Since both
gray matter loss and white matter changes are found, it is reasonable to
conclude that not only the intrinsic activity of single areas is modified, but
also the interplay of different brain areas, in particular in frontal and
temporal cortex. It has been argued that these alterations in long range
connectivity are responsible for a range of disease symptoms that are not
attributable to single areas (Friston and Frith, 1995). Taking this disconnect
hypothesis as a starting point, we can reach the working hypothesis that these
changes are also reflected in the functional activity of the brain, and that
fMRI images can be used to distinguish schizophrenic from healthy patients. We
therefore conclude that schizophrenia is an appropriate condition to
demonstrate our approach.

We requested data of schizophrenic and healthy patients from
Schizconnect111http://schizconnect.org/, a virtual database for public
schizophrenia neuroimaging data. In our request we asked for resting state
T2*-weighted functional (rfMRI) and T1-weighted structural magnet resonance
images (MRI) from patients participating in the COBRE study either with no
known disorder or diagnosed as schizophrenic according to the Diagnostic and
Statistical Manual of Mental Disorders (DSM) IV, excluding schizoaffective
disorders. In the COBRE study, the voluntary and informed participation of the
subjects was ensured by the institutional guidelines at the University of New
Mexico Human Research Protections Office. The ensuing dataset comprised 91
healthy patients and 74 schizophrenic patients. Out of these we randomly
selected 54 healthy and 49 schizophrenic subjects, to permit demonstration our
method on a small dataset with unequal group size. A detailed description on
the exact experimental design and the MRI scanning is provided by Çetin et al.
(2014).

###  4.2 Calculation of probabilities: exchangeability

Let us describe our context more precisely and set up some mathematical
notation. We have:

  * •

A number of possible health conditions, in our example healthy (h) and
schizophrenic (s). The variable $c$ denotes health condition.

  * •

A space of possible fMRI data. They are vectors with $10^{7}$–$10^{8}$ or more
positive components (Lindquist, 2008). The variable $\bm{f}$ denotes an fMRI
result.

  * •

A set of $n$ patients, labelled in some way, the variable $i$ denoting their
labels. These labels may reflect information about the times the patients were
examined, or about their geographical location. This possibility is important
in the considerations to follow. In our study $n=104$.

  * •

Knowledge of the health condition and of the fMRI result of each patient. Let
us use the propositions

$\displaystyle C_{i}^{c}$
$\displaystyle\coloneqq\text{\leavevmode\ltxml@oqmark@open\textquotedblleft\penalty
10000\hskip-0.0002pt\hskip 0.0002ptPatient $i$ has health condition
$c$\textquotedblright\ltxml@oqmark@close{}},$ (4) $\displaystyle
F_{i}^{\bm{f}}$
$\displaystyle\coloneqq\text{\leavevmode\ltxml@oqmark@open\textquotedblleft\penalty
10000\hskip-0.0002pt\hskip 0.0002ptThe fMRI of patient $i$ gives
$\bm{f}$\textquotedblright\ltxml@oqmark@close{}},$

the latter to be understood within a very small interval
$(\bm{f},\bm{f}+\mathrm{d}\bm{f})$. In our study we have $n_{\textsc{h}}=55$
healthy and $n_{\textsc{s}}=49$ schizophrenic patients.

For brevity we denote by $C^{c}$ the conjunction of the propositions
$C_{i}^{c}$ for all patients having health condition $c$, i.e. our knowledge
about which patients have that health condition; and analogously for $F^{c}$.
By $D^{c}$ we denote all data about patients with health condition $c$; by $D$
we denote all our data.

  * •

An imaginary patient, labelled “$0$”, whose fMRI result $\bm{f}$ is known, but
whose health condition is not.

  * •

Other pre-test information, denoted by $I$; for example the clinician’s
initial diagnosis of the health condition of patient $0$, and the results of
any other diagnostic tests.

  * •

The probabilities $(\hat{p}_{\textsc{h}},\hat{p}_{\textsc{s}})$ for the health
condition of patient $0$, conditional on the pre-test information, including
the results from other tests. We call these _pre-test probabilities_. Note
that they may differ from the _initial_ probabilities of eqs (1)–(2), because
they may include the likelihoods from other tests.

  * •

A set of decisions about the patient $0$ and their utilities conditional on
the patient’s health condition. We shall simply consider two decisions:
dismiss (d) or treat (t). See § 6.3 for a summary of decision theory.

Our goal is to assign numerical values to the likelihoods for the health
conditions (3): the conditional probability distribution that the fMRI result
of patient “$0$” is $\bm{f}$ given the health condition of that patient and
all other data. In our notation,

$\displaystyle\mathrm{p}(F_{0}^{\bm{f}}\mathpunct{|}C_{0}^{c}\mathbin{\ \land\
}C_{1}^{c_{1}}\land F_{1}^{\bm{f}_{1}}\mathbin{\ \land\ }\dotsb\mathbin{\
\land\ }C_{n}^{c_{n}}\land F_{n}^{\bm{f}_{n}}\mathbin{\ \land\
}I)\,\mathrm{d}\bm{f}$ (5) or just
$\displaystyle\mathrm{p}(F_{0}^{\bm{f}}\mathpunct{|}C_{0}^{c}\land D\land
I)\,\mathrm{d}\bm{f}.$

A natural assumption helps us restrict the values the distribution above may
have. Within a group of patients _having the same health condition_ we assume
that the probability that a patient shows a particular fMRI $\bm{f}_{i}$ does
not depend on the particular value of the patient’s label $i$, no matter how
many patients we have or may later add in that health group. This assumption
is called _partial exchangeability_ (de Finetti, 1938; Diaconis and Freedman,
1981; Aldous, 1985; Diaconis, 1988). If the labels carry e.g. temporal or
geographical information, partial exchangeability means that we do not expect
to observe particular kinds of fMRI results more often in the future than in
the past, or more frequently in one location than another. As a concrete
example: fix three possible fMRI results $\bm{f}_{1}$, $\bm{f}_{2}$,
$\bm{f}_{3}$ (each is a vector with $10^{7}$–$10^{8}$ positive components) and
consider the fMRI tests of three schizophrenic patients: say, one from five
years ago in Germany, one from last week in Scotland, and one to be done six
months from now in Italy. Partial exchangeability means that the probability
that the German patient’s test gave $\bm{f}_{1}$, the Scottish’s gave
$\bm{f}_{2}$, and the Italian’s will give $\bm{f}_{3}$, is numerically equal
to the probability that the German’s gave $\bm{f}_{2}$, the Scottish’s
$\bm{f}_{3}$, and the Italian’s will give $\bm{f}_{1}$; and likewise for all
six possible permutations of the three results. Keeping the same fixed fMRI
results $\bm{f}_{1}$, $\bm{f}_{2}$, $\bm{f}_{3}$, we now consider three
healthy patients instead, who may also live in different times and places.
Partial exchangeability means that also in this case the values of the six
possible joint probabilities obtained by permutation must all be equal – but
this value can be different from the one for the schizophrenic patients
considered before. Hence the term “partial”: we can freely exchange the joint
results within the schizophrenic group and within the healthy group without
altering their probabilities, but not across groups. This assumption extends
in an analogous way to more patients.

The assumption of partial exchangeability might not be completely true when we
consider geographical or epochal differences, but we may still consider it as
a good approximation. We are not making any exchangeability assumptions about
the probabilities of the health conditions of our patients, though, because
the incidence of a disease does often change with time and can depend heavily
on geographical location.

To express partial exchangeability mathematically, suppose that the patients
$i=1,2,3,\dotsc$ have health condition $c=\textsc{h}$ and the patients
$i^{\prime}=1^{\prime},2^{\prime},3^{\prime},\dotsc$ health condition
$c=\textsc{s}$. Then the joint distribution for their fMRI results satisfies

$\mathrm{p}\Bigl{(}\mathop{\textstyle\bigwedge}\nolimits_{i}F_{i}^{\bm{f}_{i}}\
\mathop{\textstyle\bigwedge}\nolimits_{i^{\prime}}F_{i^{\prime}}^{\bm{f}_{i^{\prime}}}\mathpunct{\big{|}\
}\mathop{\textstyle\bigwedge}\nolimits_{i}C_{i}^{\textsc{h}}\
\mathop{\textstyle\bigwedge}\nolimits_{i^{\prime}}C_{i^{\prime}}^{\textsc{s}}\mathbin{\
\land\
}I\Bigr{)}=\mathrm{p}\Bigl{(}\mathop{\textstyle\bigwedge}\nolimits_{i}F_{i}^{\bm{f}_{\pi(i)}}\
\mathop{\textstyle\bigwedge}\nolimits_{i^{\prime}}F_{i^{\prime}}^{\bm{f}_{\pi^{\prime}(i^{\prime})}}\mathpunct{\big{|}\
}\mathop{\textstyle\bigwedge}\nolimits_{i}C_{i}^{\textsc{h}}\
\mathop{\textstyle\bigwedge}\nolimits_{i^{\prime}}C_{i^{\prime}}^{\textsc{s}}\mathbin{\
\land\ }I\Bigr{)}\\\ \text{for all permutations $\pi$ of
$\\{i\\}\equiv\\{1,2,\dotsc\\}$, and all permutations $\pi^{\prime}$ of
$\\{i^{\prime}\\}\equiv\\{1^{\prime},2^{\prime},\dotsc\\}$.}$ (6)

The assumption of partial exchangeability is simple and quite natural – and
very powerful: it implies, as shown by de Finetti 1938; Diaconis, 1988, § 3;
Bernardo and Smith, 2000, § 4.6, that the joint distributions above must have
the form

$\mathrm{p}\Bigl{(}\mathop{\textstyle\bigwedge}\nolimits_{i}F_{i}^{\bm{f}_{i}}\
\mathop{\textstyle\bigwedge}\nolimits_{i^{\prime}}F_{i^{\prime}}^{\bm{f}_{i^{\prime}}}\mathpunct{\big{|}\
}C\land
I\Bigr{)}=\iint\Bigl{[}\mathop{\textstyle\prod}_{i}q_{\textsc{h}}(\bm{f}_{i})\Bigr{]}\,\Bigl{[}\mathop{\textstyle\prod}_{i^{\prime}}q_{\textsc{s}}(\bm{f}_{i^{\prime}})\Bigr{]}\;\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)\;\mathrm{d}q_{\textsc{h}}\,\mathrm{d}q_{\textsc{s}}$
(7)

where $q_{\textsc{h}}$, $q_{\textsc{s}}$ are distributions over the possible
values of $\bm{f}$, and
$\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)$ is a
“hyperdistribution” over such distributions, determined by the assumptions
$I$. The double integral (which can be understood as a generalized Riemann
integral: Lamoreaux and Armstrong, 1998; Swartz, 2001; Kurtz and Swartz,
2004), is over all distributions $q_{\textsc{h}}$, $q_{\textsc{s}}$. In other
words, de Finetti’s theorem say that the joint probability distribution for
the fMRIs of healthy and schizophrenic patients can be seen as the product of
independent distributions, identical for healthy cases and identical for
schizophrenic cases but different for the two cases, mixed over all possible
such pairs of distributions with weight
$\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)$.

As a very cursory example, suppose we want the joint probability that a
healthy patient has fMRI result $\bm{f}$ and a schizophrenic one
$\bm{f}^{\prime}$. De Finetti’s formula first tells us to consider all
possible distributions over positive vectors. As usual with infinities, “all”
must be made precise by specifying a topology (for details see e.g. de
Finetti, 1938; Diaconis and Freedman, 1981; Aldous, 1985; Diaconis, 1988); but
intuitively these distributions comprise, e.g., multivariate truncated
normals, gammas, exponentials, truncated Cauchys… and innumerable
distributions that we can imagine and don’t have a specific name for; all with
their possible parameter values. De Finetti’s formula tells us to choose one
distribution $q_{\textsc{h}}$, from all those possible ones, for the healthy
case and one $q_{\textsc{s}}$ for the schizophrenic case, and to attach a
weight to this pair,
$\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)$; then to calculate
this pair at the values $\bm{f}$, $\bm{f}^{\prime}$ and multiply them:
$q_{\textsc{h}}(\bm{f})\times q_{\textsc{s}}(\bm{f}^{\prime})$. Then we
consider a new pair of distributions, attach a weight to them, and again
multiply their values at $\bm{f}$ and $\bm{f}^{\prime}$. And so on, until all
possible pairs are considered. Finally we calculate the sum of all such
products, weighted accordingly: $\int
q_{\textsc{h}}(\bm{f})q_{\textsc{s}}(\bm{f}^{\prime})\,\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)\,\mathrm{d}\bm{f}\,\mathrm{d}\bm{f}^{\prime}$.

The generalization of the formulae above to more than two health conditions,
or when only one health condition is considered, is straightforward.

As the cursory example above made quite clear, an integral over probability
distributions is a mathematically complicated object (cf. Ferguson, 1974) and
may not seem a great advancement in assigning a value to the distribution (5).
In defence of de Finetti’s formula we must say that it is completely
manageable with discrete data spaces and provides a great insight in the way
we reason about probability in relation to repeated events (de Finetti, 1937;
Lindley and Phillips, 1976; Kingman, 1978; Koch and Spizzichino, 1982; Dawid,
2013; Bernardo and Smith, 2000, § 4.2). It also has several important
consequences for our inference, which we now discuss.

Using de Finetti’s formula and the definition of conditional probability we
can rewrite our goal plausibility (5) as

$\displaystyle\mathrm{p}(F_{0}^{\bm{f}}\mathpunct{|}C_{0}^{c}\land D\land
I)=\int
q_{c}(\bm{f})\,\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}D\land
I)\,\mathrm{d}q_{\textsc{h}}\,\mathrm{d}q_{\textsc{s}}$ (8a)
$\displaystyle\text{with}\quad\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}D\land
I)=\frac{\bigl{[}\mathop{\textstyle\prod}_{i}q_{c_{i}}(\bm{f}_{i})\bigr{]}\,\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)}{\int\bigl{[}\mathop{\textstyle\prod}_{i}q_{c_{i}}(\bm{f}_{i})\bigr{]}\,\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)\,\mathrm{d}q_{\textsc{h}}\,\mathrm{d}q_{\textsc{s}}}.$
(8b)

The latter is called posterior distribution since it is conditional on all
data $D$.

Excluding pathological prior distributions (Diaconis and Freedman, 1986), this
posterior distribution becomes more and more concentrated on two particular
distributions $\bigl{(}\hat{q}_{\textsc{h}},\hat{q}_{\textsc{s}}\bigr{)}$,
fully determined by the data, as our data $D$ comprise a larger and larger
number of patients. This concentration occurs independently of the original
shape of the distribution
$\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)$. In this limit our
probability distribution (5) becomes

$\mathrm{p}(F_{0}^{\bm{f}}\mathpunct{|}C_{0}^{c}\land D\land
I)\approx\hat{q}_{c}(\bm{f})$ (9)

with $\hat{q}_{c}$ completely determined by the data $D$. This would solve our
plausibility assessment (5).

De Finetti’s formula therefore tells us also the theoretical limit by which
the pre-test probability for the health condition of the patient,
$\mathrm{P}(C_{0}^{c}\mathpunct{|}D\land I)$, can be improved by the fMRI
result. For example, if $\hat{q}_{c}(\bm{f})$ is more or less uniform in
$\bm{f}$ or has the same peaks in $\bm{f}$ for each $c$, then the fMRI is of
no use for discriminating the health condition of the patient. This result
follows mathematically from the assumption of exchangeability (6) and the
rules of probability calculus, hence no amount of ingenuity could overcome
this limit.

In our case the amount of data $D$ is not enough to allow the use of the
approximation (9). We should use the general formula (8), but it is unwieldy
in two respects. First, the fMRI result $\bm{f}$ of a patient is a positive-
valued vector with $10^{7}$–$10^{8}$ components or more (Lindquist, 2008), so
the distributions $q_{\textsc{h}}$, $q_{\textsc{s}}$ are highly
multidimensional. Second, the formula asks us to consider in principle _all_
such distributions, as explained in the example above.

We tame this double unwieldiness in two ways. First, it is conceivable that
not all information contained in the fMRI result $\bm{f}$ of a patient be
relevant to discriminate the patient’s health condition $c$. The integral in
eq. (8), if it could be performed, would automatically winnow out the relevant
information (Jaynes, 2003, ch. 17), possibly reducing the problem to a lower-
dimensional set in the space of fMRI data. Being unable to perform the
integral, we must try to apply heuristics based on our understanding of the
target conditions and perform such dimensional reduction by hand. For example,
by employing the hypothesis that in schizophrenic patients the time
correlation between brain regions is altered with respect to healthy ones. We
can thus address the first problem by reducing fMRI data $\bm{f}$ to a
manageable set of graph properties $\bm{f}$, and applying our inference
directly on these, as explained in the next section.

Second, we entertain a working assumption about which features of our graph
data $\\{\bm{f}_{i}\\}$ from a set of patients are relevant for inferences
about new patients. We can, for example, assume that only the first and second
moments are relevant for making predictions about the graph quantities to be
observed in the new patients; these moments are then called _sufficient
statistics_. Assumptions of this kind reduce the infinite-dimensional space
all possible distributions $(q_{\textsc{h}},q_{\textsc{s}})$ to a finite-
dimensional space of a parametric family of distributions of exponential type,
as explained in § 4.4.

###  4.3 Trimming the data space: functional connectivity

The preprocessed functional image in standard space, in which the activity for
each voxel is recorded, consists of approximately $10^{17}$ time series of
$140$ time points. We reduce this huge data space by the following steps.

First, we consider only the activity of the voxels belonging to the $94$
regions defined by the lateral cortical Oxford atlas (see § 4.1 and Desikan et
al., 2006) and average the activity of all voxels in a region (details in §
6.1). Second, we measure the functional connectivity defined by the Pearson
correlation coefficient between pairs of regions, obtaining $94\times
93/2=4371$ connectivity weights. This is still a considerable data space for
our computational resources, so we select $d=40$ connectivity weights that
exhibit the greatest difference in their connection weight average across the
schizophrenic and healthy groups.

The resulting distributions for four of these connectivity weights are
depicted in fig. 1. Note that for each considered brain connection, the
histograms of the healthy group and the schizophrenic group display
significant overlap, such that none of them could be used in isolation to
reliably discriminate between two groups.

Figuro 1: Distributions of functional connectivities of schizophrenic and
healthy patients. Normalized density histograms of connectivity weights for
healthy (h, yellow, solid) and schizophrenic (s, red, dashed) patients of four
cortical connections selected to demonstrate our statistical framework.
Empirical means are shown as vertical lines. All connectivities for healthy
and schizophrenic patients have substantial overlap, evidenced by the darker
regions in the histograms.

Our data space is therefore vastly reduced, from $[0,\infty[^{10^{17}}$

to $[-1,1]^{40}$. We let the symbol $\bm{f}$ stand for the set of connectivity
weights extracted from the fMRI data, rather than the full fMRI data
themselves. With this new meaning of the symbol $\bm{f}$, the likelihoods for
the health conditions (5) and de Finetti’s formulae (7)–(8) remain formally
unchanged, but now involve a data space with much fewer dimensions.

###  4.4 Trimming the distribution space: models by sufficiency and
generalized normals

####  4.4.1 Parametric models

The integrals in de Finetti’s formulae (7)–(8) still represent a mixture of
all imaginable pairs of probability distributions
$(q_{\textsc{h}},q_{\textsc{s}})$ over the space $[-1,1]^{d}$, and are
therefore extremely complex. We now examine two ways to reduce this integral
to a manageable set of distributions and to obtain analytically tractable
formulae.

In the Bayesian literature, the complication of considering all possible
distributions $q_{c}(\bm{f})$ of the quantity $\bm{f}$ is typically
sidestepped by restricting them to a finite-dimensional set of distributions
$L_{c}(\bm{f}\mathpunct{|}\bm{\theta}_{c})$, identified or indexed by a finite
number of parameters $\bm{\theta}_{c}$. For this reason, such a set is called
a parametric family of distributions. An example of parametric family is the
set of $d$-variate normal distributions parameterized by their mean $\bm{\mu}$
and covariance matrix $\bm{\varSigma}$. With this restriction, the integrals
in de Finetti’s formulae (7) and (8) represent mixtures of distributions
within the parametric family, the weight for each distribution being
represented by a weight for its parameters. That is, we are no longer
considering mixtures of all possible multivariate truncated normals, gammas,
exponentials, etc., as in the example of § 4.2, but only mixtures of truncated
normals, say, with different means and covariance matrices. These integrals
are thus ordinary finite-dimensional integrals. What happens to formulae
(7)–(8) is that

$q_{c}(\bm{f})\text{ is replaced by
}L_{c}(\bm{f}\mathpunct{|}\bm{\theta}_{c}),\qquad\mathrm{p}(q_{\textsc{h}},q_{\textsc{s}}\mathpunct{|}I)\,\mathrm{d}q_{\textsc{h}}\,\mathrm{d}q_{\textsc{s}}\text{
is replaced by
}\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}M,I)\,\mathrm{d}\bm{\theta}_{\textsc{h}}\,\mathrm{d}\bm{\theta}_{\textsc{s}}.$
(10)

The distribution $L_{c}$ is called the likelihood of the parameters
$\bm{\theta}_{c}$ , and
$\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}M,I)$
is the prior parameter distribution. A parametric family and a prior
distribution over its parameters are jointly called a parametric statistical
model, which we denote by $M$. The term “model” is justly criticized by some
probability theorists (see e.g. Besag & Kalman in Besag et al., 2002) but
widely used, so we shall adopt it here.

In our present problem, the parametric statistical model needs not be the same
for all health conditions: for example, we could use normal distributions for
one condition and beta distributions for another, if that choice better
reflected the distributions of connectivity weights under the two different
health conditions. For this reason, we use the subscript “$c$” in the formulae
above. The likelihood we want to determine, eq. (5), thus becomes, from eq.
(8) with the replacements (10),

$\displaystyle\mathrm{p}(F_{0}^{\bm{f}}\mathpunct{|}C_{0}^{c}\land D\land
M\land I)=\int
L_{c}(\bm{f}\mathpunct{|}\bm{\theta}_{c})\,\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}D,M,I)\,\mathrm{d}\bm{\theta}_{\textsc{h}}\,\mathrm{d}\bm{\theta}_{\textsc{s}}$
(11a)
$\displaystyle\text{with}\quad\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}D,M,I)=\frac{\bigl{[}\mathop{\textstyle\prod}_{i}L_{c_{i}}(\bm{f}_{i}\mathpunct{|}\bm{\theta}_{c_{i}})\bigr{]}\,\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}M,I)}{\int\bigl{[}\mathop{\textstyle\prod}_{i}L_{c_{i}}(\bm{f}_{i}\mathpunct{|}\bm{\theta}_{c_{i}})\bigr{]}\,\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}M,I)\,\mathrm{d}\bm{\theta}_{\textsc{h}}\,\mathrm{d}\bm{\theta}_{\textsc{s}}}.$
(11b)

####  4.4.2 Models by sufficient statistics

The choice of a statistical model often appears as an art and a matter of
experience. Notable statisticians have voiced concerns over the esoteric
character of this choice. Dawid (1982a, p. 220) says: “Where do probability
models come from? To judge by the resounding silence over this question on the
part of most statisticians, it seems highly embarrassing”. And Diaconis (1988,
§ 8, p. 121) remarks:

> de Finetti’s alarm at statisticians introducing reams of unobservable
> parameters has been repeatedly justified in the modern curve fitting
> exercises of today’s big models. These seem to lose all contact with
> scientific reality focusing attention on details of large programs and
> fitting instead of observation and understanding of basic mechanism. It is
> to be hoped that a fresh implementation of de Finetti’s program based on
> observables will lead us out of this mess.

Authors like these have also tried to develop intuitive methods to choose a
model, for example by proving that a parametric family can be uniquely
determined by some symmetry assumptions about our inferences, or by other
information-theoretical properties Bernardo and Smith, 2000, ch. 4; Lindley,
2008, § 5.5; an enlightening discussion of this topic is given by Dawid, 2013.
In the present study we want to emphasize, as Cifarelli & Regazzini (1982)
did, that a statistical model can be chosen by selecting a _sufficient
statistics_ (Kolmogorov, 1942; Freedman, 1962; Diaconis and Freedman, 1980,
1981; Cifarelli and Regazzini, 1982; Lauritzen, 1988; Diaconis, 1992;
Kallenberg, 2005, and the textbook references above). Here is an example.

Imagine that we have patients labelled
$i^{\prime}\in\\{1^{\prime},2^{\prime},\dotsc\\}$, and $n$ patients labelled
$i\in\\{1,2,\dotsc\\}$, all with the same health condition. Of the second set
of patients we also know the connectivity weights $\\{\bm{f}_{i}\\}$ obtained
by fMRI. We want to specify the joint probability distribution
$\mathrm{p}(\\{\bm{f}_{i^{\prime}}\\}\mathpunct{|}\\{\bm{f}_{i}\\},I)$ that
the fMRIs of the patients $\\{i^{\prime}\\}$ yield connectivity weights
$\\{\bm{f}_{i^{\prime}}\\}$, conditional on our knowledge of the connectivity
weights of the $n$ patients $\\{i\\}$. Now assume that the probabilities for
the fMRI results are exchangeable, so that de Finetti’s formulae (7) and (8)
hold. Also assume that in order to specify the joint distribution we do not
need the full set of data $\\{\bm{f}_{i}\\}$, but only their number $n$ and
some statistics, e.g. the empirical mean and covariance matrix of these data,

${\mkern 1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern
1.5mu}\coloneqq\frac{1}{n}\mathop{\textstyle\mathop{\textstyle\sum}}\nolimits_{i}\bm{f}_{i},\qquad\operatorname{Cov}(\bm{f})\coloneqq\frac{1}{n}\mathop{\textstyle\mathop{\textstyle\sum}}\nolimits_{i}(\bm{f}_{i}-{\mkern
1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu})(\bm{f}_{i}-{\mkern
1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu})^{\intercal};$ (12)

the rest of the details of the data $\\{\bm{f}_{i}\\}$ being irrelevant. In
other words, we are assuming that the statistics above are _sufficient_ for us
to make predictions as if we had the full data. In symbols,

$\mathrm{p}(\\{\bm{f}_{i^{\prime}}\\}\mathpunct{|}\\{\bm{f}_{i}\\},I)=\mathrm{p}(\\{\bm{f}_{i^{\prime}}\\}\mathpunct{|}n,{\mkern
1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu},\operatorname{Cov}{\bm{f}},I).$ (13)

If this is true no matter the number of patients $\\{i^{\prime}\\}$ and
$\\{i\\}$, then these statistics are called _(predictive) sufficient
statistics_. There are several notions of sufficiency, including the
traditional one by Fisher (1922) and Neyman (1935), but they all are more or
less equivalent (Bernardo and Smith, 2000, § 4.5.2).

The assumption of the existence of sufficient statistics has a very important
consequence for de Finetti’s formulae (7) and (8): the space of possible prior
distributions is hugely reduced, constrained to be non-zero only over a
parametric family of distributions that is determined by the sufficient
statistics. The replacement (10) takes place, leading to the simpler formula
(11) for the likelihoods of the health conditions. The number of parameters is
equal to that of the sufficient statistics. The proof of this reduction was
given by Pitman and Koopman Koopman, 1936; Pitman, 1936; Darmois, 1935; for
generalizations, e.g. to the discrete case, see Hipp, 1974; Andersen, 1970;
Denny, 1967; Fraser, 1963; Barankin and Maitra, 1963. When the sufficient
statistics are the mean and covariance matrix, as above, the likelihoods turn
out to be (truncated) multivariate normal distributions.

In the slightly more complicated case of two or more health conditions and the
assumption of partial exchangeability (6), this theorem leads to formula (11)
with likelihoods $L_{c}$ determined by the sufficient statistics that we have
chosen for the different health conditions (Bernardo and Smith, 2000, § 4.6).
The prior distribution over the parameters of the likelihoods,
$\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}I)$,
is not determined by the theorem, but has to be determined by other
consideration that can again involve symmetry and information theory.

As we mentioned in the introduction, the notion of sufficient statistics can
be a helpful bridge between biophysical considerations and the specification
of probabilities. It may be easier to conceive and understand a connection
between biophysical considerations and some statistics of our measurements,
than between biophysical considerations and an abstract multidimensional
distribution function. Once such statistics are selected, they in turn
uniquely select a probability distribution for us. Vice versa, if a
statistical model based on some sufficient statistics proves to be very
reliable in its predictions, we may conclude that its sufficient statistics
must have an important biological meaning.

In the rest of this study we shall use three statistical models determined by
three different sufficient statistics. Our choice of statistics is
unfortunately not biologically motivated, as such models have yet to be
determined for fMRI data. However, they are adequate to demonstrate the
approach and we hope that authors with more experience will pursue this line
of thought and derive better-motivated sufficient statistics.

####  4.4.3 Edgeworth’s “method of translation”: generalized normal models

The assumption of a sufficient statistics makes the integrals in de Finetti’s
formulae (7) and (8) finite-dimensional, but these integrals and other
expressions that depend on them, like the post-test probabilities, may still
lack a closed form and be analytically intractable. In this case we could use
numerical methods, a computationally costly possibility we consider in the
Discussion, § 5.3. In the present work we choose models with closed-form
formulae instead; their swift calculation facilitates the model comparison to
be illustrated later.

Our starting point is an analytically tractable model by sufficient statistics
that has been the subject of much study Gelman et al., 2014, § 3.6; Minka,
2001; Murphy, 2007: it has a normal likelihood, with mean $\bm{\lambda}$ and
covariance matrix $\bm{\varLambda}$ as parameters, and a normal-inverse-
Wishart prior distribution over these parameters. This parameter prior
maintains the same functional form when updated with training data; this kind
of prior is called _conjugate_ DeGroot, 2004, ch. 9; Diaconis and Ylvisaker,
1979. This model is outlined more in detail in § 6.2.

We try to make the normal + normal-inverse-Wishart model more flexible by
combining it with an idea that Edgeworth (1898) called “Method of
Translation”, discussed also by Johnson (1949) and Mead (1965): the
transformation of a quantity into a normally distributed one. That is, instead
of considering the connectivity weights $\bm{f}$, we consider transformed
quantities $l(\bm{f})$, where $l$ is a component-wise monotonic function, and
suppose the latter quantities to be normally distributed. This leads to
generalized-normal likelihoods of the form

$\mathrm{N}[l(\bm{f})\mathpunct{|}\bm{\lambda},\bm{\varLambda}]\,l^{\prime}(\bm{f})\,\mathrm{d}\bm{f}$
(14)

where $\mathrm{N}$ is the normal distribution with mean $\bm{\lambda}$ and
covariance matrix $\bm{\varLambda}$, and $l^{\prime}$ is the Jacobian
determinant of $l$.

This simple idea has an amazing scope: it allows us to explore a vast range of
non-normal likelihoods – in particular likelihoods defined on bounded domains
such as $[-1,1]^{d}$ – and to keep the low computational costs of the
conjugate prior. In our present problem it has also an additional convenient
feature: in the calculation of the post-test probability, the Jacobian
determinant $l^{\prime}$ disappears, as eq. (19) below shows.

These generalized-normal models, which we denote by $M_{l}$, are also
determined by a choice of sufficient statistics, analogous to eq. (12): the
mean and covariance matrix of the transformed data $\\{l(\bm{f}_{i})\\}$,

${\mkern 1.5mu{\overline{\mkern-1.5mul(\bm{f})\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu},\qquad\operatorname{Cov}[l(\bm{f})].$ (15)

In our partially exchangeable case, with formulae (11), we need to specify a
likelihood $L_{c}$ for each health condition $c$. We assume these two
likelihoods $L_{\textsc{h}}$, $L_{\textsc{s}}$ to be functionally identical
generalized normals, i.e. the function $l$ is the same for the healthy and the
schizophrenic case; but their means and covariance matrices
$(\bm{\lambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{h}})$ and
$(\bm{\lambda}_{\textsc{s}},\bm{\varLambda}_{\textsc{s}})$ can be different.

We also need to specify a joint parameter prior for
$(\bm{\lambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{h}};\bm{\lambda}_{\textsc{s}},\bm{\varLambda}_{\textsc{s}})$.
To use the analytic advantage of the conjugate prior, we assume that the
distribution for these parameters is a product of independent distributions:

$\mathrm{p}(\bm{\lambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{h}};\bm{\lambda}_{\textsc{s}},\bm{\varLambda}_{\textsc{s}}\mathpunct{|}M_{l},I)=\mathrm{p}(\bm{\lambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{h}}\mathpunct{|}M_{l},I)\times\mathrm{p}(\bm{\lambda}_{\textsc{s}},\bm{\varLambda}_{\textsc{s}}\mathpunct{|}M_{l},I),$
(16)

each of them being a normal-inverse-Wishart distribution described in § 6.2.
This independence assumption is quite strong and has an important consequence:
_the likelihood for a health condition only depends on the data from previous
patients having that same health condition_.

With the assumptions above, the likelihood for the health condition needed by
the clinician has a closed form for this model (see § 6.2). The likelihood for
patient $0$’s being healthy, in view of the patient’s measured connectivity
weights $\bm{f}$, and given the data $(\bm{f}_{i})$ from previous
$n_{\textsc{h}}$ healthy patients, is

$\mathrm{p}(\text{fMRI result $\bm{f}$}\mathpunct{\,|\
}\text{healthy}\mathbin{\ \land\ }\text{prior
info})=\operatorname{t}\Bigl{[}l(\bm{f})\mathpunct{\big{|}\
}\nu_{\textsc{h}}-d+1,\bm{\delta}_{\textsc{h}},\tfrac{\kappa_{\textsc{h}}+1}{\kappa_{\textsc{h}}\,(\nu_{\textsc{h}}-d+1)}\bm{\varDelta}_{\textsc{h}}\Bigr{]}\,\prod_{k}l^{\prime}(f_{k}),$
(17)

where $\operatorname{t}$ is a multivariate t distribution with
$\nu_{\textsc{h}}-d+1$ degrees of freedom, mean $\bm{\delta}_{\textsc{h}}$,
and scale matrix
$\frac{\kappa_{\textsc{h}}+1}{\kappa_{\textsc{h}}\,(\nu_{\textsc{h}}-d+1)}\bm{\varDelta}_{\textsc{h}}$.
This distribution has covariance matrix
$\frac{\kappa_{\textsc{h}}+1}{\kappa_{\textsc{h}}\,(\nu_{\textsc{h}}-d-1)}\bm{\varDelta}_{\textsc{h}}$
(note the different denominator from the scale matrix), and approaches a
generalized normal for large $\nu_{\textsc{h}}$. The final factor is the
Jacobian determinant of $l$.

The most important feature of this likelihood is the dependence of the
coefficients
$(\kappa_{\textsc{h}},\bm{\delta}_{\textsc{h}},\nu_{\textsc{h}},\bm{\varDelta}_{\textsc{h}})$
on the data $(\bm{f}_{i})$ of the previous $n_{\textsc{h}}$ healthy patients:

$\displaystyle\kappa_{\textsc{h}}$
$\displaystyle=\kappa_{0}+n_{\textsc{h}},\quad$
$\displaystyle\nu_{\textsc{h}}$ $\displaystyle=\nu_{0}+n_{\textsc{h}},$ (18)
$\displaystyle\bm{\delta}_{\textsc{h}}$
$\displaystyle=\frac{\kappa_{0}\,\bm{\delta}_{0}+n_{\textsc{h}}\,{\mkern
1.5mu{\overline{\mkern-1.5mul(\bm{f})\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}}{\kappa_{0}+n_{\textsc{h}}},\quad$
$\displaystyle\bm{\varDelta}_{\textsc{h}}$
$\displaystyle=\bm{\varDelta}_{0}+n_{\textsc{h}}\,\operatorname{Cov}[l(\bm{f})]+\frac{\kappa_{0}\,n_{\textsc{h}}}{\kappa_{0}+n_{\textsc{h}}}\bigl{[}{\mkern
1.5mu{\overline{\mkern-1.5mul(\bm{f})\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}-\bm{\delta}_{0}\bigr{]}\bigl{[}{\mkern
1.5mu{\overline{\mkern-1.5mul(\bm{f})\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}-\bm{\delta}_{0}\bigr{]}^{\intercal},$

where $(\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0})$ are prior
coefficients that represent the clinician’s knowledge before any patients were
observed. As the number $n_{\textsc{h}}$ of observed healthy patients
increases, the probability for the transformed data $l(\bm{f})$ tends to a
normal distribution with mean and covariance matrix equal to the empirical
average and covariance matrix of the transformed data. The formulae above show
that previous data enter only through the sufficient statistics ${\mkern
1.5mu{\overline{\mkern-1.5mul(\bm{f})\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}$ and $\operatorname{Cov}[l(\bm{f})]$.

An analogous formula holds for the likelihood for the patient’s being
schizophrenic, with coefficients
$(\kappa_{\textsc{s}},\bm{\delta}_{\textsc{s}},\nu_{\textsc{s}},\bm{\varDelta}_{\textsc{s}})$
that depend on the data of previous schizophrenic patients and some initial
coefficients. The function $l$ and the prior coefficients
$(\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0})$ could be different
for the healthy and schizophrenic cases, but for simplicity we assume them
identical for both health conditions.

If $(\hat{p}_{\textsc{h}},\hat{p}_{\textsc{s}})$ is the pre-test probability
distribution for the health condition of patient $0$, his post-test
probability to be healthy is

$\mathrm{P}(\text{healthy}\mathpunct{\,|\ }\text{fMRI result}\mathbin{\ \land\
}\text{prior info})={}\\\\[6.0pt]
\frac{\operatorname{t}\Bigl{[}l(\bm{f})\mathpunct{\big{|}\
}\nu_{\textsc{h}}-d+1,\bm{\delta}_{\textsc{h}},\tfrac{\kappa_{\textsc{h}}+1}{\kappa_{\textsc{h}}\,(\nu_{\textsc{h}}-d+1)}\bm{\varDelta}_{\textsc{h}}\Bigr{]}\,\hat{p}_{\textsc{h}}}{\operatorname{t}\Bigl{[}l(\bm{f})\mathpunct{\big{|}\
}\nu_{\textsc{h}}-d+1,\bm{\delta}_{\textsc{h}},\tfrac{\kappa_{\textsc{h}}+1}{\kappa_{\textsc{h}}\,(\nu_{\textsc{h}}-d+1)}\bm{\varDelta}_{\textsc{h}}\Bigr{]}\,\hat{p}_{\textsc{h}}+\operatorname{t}\Bigl{[}l(\bm{f})\mathpunct{\big{|}\
}\nu_{\textsc{s}}-d+1,\bm{\delta}_{\textsc{s}},\tfrac{\kappa_{\textsc{s}}+1}{\kappa_{\textsc{s}}\,(\nu_{\textsc{s}}-d+1)}\bm{\varDelta}_{\textsc{s}}\Bigr{]}\,\hat{p}_{\textsc{s}}}.$
(19)

Note that the Jacobian determinants $l^{\prime}$ do not appear in this
formula.

####  4.4.4 Generalized normal models in our study

In the rest of our study we compare three different transformations $l$ of the
connectivity weights $\bm{f}$, with one set of prior coefficients
$(\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0})$ each:

Logit-normal model:

A slightly modified logit transformation

$l(f_{k})\coloneqq\ln\frac{1+f_{k}}{1-f_{k}},\qquad
l^{\prime}(f_{k})=\frac{2}{1-f_{k}^{2}},$ (20)

with prior coefficients

$\kappa_{0}=1,\quad\bm{\delta}_{0}=0,\quad\nu_{0}=d+1,\quad\bm{\varDelta}_{0}=(d+2)\bm{I}_{d}.$
(21)

Tangent-normal model:

A tangent transformation

$l(f_{k})\coloneqq\tan\frac{\uppi f_{k}}{2},\qquad
l^{\prime}(f_{k})=\frac{\uppi}{1+\cos\uppi f_{k}},$ (22)

with prior coefficients

$\kappa_{0}=1,\quad\bm{\delta}_{0}=0,\quad\nu_{0}=d+1,\quad\bm{\varDelta}_{0}=\frac{d+2}{4}\bm{I}_{d}.$
(23)

Normal model:

An identity transformation (that is, no transformation at all)

$l(f_{k})\coloneqq f_{k},\qquad l^{\prime}(f_{k})=1,$ (24)

with prior coefficients

$\kappa_{0}=1,\quad\bm{\delta}_{0}=0,\quad\nu_{0}=d+1,\quad\bm{\varDelta}_{0}=10\bm{I}_{d}.$
(25)

For brevity we shall denote $l(\bm{f})\coloneqq\bigl{(}l(f_{i})\bigr{)}$.

The first two transformations, plotted in the upper panel of fig. 2, map the
bounded domain $]-1,1[$ of the connectivity weights into the reals, and thus
restrict the generalized-normal likelihood to meaningful values of the
connectivity weights. The last model instead allows for connectivity weights
outside their meaningful bounds. It can be conceived as the model of a person
who has no precise knowledge of what the quantities $\bm{f}$ are. Since the
clinician’s final predictions concern health conditions given data $\bm{f}$,
not the data $\bm{f}$ themselves, this model can still be meaningfully used.
The probabilities for the connectivity weight $f_{i}$ conditional on the prior
coefficients above are shown in the lower panel of fig. 2.

Figuro 2: General normalized models. Upper panel: Generalized logit and
tangent transformation functions as defined in the text. Lower panel: Prior
distributions for any connectivity weight $f$ conditional on the three
generalized normal models defined in the text

The prior coefficients are chosen by the following criteria: uniform marginal
distributions of the correlations between connectivity weights (leading to
$\nu_{0}=d+1$ as explained before); large uncertainty in the location
parameters ($\kappa_{0}=1$); symmetry with respect to the origin
($\bm{\delta}_{0}=0$); a prior distribution for the connectivity as flat as
possible (its second derivative vanishes at the origin, leading to the values
of $\bm{\varDelta}_{0}$ above). In the case of the identity transformation we
have chosen a $\bm{\varDelta}_{0}$ that somewhat concentrates the prior around
the true range of the connectivity weights, $[-1,1]$.

The numerical values of the main quantities used throughout this study are
summarized in table 1.

$n_{\textsc{h}}=55$ | healthy patients
---|---
$n_{\textsc{s}}=49$ | schizophrenic patients
$d=40$ | graph parameters
$\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0}$ | prior coefficients: see eqs (21), (23), (25)
Tabelo 1: Numerical values in our study

###  4.5 Model comparison and selection

####  4.5.1 Criteria for model comparison

Any two statistical models differ in two main characteristics: their
predictive power and their learning speed. Predictive power is a model’s
capacity to give high probability to propositions that turn out to be true,
during and especially after its learning phase (cf. Dawid, 1982b). Learning
speed is how quickly a model reaches unchanging, stable predictive
probabilities as it gets updated with new data; note, however, that a model
may also never reach stable probabilities (see e.g. Bruno, 1964; Berk, 1966);
“Alas, this seems like a model of the way things work in practical inference –
as more data comes in, one admits a richer and richer variety of explanatory
hypothesis” (Diaconis, 1988, § 3, p. 113). Thus, our choice of a model depends
on the relative importance we give to these two characteristics.

These two characteristics need not go hand in hand: a model can quickly learn
with very little data but settle on probabilities with poor predictive power;
conversely, it can reach great predictive power but only after a long learning
phase with a huge amount of data. A model can also “unlearn”, i.e. its
predictive power initially increases and then decreases before stabilizing.
The latter phenomenon can happen because every model initially makes its
prediction through a mixture of likelihoods – the integral (11), in our case
the t distribution (36) – but as the training continues the mixture is
replaced by a single likelihood, in our case a generalized normal (14), as
explained in § 4.4.3. It may happen that the mixture of likelihoods has higher
predictive power than a single likelihood, and in this case a decline in
predictive power will be observed. A model with such behaviour is obviously
unfit for the clinician’s goal; this also means that the sufficient statistics
on which it is based capture very poorly the differences in connectivity
weights between health conditions.

When only a small amount of training data is available, it can be difficult to
assess which of two models has or will have the greater predictive power. The
first model can initially reach a greater predictive power than the second,
but the second model may eventually reach greater predictive power than the
first, with further training data. Models having the same likelihood, however,
have the same final predictive power; their learning speeds depend on their
parameter priors.

In a diagnostic problem like that faced by our clinician, the choice of a
model is ultimately dictated by the predictive power of the post-test
probabilities given by the model; but if we have little training data, the
learning speed of the model is also of some importance. Several quantitative
criteria can be conceived to assess these two characteristics:

  1. 1.

the post-test probabilities the model gives to the correct health conditions
for all training data, i.e. $\mathrm{P}(c_{1},c_{2},\dotsc)$;

  2. 2.

the post-test probabilities the model gives to the correct health conditions
in the final phase of the training only, i.e.
$\mathrm{P}(c_{\text{last}}\mathpunct{|}c_{1},c_{2},\dotsc)$;

  3. 3.

the expected utility the model yields for all training data;

  4. 4.

the expected utility the model yields in the final phase of the training only.

Post-test probabilities (criteria 1, 2) are important for obvious reasons, and
utilities (criteria 3, 4) are important because the clinician’s overall
problem is one of decision, as emphasized in the Introduction. Consideration
of all data (criteria 1, 3) is important if we are interested in the
performance of the model for the whole set of patients; but consideration of
the final data only, conditional on the previous ones (criteria 2, 4), tells
us how much the model has learned (compare with a similar remark in model
comparison using Bayes factors by Berger and Pericchi, 1996).

We must keep in mind that these criteria assess a statistical model not by
itself but in combination with other factors, since they also depend on pre-
test probabilities (which can be influenced by other diagnostic tests) or
utilities.

Applied to our models, each of these four criteria gives a very similar
picture. We shall calculate the results for criteria 2 and 4, the latter with
two different utility tables. This calculation can be explained in very
intuitive terms:

Imagine that the $n_{\textsc{h}}=55$ healthy and $n_{\textsc{s}}=49$
schizophrenic patients visit the clinician in turn, in an unknown order. For
each patient, let us further assume that the clinician has pre-test
probabilities $(\hat{p}_{\textsc{h}},\hat{p}_{\textsc{s}})=(0.5,0.5)$, i.e.
she is completely uncertain about the patient’s health condition. The
incidence in the population is much lower than 50%, of course, but the
patients presenting themselves for diagnosis are not representative of the
full population.

As stated in § 4.2, pre-test probabilities represent the clinician’s
uncertainty before the fMRI test is made; they can be based for example on a
first diagnosis considering symptoms and medical history of the patient and of
the patient’s family, on psychological evaluations, and on other diagnostic
tests. Here we assume complete uncertainty for demonstration purposes.

The clinician acquires the fMRI result $\bm{f}$ for that patient, and uses the
statistical model, trained with the data from all the patients that previously
visited her, to update to a post-test probability for schizophrenia
$p_{\textsc{s}}$, given by eq. (19); obviously
$p_{\textsc{h}}=1-p_{\textsc{s}}$. Now the clinician must make a decision –
say, treat or dismiss – based on the expected utilities of the decisions
available. Each decision has a different utility depending on the patient’s
true health condition, as summarized by a table. We consider two tables: a
symmetric one

(symmetric) healthy schizophrenic dismiss $1$ $0$ treat $0$ $1$ (26)

and an asymmetric one

(asymmetric) healthy schizophrenic dismiss $1$ $-2$ treat $-1$ $2$ (27)

In order to maximize the expected utility, as explained in § 6.3, the
clinician dismisses the patient if $p_{\textsc{s}}<1/2$ in the case of the
symmetric utility table, and if $p_{\textsc{s}}<1/3$ in the case of the
asymmetric one; and treats the patient otherwise. After the clinician’s
decision is made, the patient’s true health condition is revealed and we
record the actual utility gained for each table, and the post-test probability
the clinician assigned to the true health condition, or its logarithm, usually
called “negative surprise” (Bartlett, 1952; Good, 1956, 1957a). The health
condition and fMRI data of this patient are used to update the model. The next
patient is received, and so on, until all patients have been examined.

The particular sequence of utilities and log-probabilities recorded in the
manner just described depends on the exact order by which the 104 patients
visit the clinician. We take an approximate average over all possible
$104!\approx 10^{166}$ orders by randomly sampling $520\,000$ of them. The
values of these averages for the final patient constitute the quantitative
criteria 2 and 4.

With the symmetric utility table (26), the average utility is also the average
number of schizophrenic patients for which the model yields
$p_{\textsc{s}}>1/2$. The average utility for the last patient, when the model
has been trained with the rest of the patients, is therefore a form of leave-
one-out cross-validation (Allen, 1974; Stone, 1974; Kotz et al., 2006, vol. 2,
pp. 1454–1458). The asymmetric table (27), slightly more realistic, tells us
that dismissing a schizophrenic patient has worse consequences than treating a
healthy one, and treating a schizophrenic patient has better consequences than
dismissing a healthy one (McKenzie, 2014; Ho et al., 2000). For this reason
the patient is dismissed, more conservatively, only if $p_{\textsc{s}}<1/3$.
Note that scaling a utility table by a positive factor or shifting its values
by a constant represent changes in the unit of measure and in the zero of
utilities, and therefore do not affect our relative comparison of the
statistical models.

####  4.5.2 Results for our three models

Figuro 3: Averaged sequence of utilities, with utility tables (26) and (27),
and of log-probabilities for our three models. The standard deviations of the
averages are smaller than the markers’ size. The average values for the first
and last patients are exact.

The averaged sequences of utilities and log-probabilities calculated as in §
4.5.1 are shown in fig. 3. The R code for the calculation is publicly
available (Porta Mana et al., 2018). The results, summarized in table 2, are
qualitatively identical by the two criteria and two utility tables we chose:
the normal model gives the best values for the final patient, followed by the
logit-normal model; the tangent-normal model has the worst final predictive
power, at or below chance level.

symmetric utility (26) asymmetric utility (27) log-probability normal model
$0.80$ $0.70$ $-0.29$ logit-normal model $0.75$ $0.66$ $-0.34$ tangent-normal
model $0.56$ $0.03$ $-0.60$ chance $0.50$ $0.41$ $-0.69$

Tabelo 2: Final results for the three models

The plots of fig. 3 illustrate the points made at the beginning of § 4.5.1:
one model can initially learn faster than another and yet be overtaken in the
later stages of learning; this is the case for the logit-normal and normal
models. The tangent-normal model shows strong unlearning; this means that a
tangent-normal likelihood and its sufficient statistics do not distinguish
well between healthy and schizophrenic conditions. The slight downward bends
at the final stages of the logit-normal and normal models raise the suspicion
that they might also show some unlearning if further training data were
supplied.

Figuro 4: Updated distributions of the logit-normal model for the location
parameters $(\bm{\lambda}_{\textsc{h}},\bm{\lambda}_{\textsc{s}})$ (left),
scale parameters $(\bm{\varLambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{s}})$
(right), and connectivity weights $f_{\textsc{h}}$, $f_{\textsc{s}}$ (bottom,
superposed on the empirical distributions) for the connectivity between left
superior parietal lobule and left lingual gyrus, corresponding to the bottom
left panel of fig. 1. The vertical lines in the first two plots indicate the
corresponding empirical statistics from the data.

The trends of the logit-normal and normal models suggest that the learning
phase is not finished: more patients are needed before their predictive
probabilities become stable. This is also evident from the updated marginal
distributions of their parameters
$(\bm{\lambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{h}};\bm{\lambda}_{\textsc{s}},\bm{\varLambda}_{\textsc{s}})$,
for example those for the connectivity weight $f$ between the left superior
parietal lobule and left lingual gyrus, shown in fig. 4 for the logit-normal
model. The distributions of the location parameters
$(\bm{\lambda}_{\textsc{h}},\bm{\lambda}_{\textsc{s}})$ have reached the
empirical means of the data, but those of the scale parameters
$(\bm{\varLambda}_{\textsc{h}},\bm{\varLambda}_{\textsc{s}})$ are still very
far away from the empirical variances. The reason is that the prior for the
scale parameters had a peak at a very large value of $\varLambda\approx 20$.
The $55$ data for healthy patients and $49$ for schizophrenic ones have
shifted this peak to $\varLambda_{\textsc{h}}=1.3$ and
$\varLambda_{\textsc{s}}=1.5$, but more data are needed to shift these peaks
to even smaller values – provided that in the meantime the empirical values do
not change too much as new data are gathered.

The peak at high values of $\bm{\varLambda}$ is a known inconvenient feature
of the normal-inverse-Wishart conjugate prior, related to the correlation
between correlation and variance components of $\bm{\varLambda}$
characteristic of this prior (e.g. Barnard et al., 2000).

####  4.5.3 Contrast with other model-comparison criteria

Common Bayesian model-comparison criteria are based on the joint probability
that the model gives to training data; especially its logarithm, called
“weight of evidence” or “marginal log-likelihood”, or the ratio of such
logarithms, called Bayes factors Jeffreys, 2003, chs V, VI, A; Good, 1950;
MacKay, 1992a; Kass and Raftery, 1995. The simple reason is Bayes’ theorem:

$\mathrm{P}(\text{model}\mathpunct{\,|\ }\text{data}\mathbin{\ \land\
}\text{prior info})\propto\mathrm{P}(\text{data}\mathpunct{\,|\
}\text{model}\mathbin{\ \land\ }\text{prior
info})\times\mathrm{P}(\text{model}\mathpunct{\,|\ }\text{prior info}),$ (28)

the latter probability usually assumed the same for all models (but see Porta
Mana, 2017). A higher weight of evidence means that the model is more
probable.

In our study, however, we have two kinds of data: health conditions and fMRI
results. Since the likelihood for the health condition, used by the clinician
to arrive at a post-test probability, gives the probability for the fMRI
results $\\{\bm{f}_{i}\\}$, it seems intuitive to calculate the weights of
evidence of our models based on these data. The result is the opposite of what
we obtain with the averaged-utility criterion or any of other three mentioned
above. We obtain:

$\displaystyle\ln\mathrm{p}(\text{fMRI results}\mathpunct{\,|\ }\text{logit-
normal model}\mathbin{\ \land\ }\text{health conditions})=-1913,$ (29)
$\displaystyle\ln\mathrm{p}(\text{fMRI results}\mathpunct{\,|\ }\text{tangent-
normal model}\mathbin{\ \land\ }\text{health conditions})=-1858,$
$\displaystyle\ln\mathrm{p}(\text{fMRI results}\mathpunct{\,|\ }\text{normal
model}\mathbin{\ \land\ }\text{health conditions})=-2488,$

which gives the normal model a much smaller probability than the other two,
and the tangent-normal model the highest.

This discrepancy with the averaged-utility criterion is not completely
surprising, though. Imagine a disease that leads to no differences at all
between the fMRI results of patients with the disease and those of healthy
controls. If we found a statistical model that predicted the fMRI results with
certainty, this model would thus have a the highest weight of evidence (zero),
and yet its final average utility would be at chance level, since it could not
help us at all in telling healthy from diseased patients. For our problem the
right comparison and selection criterion is the utility or one of the other
three criteria previously listed.

####  4.5.4 Final assessment of models

The average-utility criterion clearly excludes the tangent-normal model, which
even shows a rapid unlearning. The logit-normal and normal models have almost
similar performances, at around $75$–$80\%$ of final patients correctly
treated. We can also plot the sequence of utilities averaged over healthy and
schizophrenic patients separately, as in fig. 5, which gives us an idea of the
ratio between true and false negatives, and true and false positives. Both
models show around $35\%$ false positives (dismissed schizophrenic patients),
with the normal model giving slightly higher final rates of true negatives,
$93\%$, and true positives, $65\%$, than the normal, $85\%$ and $63\%$.

Figuro 5: Sequence utilities averaged over healthy (left) and schizophrenic
(right) patients separately, for the logit-normal and normal models

However, we emphasize that the features used as inputs to the model were
selected using a very simple heuristic (maximum difference in means between
the two groups, see § 4.3), and the statistical model was selected for its
computational properties rather than its fit to the distributions of
connection weights derived from fMRI data. This notwithstanding, we believe
that with further training, the logit-normal model could reach a higher
predictive power. The reason is that some empirical distributions of
connectivity weights, like the one for schizophrenic patients shown in red in
fig. 4, seem to be bimodal; and the logit-normal likelihood, unlike the normal
one, is capable of bimodality, thus fitting these distributions better.

Our assessment, however, is just an illustrative example for the general
method discussed in this paper, and we are not earnestly proposing the logit-
normal model (nor any other specific model) as the optimal one to use in the
problem of diagnosing schizophrenia. We note that any model assessment and
selection using the average-utility metric depends on several important
quantities and assumptions:

  1. A.

the pre-test probabilities given by the clinician; we assumed these to be
$(0.5,0.5)$;

  2. B.

the clinician’s range of decisions; we assumed it to be simply “treat or
dismiss”, but it could comprise several different kinds of treatments;

  3. C.

the utilities of the clinician’s decisions; we assumed these to be as in
formula (37);

  4. D.

the ratio between the numbers of healthy and schizophrenic training data;
$55/49=1.12$ in our case;

  5. E.

the clinician diagnoses one patient at a time.

For a proper model assessment we should therefore investigate and consider
more realistic rates of healthy vs schizophrenic cases that visit a particular
clinician, in order to have better-informed pre-test probabilities; and we
should consider more realistic decisions available to the clinician, together
with realistically quantified utilities.

Assumption E. deserves some explanation as it may mistakenly appear that it
doesn’t matter whether patients visit the clinician simultaneously or one at a
time. Suppose two patients, Tom and Joe, visit the clinician together, and the
clinician obtains fMRI data for both, $\bm{f}_{\text{T}}$ and
$\bm{f}_{\text{J}}$. The joint post-test probability for their health
conditions $c_{\text{T}}$ and $c_{\text{J}}$ is different from the one
obtained first calculating Joe’s one, say, and then Tom’s using Joe’s results:

$\mathrm{p}(c_{\text{T}},c_{\text{J}}\mathpunct{|}\bm{f}_{\text{T}},\bm{f}_{\text{J}},D,M_{l})\neq\mathrm{p}(c_{\text{T}}\mathpunct{|}\bm{f}_{\text{T}},c_{\text{J}},\bm{f}_{\text{J}},D,M_{l})\times\mathrm{p}(c_{\text{J}}\mathpunct{|}\bm{f}_{\text{J}},D,M_{l}).$
(30)

This inequality can be easily verified by applying the probability product
rule to the left side, and can be understood as follows. Suppose the clinician
first wants to calculate the likelihood for Joe’s being healthy. If Tom is
schizophrenic then his fMRI result is unimportant for Joe’s likelihood, owing
to our assumption of independent priors (16). But if Tom is healthy, then his
fMRI result, _which is known to the clinician_ , should lead to an updated
model for Joe’s likelihood. The likelihood for Joe’s being healthy is
therefore a mixture of these two possible likelihoods, with weights
proportional to the post-test probability for Tom’s health condition. Thus,
Joe’s likelihood is affected by Tom’s fMRI result even if Joe’s is calculated
first and Tom’s health condition is not yet known. More generally, if several
patients visit the clinician simultaneously, she should order diagnostic fMRI
tests for all of them at once and calculate a joint post-test probability for
them, in order to make the best-informed prediction for each.

##  5 Discussion

###  5.1 Summary

The diagnosis of a medical condition is a complex process that takes in a
variety of judgements and evidence from the clinician and from any diagnostic
tests available to her. Bayesian probability theory has found wider acceptance
in medicine because it can consistently combine and frame such judgements and
evidence (Goodman, 1999; Davidoff, 1999; Greenland, 1998). Formulae (1)–(3)
show the basic scheme of how the clinicians’ judgements and the results of
diagnostic tests are combined (Sox et al., 2013). The role of a diagnostic
test is not simply to give a dichotomic answer, e.g. healthy/ill, but a
_likelihood_ for each health condition, to be combined into this scheme
together with the likelihoods from other tests. The final probability obtained
from these likelihoods is finally used by the clinician to decide upon a
course of action, e.g. dismiss/treat (§ 6.3).

The values of likelihoods from such tests need to be determined from a set of
training set of data for each health condition. In this work we have discussed
how to determine the likelihoods when the diagnostic test and training set
consist of fMRI data, considering for concreteness the case of schizophrenia
as the disease in question, and by using real fMRI data from healthy and
schizophrenic patients (§ 4.1). We derived them step by step from first
principles through a sequence of assumptions:

  1. (a)

Partial exchangeability with respect to the health conditions, explained in §
4.2. We believe this assumption to be very natural in medical diagnostics. By
itself it already leads to a specific expression for the likelihood, eq. (8),
although this expression is very difficult to compute.

  2. (b)

Sufficiency of an empirical statistics of a reduced set of fMRI data (§§
4.3–4.4). We believe such kind of assumption to be sound and at least
approximately true when neurologically motivated, and moreover it provides a
bridge between biophysical considerations and the specification of
probabilities. It leads to a likelihood, eq. (11), amenable to numerical or
analytic computation.

  3. (c)

Prior knowledge of the empirical statistics for the different health
conditions (§ 4.4.4). An assumption of this kind is always necessary,
especially with small training data sets; but it affects our inferences less
and less as our training data accumulate.

In our study, for b we specifically assumed the sufficiency of the first and
second moments of some functions (§ 4.4) of the functional-connectivity
weights obtained from the fMRI data (§ 4.3). Our focus on connectivities was
neurologically motivated, but our choice of first and second moments of
particular functions was made for mathematical simplicity, in order to
illustrate the method. Our choice of roughly flat, conjugate priors for
assumption c was also made for the sake of mathematical simplicity and
illustration (§ 4.4.4).

Notwithstanding our simple, mainly illustrative choices, we obtained good
diagnostic performances, briefly discussed in the next section. To assess
these performances and the relative predictive power of different choices in
assumption b above, we presented several criteria, based on decision theory (§
4.5). These criteria also help in understanding whether the training of the
likelihoods has stabilized (§ 4.5). We observed that in this kind of study
decision-theoretical criteria can yield results in seeming contrast with
Bayesian model-comparison criteria, like weights of evidence and Bayes factors
(§ 4.5.3); but this contrast is understandable and is not a sign of
inconsistency.

We emphasize that the analysis and conclusions presented here are just
illustrations of the general method, and are not meant to be used for real
diagnoses: as explained in §§ 4.4.2 and 4.5.4, for a real application we would
first need to investigate better-informed sufficient statistics, realistic
decisions available to the clinician, the latter’s utilities, and realistic
pre-test probabilities. In this study we used simple default values for
illustrative purposes only.

###  5.2 Comparison with other studies and methods

In this study we used a rather naive approach, explained in § 4.3, to select a
subset of brain connections for our analysis. That approach can be criticized
in two different ways. First, it ignores the fact that if distributions are
narrow, the means can be close together without much overlap, and that such
distributions are likely to be more informative for the purposes of
discrimination. This could be improved, for example, by taking the minimal
area of the distributions’ overlap (dark red area in fig. 1) instead. Second,
we did not restrict our search for suitable connections and associated brain
areas to the areas that are known to be part of resting-state networks
identified in previous studies, like the default mode network. Our lack of
specificity, though, was motivated by previous studies which have demonstrated
that functional connectivity can be also measured in other task-related
networks, induced by spontaneous activity (He et al., 2009), and that in
resting state different activity patterns can appear in schizophrenic
patients, owing e.g. to hallucinations during the scan (e.g. Shergill et al.,
2000).

Despite our simplified and possibly unrealistic choices of connectivity
weights, sufficient statistics, parameter priors, pre-test probabilities, and
utility functions, we obtained a diagnostic performance of $80\%$, measured by
leave-one-out cross-validation (§ 4.5), comparable to classification results
based on fMRI data reported in previous studies e.g., Venkataraman et al.,
2012, 18 healthy + 18 schizophrenic patients; Cetin et al., 2016, 45 + 46
patients; Demirci et al., 2008, 36 + 34 patients .

Our results also compare with classification results that we achieved using
machine-learning methods. Cross-validation tests using support-vector machines
(using $80\%$ of the data for training and $20\%$ for testing in a randomized
iterative way) also yielded around $80\%$ of correct diagnoses (data not
shown).

But, as explained in the Introduction (points I., II.), methods like these,
which simply classify or are deterministic, do not fit the clinician’s
decision-theoretical problem: they cannot be combined with other diagnostic
tests and do not fit a general decision-theoretic approach – cf. §§ 4.5.1,
6.3. Most machine-learning methods (Bishop, 2006; Murphy, 2012) are thus ruled
out.

There is no real contrast, however, between machine-learning methods and the
method presented here: machine-learning algorithms can be interpreted as
convenient, fast approximations of Bayesian methods (see e.g. the explicative
image in Huszár, 2017), often combined with default utility functions and
decision rules (Murphy, 2012; MacKay, 2003, 1992a, 1992e, 1992d, 1992c,
1992b). A machine-learning classification algorithm that gives a good
performance can suggest good likelihoods or parameter priors to be used in a
statistical model. For example, the simplest version of a support-vector
machine Bishop, 2006, ch. 7; Murphy, 2012, § 14.5 can be interpreted as a
model where the likelihood
$L_{\textsc{h}}(\bm{f}\mathpunct{|}\bm{\theta}_{\textsc{h}})$ for one health
condition is very large in a half of the dataspace $\bm{f}\in[-1,1]^{d}$ and
zero in the rest, the hyperplane separating these half-spaces being determined
by the parameter $\bm{\theta}_{\textsc{h}}$. The likelihood
$L_{\textsc{s}}(\bm{f}\mathpunct{|}\bm{\theta}_{\textsc{s}})$ for the other
health condition is likewise large or zero in two half-spaces separated by a
hyperplane determined by $\bm{\theta}_{\textsc{s}}$; see eqs (11) in § 4.4.1.
In this model the parameter prior
$\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}M,I)$
correlates the two parameters
$(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}})$ in such a way that the
two hyperplanes coincide and the two likelihoods have support on opposite
sides. As the model is trained and the parameter prior is updated, eq. (11b),
the hyperplane moves in space in a way to maximize the product of the
likelihoods. This corresponds to the search of an optimal separation
hyperplane by the support-vector machine.

###  5.3 Possible improvements

Besides using more realistic pre-test probabilities, and utility values, the
method presented here could be improved in several other respects, especially
with regard to assumptions b and c summarized in § 5.1.

In point b we assumed that particular connectivity weights are sufficient to
distinguish among schizophrenic and healthy patients. These connectivities
were calculated as in §§ 4.3 and 6.1. More sophisticated choices of Regions Of
Interests and functional-connectivity measures (Marrelec and Fransson, 2011;
Smith et al., 2011; Wang et al., 2014; Gheiratmand et al., 2017; Demirci et
al., 2008) or even integration of functional and structural imaging (Michael
et al., 2010) could obviously lead to an increased predictive power. A
different choice of sufficient statistics could also improve the performance.
With increased computational power it would even be possible to use the full
set of connectivities rather than sufficient statistics thereof – so-called
“nonparametric” models (cf. Zhang et al., 2014, 2016; Nielsen et al., 2016;
Kook et al., 2017).

Regarding assumption c, in § 4.4.3 we mentioned that parametric models may
lead to probabilities that lack a closed form and are analytically
intractable. The models we chose, with conjugate prior, have the advantage of
having closed-form formulae, but they also restrict our choice of sufficient
statistics and parameter priors. Higher predictive power could be achieved by
using other kinds of sufficient statistics, leading to likelihoods that are
not generalized normals, or by using non-conjugate priors, e.g. treating
means, correlations, and variances independently, as discussed by Barnard et
al. (2000). In this case numerical methods are needed, such as Markov-Chain
Monte Carlo sampling and integration MacKay, 2003, ch. IV; Murphy, 2012, chs
23–24. It must be kept in mind, though, that numerical methods can be
computationally vastly more expensive than analytic ones. In preliminary
studies that led to our present work we considered a couple of statistical
models that require Monte Carlo sampling: a truncated normal and a product of
beta distributions among them. The calculation of the relevant integrals for
these models has vastly higher time costs than for the closed-form models
presented here. For example, calculation of a posterior parameter distribution
for the truncated-normal model required $17\>\mathrm{h}$ on a computer
cluster; whereas the corresponding calculation for the models in the present
work takes a fraction of a second on a laptop. To assess and select a model
for our clinician to use, such integrals need to be computed over and over, as
we explained in § 4.5.1. The assessment of statistical models that require
Monte Carlo methods can therefore lead to months of computation. Further
explorations are needed in this direction.

The comparison with support-vector machines sketched at the end of the
previous section shows one important assumption of our statistical models: the
independence of the prior parameter distributions for the two health
conditions:
$\mathrm{p}(\bm{\theta}_{\textsc{h}},\bm{\theta}_{\textsc{s}}\mathpunct{|}I)=\mathrm{p}(\bm{\theta}_{\textsc{h}}\mathpunct{|}I)\times\mathrm{p}(\bm{\theta}_{\textsc{s}}\mathpunct{|}I)$,
see eq. (16). Statistical models of which support-vector machines are
approximations clearly cannot make this assumption. The performance of the
model might improve by using a non-independent joint prior distribution, which
allows training data for one health condition to influence the parameter
distribution for the other. In the case of the generalized-normal models we
examined, this can be achieved – whilst preserving their computational
convenience – by taking a weighted average of several values of prior
coefficients (33). This is known as a hierarchical model (Bernardo and Smith,
2000, § 4.6.5).

The possibilities for improvement listed in these last paragraphs suggest that
the method we have presented here, hinging on first principles, has great
potential for applications and for development in different directions;
moreover this is not to the exclusion of other methods, but assimilating their
principles and advantages.

##  6 Methods

###  6.1 Data preprocessing

Preprocessing of the rfMRI images is carried out using the fmrib Software
Library tools (fsl, v5.08: Jenkinson et al., 2012; Smith et al., 2005) and
consists of the following steps: removal of the first ten image volumes,
leaving the remaining 130 volumes for further data processing; removing non-
brain tissue (Bet: Smith, 2002); motion correction (mcflirt: Jenkinson et al.,
2002); spatial smoothing with a $6\;\mathrm{mm}$ full width at half maximum
normal kernel; temporal low-pass filtering with a cut-off frequency of
$0.009\;\mathrm{Hz}$; white matter and cerebrospinal fluid regression (fsl
regfilt, melodic).

For each subject we first linearly register the rfMRI image first to the
structural, skull-removed image (image segmentation for skull removing with
spm8, Wellcome Department of Cognitive Neurology, London, ukfsl; linear
registration with fsl/flirt: Jenkinson and Smith 2001; Jenkinson et al. 2002)
and then, through a non-linear mapping, to the MNI standard brain (non-linear
registration with Advanced Normalization Tools (ANTs: Avants et al., 2011);
MNI 152 standard brain, non-linear 6th generation (Grabner et al., 2006).
Regions of interest (ROIs) of the resulting functional image in standard space
are extracted such that they match the 94 regions identified by the Oxford
lateral cortical atlas with a probability above $50\%$ (Desikan et al., 2006).
The temporal mean signals across the voxels in each ROI are used to calculate
the functional connectivity measured based on the Pearson correlation
coefficient.

###  6.2 The normal model with conjugate prior

This statistical model, denoted in this section by $M$, is amply discussed in
the literature Gelman et al., 2014, § 3.6; Minka, 2001; Murphy, 2007; here we
only give a summary.

Its likelihood is a normal distribution

$L(\bm{f}\mathpunct{|}\bm{\lambda},\bm{\varLambda},M,I)=\mathrm{N}[l(\bm{f})\mathpunct{|}\bm{\lambda},\bm{\varLambda}]\,l^{\prime}(\bm{f})$
(31)

with mean $\bm{\lambda}$ and covariance matrix $\bm{\varLambda}$.

The prior distribution for the parameters $(\bm{\lambda},\bm{\varLambda})$ is
a normal-inverse-Wishart distribution, i.e. the product of a normal
distribution for $\bm{\lambda}$ and an inverse-Wishart matrix distribution
Gupta and Nagar, 2000, § 3.4; Tiao and Zellner, 1964; Bernardo and Smith,
2000, § 3.2.5 for $\bm{\varLambda}$:

$\displaystyle\mathrm{p}(\bm{\lambda},\bm{\varLambda}\mathpunct{|}\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0},M_{l},I)=\mathrm{p}(\bm{\lambda}\mathpunct{|}\bm{\varLambda},\kappa_{0},\bm{\delta}_{0},M_{l},I)\times\mathrm{p}(\bm{\varLambda}\mathpunct{|}\nu_{0},\bm{\varDelta}_{0},M_{l},I),$
(32) with $\displaystyle\begin{aligned}
\mathrm{p}(\bm{\lambda}\mathpunct{|}\bm{\varLambda},\kappa_{0},\bm{\delta}_{0},M_{l},I)&=\mathrm{N}(\bm{\lambda}\mathpunct{|}\bm{\delta}_{0},\bm{\varLambda}/\kappa_{0}),\\\
\mathrm{p}(\bm{\varLambda}\mathpunct{|}\nu_{0},\bm{\varDelta}_{0},M_{l},I)&=\operatorname{Wishart^{-1}}(\bm{\varLambda}\mathpunct{|}\nu_{0},\bm{\varDelta}_{0})\propto\det(\bm{\varLambda})^{-\frac{\nu_{0}+d+1}{2}}\,\exp\bigl{(}-\tfrac{1}{2}\operatorname{tr}\bm{\varDelta}_{0}\bm{\varLambda}^{-1}\bigr{)}.\end{aligned}$
(33)

It should be noted how $\bm{\varLambda}$ appears as parameter in the
distribution for $\bm{\lambda}$, so their distributions are not independent.
The composite distribution depends on two scalar, one vector, and one matrix
coefficients $(\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0})$.

This prior parameter distribution retains the same form when it is conditioned
on the data $(\bm{f}_{i})$ of $n$ patients, becoming a posterior parameter
distribution with updated coefficients
$(\kappa,\bm{\delta},\nu,\bm{\varDelta})$ depending on the prior ones and on
the sufficient statistics:

$\displaystyle\kappa$ $\displaystyle=\kappa_{0}+n,\quad$ $\displaystyle\nu$
$\displaystyle=\nu_{0}+n,$ (34) $\displaystyle\bm{\delta}$
$\displaystyle=\frac{\kappa_{0}\,\bm{\delta}_{0}+n\,{\mkern
1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}}{\kappa_{0}+n},\quad$
$\displaystyle\bm{\varDelta}$
$\displaystyle=\bm{\varDelta}_{0}+n\,\operatorname{Cov}(\bm{f})+\frac{\kappa_{0}\,n}{\kappa_{0}+n}\bigl{(}{\mkern
1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}-\bm{\delta}_{0}\bigr{)}\bigl{(}{\mkern
1.5mu{\overline{\mkern-1.5mu\bm{f}\mkern-1.5mu\mkern
2.0mu}\mkern-2.0mu}{}\mkern 1.5mu}-\bm{\delta}_{0}\bigr{)}^{\intercal}.$

The main features of the normal-inverse-Wishart distribution for
$(\bm{\lambda},\bm{\varLambda})$ are these:

$\displaystyle\text{$\bm{\lambda}$:}\quad\;\hphantom{\Biggl{\\{}}\text{mean
\&\
mode}=\bm{\delta},\qquad\text{covariances}=\frac{1}{\kappa\,(\nu-d-1)}\bm{\varDelta};$
(35) $\displaystyle\text{$\bm{\varLambda}$:}\quad\left\\{\\!\begin{aligned}
&\text{mean}=\frac{1}{\nu-d-1}\bm{\varDelta},\qquad\text{mode}=\frac{1}{\nu+d+2}\bm{\varDelta},\\\
&\text{diagonal
variances}=\frac{2}{(\nu-d-3)(\nu-d-1)^{2}}{(\bm{\varDelta})_{kk}}^{2}.\end{aligned}\right.$

These formulae above say that the uncertainty in the location parameter
$\bm{\lambda}$ decreases as $\kappa$ and $\nu$ increase for fixed
$\bm{\varDelta}$, and the uncertainty in the matrix scale parameter
$\bm{\varLambda}$ decreases with increasing $\nu$. When $\nu=d+1$ the marginal
distributions for the correlations of $\bm{\lambda}$ are uniform Gelman et
al., 2014, § 3.6; Barnard et al., 2000, § 2.2. Because of these properties, a
“vaguely informative” parameter distribution should have small $\kappa_{0}$
and $\nu_{0}$ (Minka, 2001; Murphy, 2007).

When the likelihood (31) and the parameter prior (32), updated with (34), are
multiplied and the parameters are integrated, the resulting distribution for
$\bm{f}$ is a multivariate t distribution (Kotz and Nadarajah, 2004; Minka,
2001; Murphy, 2007)

$\mathrm{p}[\bm{f}\mathpunct{|}(\bm{f}_{i}),\kappa_{0},\bm{\delta}_{0},\nu_{0},\bm{\varDelta}_{0},M_{l},I]\equiv\mathrm{p}(\bm{f}\mathpunct{|}\kappa,\bm{\delta},\nu,\bm{\varDelta},M_{l},I)=\operatorname{t}\Bigl{[}l(\bm{f})\mathpunct{\big{|}\
}\nu-d+1,\bm{\delta},\tfrac{\kappa+1}{\kappa\,(\nu-d+1)}\bm{\varDelta}\Bigr{]}$
(36)

with $\nu-d+1$ degrees of freedom, mean $\bm{\delta}$, scale matrix
$\frac{\kappa+1}{\kappa\,(\nu-d+1)}\bm{\varDelta}$, and covariance matrix
$\frac{\kappa+1}{\kappa\,(\nu-d-1)}\bm{\varDelta}$.

###  6.3 Decision theory and utility

Once we have the post-test probabilities $(p_{\textsc{h}},p_{\textsc{s}})$ for
the possible health conditions of a patient given the fMRI data, there remains
to decide upon a course of action. This is the domain of decision theory
Raiffa and Schlaifer, 2000; Jaynes, 2003, chs 13, 14; Sox et al., 2013, chs 6,
7.

Suppose we have only two courses of action: treat t or dismiss d. A decision-
theoretical analysis needs, besides the probabilities for the health
conditions, also the utilities (or costs) of choosing an action given the
patient’s true health condition. For example, treatment of a healthy patient
could harm the latter, or it could be innocuous. With two courses of action
and two health conditions we have four utilities
$u_{\text{decision}\mathpunct{|}\text{condition}}$:

$\mbox{ \begin{tabular}[]{lcc}&healthy&schizophrenic\\\ \hline\cr
dismiss&$u_{\textsc{d}\mathpunct{|}\textsc{h}}$&$u_{\textsc{d}\mathpunct{|}\textsc{s}}$\\\\[-6.0pt]
treat&$u_{\textsc{t}\mathpunct{|}\textsc{h}}$&$u_{\textsc{t}\mathpunct{|}\textsc{s}}$\end{tabular}}\qquad\begin{aligned}
u_{\textsc{d}\mathpunct{|}\textsc{h}}&>u_{\textsc{t}\mathpunct{|}\textsc{h}},\\\
u_{\textsc{t}\mathpunct{|}\textsc{s}}&>u_{\textsc{d}\mathpunct{|}\textsc{s}}.\end{aligned}$
(37)

Typically
$u_{\textsc{d}\mathpunct{|}\textsc{h}}>u_{\textsc{t}\mathpunct{|}\textsc{h}}$
and
$u_{\textsc{t}\mathpunct{|}\textsc{s}}>u_{\textsc{d}\mathpunct{|}\textsc{s}}$,
and $u_{\textsc{d}\mathpunct{|}\textsc{h}}$,
$u_{\textsc{t}\mathpunct{|}\textsc{s}}$ are positive and
$u_{\textsc{t}\mathpunct{|}\textsc{h}}$,
$u_{\textsc{d}\mathpunct{|}\textsc{s}}$ negative if we appropriately shift the
zero of our measurement units.

The expected utilities for dismissal and treatment are therefore

$\mathrm{E}(u_{\textsc{d}})=u_{\textsc{d}\mathpunct{|}\textsc{h}}\,p_{\textsc{h}}+u_{\textsc{d}\mathpunct{|}\textsc{s}}\,p_{\textsc{s}},\qquad\mathrm{E}(u_{\textsc{t}})=u_{\textsc{t}\mathpunct{|}\textsc{h}}\,p_{\textsc{h}}+u_{\textsc{t}\mathpunct{|}\textsc{s}}\,p_{\textsc{s}}.$
(38)

Decision theory says the clinician ought to chose the action having maximum
expected utility. For example, she dismisses the patient if
$\mathrm{E}(u_{\textsc{d}})>\mathrm{E}(u_{\textsc{t}})$, that is if

$p_{\textsc{s}}<\frac{u_{\textsc{d}\mathpunct{|}\textsc{h}}-u_{\textsc{t}\mathpunct{|}\textsc{h}}}{u_{\textsc{d}\mathpunct{|}\textsc{h}}-u_{\textsc{t}\mathpunct{|}\textsc{h}}+u_{\textsc{t}\mathpunct{|}\textsc{s}}-u_{\textsc{d}\mathpunct{|}\textsc{s}}}.$
(39)

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

CB constructed and analysed the graph data from fMRI scans. PGLPM developed
the statistical model. They both made the statistical analysis of the data
from the model. The manuscript was written PGLPM, CB, AM.

## Funding

We acknowledge partial support by the Helmholtz Alliance through the
Initiative and Networking Fund of the Helmholtz Association and the Helmholtz
Portfolio theme “Supercomputing and Modeling for 830 the Human Brain”.

## Acknowledgments

PGLPM thanks Mari & Miri for continuous encouragement and affection; the kind
staff at Iris, where part of this work was done; Buster Keaton and Saitama for
filling life with awe and inspiration; and the developers and maintainers of
LaTeX, Emacs, AUCTeX, Open Science Framework, biorXiv, Hal archives, Python,
Inkscape, Sci-Hub for making a free and unfiltered scientific exchange
possible. We thank Alper Yegenoglu and Jakob Jordan for support and advice.

## Citaĵoj

  * Aldous (1985) Aldous, D. J. (1985).  Exchangeability and related topics.  In _Aldous et al._(_1985 _)__. VII, 1–198.  https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/papers.html
  * Aldous et al. (1985) Aldous, D. J., Ibragimov, I. A., and Jacod, J. (1985).  _ École d’Été de Probabilités de Saint-Flour XIII – 1983_, vol. 1117 of _Lecture notes in mathematics_ (Berlin: Springer).  Édité par P. L. Hennequin
  * Allen (1974) Allen, D. M. (1974).  The relationship between variable selection and data agumentation and a method for prediction.  _Technometrics_ 16, 125–127
  * Andersen (1970) Andersen, E. B. (1970).  Sufficiency and exponential families for discrete sample spaces.  _J. Am. Stat. Assoc._ 65, 1248–1255
  * Avants et al. (2011) Avants, B. B., Tustison, N. J., Song, G., Cook, P. A., Klein, A., and Gee, J. C. (2011).  A reproducible evaluation of ants similarity metric performance in brain image registration.  _NeuroImage_ 54, 2033–2044
  * Barankin and Maitra (1963) Barankin, E. W. and Maitra, A. P. (1963).  Generalization of the Fisher-Darmois-Koopman-Pitman theorem on sufficient statistics.  _Sankhyā A_ 25, 217–244
  * Barnard et al. (1974) Barnard, G. A., Atkinson, A. C., Chan, L. K., Dawid, A. P., Downton, F., Dickey, J., et al. (1974).  Discussion [Cross-Validatory choice and assessment of statistical predictions].  _J. Roy. Stat. Soc. B_ 36, 133–147.  See Stone (1974)
  * Barnard et al. (2000) Barnard, J., McCulloch, R., and Meng, X.-L. (2000).  Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage.  _Stat. Sinica_ 10, 1281–1311
  * Bartlett (1952) Bartlett, M. S. (1952).  The statistical significance of odd bits of information.  _Biometrika_ 39, 228–237
  * Berger and Pericchi (1996) Berger, J. O. and Pericchi, L. R. (1996).  The intrinsic Bayes factor for model selection and prediction.  _J. Am. Stat. Assoc._ 91, 109–122
  * Berk (1966) Berk, R. H. (1966).  Limiting behavior of posterior distributions when the model is incorrect.  _Ann. Math. Stat._ 37, 51–58
  * Bernardo et al. (1988) Bernardo, J.-M., DeGroot, M. H., Lindley, D. V., and Smith, A. F. M. (eds.) (1988).  _Bayesian Statistics 3_ (Oxford: Oxford University Press)
  * Bernardo and Smith (2000) Bernardo, J.-M. and Smith, A. F. (2000).  _Bayesian Theory_.  Wiley series in probability and mathematical statistics (New York: Wiley), reprint edn.  First publ. 1994
  * Besag et al. (2002) Besag, J., Bickel, P. J., Brøns, H., Fraser, D. A. S., Reid, N., Helland, I. S., et al. (2002).  What is a statistical model?: Discussion and Rejoinder.  _Ann. Stat._ 30, 1267–1310.  http://www.stat.uchicago.edu/~pmcc/publications.html. See McCullagh (2002)
  * Bishop (2006) Bishop, C. M. (2006).  _Pattern Recognition and Machine Learning_.  Information science and statistics (New York: Springer)
  * Browder (1992) Browder, F. E. (ed.) (1992).  _Mathematics into the Twenty-first Century: 1988 Centennial Symposium August 8–12_ , vol. II of _American Mathematical Society centennial publications_ (Providence, USA: American Mathematical Society)
  * Bruno (1964) Bruno, A. (1964).  On the notion of partial exchangeability.  _ Giorn. Ist. Ital. Att._ 27, 174–196.  Transl. in de Finetti (1972), ch. 10, pp. 229–246
  * Çetin et al. (2014) Çetin, M. S., Christensen, F., Abbott, C. C., Stephen, J. M., Mayer, A. R., Can̄ive, J. M., et al. (2014).  Thalamus and posterior temporal lobe show greater inter-network connectivity at rest and across sensory paradigms in schizophrenia.  _NeuroImage_ 97, 117–126
  * Cetin et al. (2016) Cetin, M. S., Houck, J. M., Rashid, B., Agacoglu, O., Stephen, J. M., Sui, J., et al. (2016).  Multimodal classification of schizophrenia patients with MEG and fMRI data using static and dynamic connectivity measures.  _Front. Neurosci._ 10, 466
  * Cifarelli and Regazzini (1982) Cifarelli, D. M. and Regazzini, E. (1982).  Some considerations about mathematical statistics teaching methodology suggested by the concept of exchangeability.  In _Koch and Spizzichino _(_1982 _)__. 185–205
  * Cox (1946) Cox, R. T. (1946).  Probability, frequency, and reasonable expectation.  _Am. J. Phys._ 14, 1–13.  http://algomagic.org/ProbabilityFrequencyReasonableExpectation.pdf
  * Cox (1961) Cox, R. T. (1961).  _The Algebra of Probable Inference_ (Baltimore: The Johns Hopkins Press)
  * Cox (1979) Cox, R. T. (1979).  Of inference and inquiry, an essay in inductive logic.  In _Levine and Tribus _(_1979 _)__. 119–167
  * Damien et al. (2013) Damien, P., Dellaportas, P., Polson, N. G., and Stephens, D. A. (eds.) (2013).  _Bayesian Theory and Applications_ (Oxford: Oxford University Press)
  * Darmois (1935) Darmois, G. (1935).  Sur les lois de probabilité à estimation exhaustive.  _Comptes rendus hebdomadaires des séances de l’Académie des sciences_ 200, 1265–1266
  * Davidoff (1999) Davidoff, F. (1999).  Standing statistics right side up.  _Ann. Intern. Med._ 130, 1019–1021.  http://www.perfendo.org/docs/bayesprobability/5.3_goodmanannintmed99all.pdf. See Goodman (1999); Sulmasy et al. (2000)
  * Dawid (1982a) Dawid, A. P. (1982a).  Intersubjective statistical models.  In _Koch and Spizzichino _(_1982 _)__. 217–232
  * Dawid (1982b) Dawid, A. P. (1982b).  The well-calibrated Bayesian.  _J. Am. Stat. Assoc._ 77, 605–610
  * Dawid (2013) Dawid, A. P. (2013).  Exchangeability and its ramifications.  In _Damien et al._(_2013 _)__ , chap. 2. 19–29
  * de Finetti (1937) de Finetti, B. (1937).  La prévision: ses lois logiques, ses sources subjectives.  _Ann. Inst. Henri Poincaré_ 7, 1–68.  Transl. in Kyburg and Smokler (1980), pp. 53–118, by Henry E. Kyburg, Jr.
  * de Finetti (1938) de Finetti, B. (1938).  Sur la condition d’équivalence partielle.  In _Colloque consacré à la théorie des probabilités. VI: Conceptions diverses_ , eds. B. de Finetti, V. Glivenko, and G. Neymann (Paris: Hermann), no. 739 in Actualités scientifiques et industrielles. 5–18.  Transl. in Jeffrey (1980), pp. 193–205, by P. Benacerraf and R. Jeffrey
  * de Finetti (1972) de Finetti, B. (1972).  _Probability, Induction and Statistics: The art of guessing_ (London: Wiley)
  * DeGroot (2004) DeGroot, M. H. (2004).  _optimal statistical decisions_.  Wiley classics library (New York: Wiley), reprint edn.
  * Demirci et al. (2008) Demirci, O., Clark, V. P., and Calhoun, V. D. (2008).  A projection pursuit algorithm to classify individuals using fMRI data: Application to schizophrenia.  _NeuroImage_ 39, 1774–1782
  * Denny (1967) Denny, J. L. (1967).  Sufficient conditions for a family of probabilities to be exponential.  _Proc. Natl. Acad. Sci. (USA)_ 57, 1184–1187
  * Desikan et al. (2006) Desikan, R. S., Ségonne, F., Fischl, B., Quinn, B. T., Dickerson, B. C., Blacker, D., et al. (2006).  An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest.  _NeuroImage_ 31, 968–980
  * Diaconis (1988) Diaconis, P. (1988).  Recent progress on de Finetti’s notions of exchangeability.  In _Bernardo et al._(_1988 _)__. 111–125.  With discussion by D. Blackwell, Simon French, and author’s reply. http://statweb.stanford.edu/~cgates/PERSI/year.html, https://statistics.stanford.edu/research/recent-progress-de-finettis-notions-exchangeability
  * Diaconis (1992) Diaconis, P. (1992).  Sufficiency as statistical symmetry.  In _Browder _(_1992 _)__. 15–26.  First publ. 1991 as technical report https://statistics.stanford.edu/research/sufficiency-statistical-symmetry
  * Diaconis and Freedman (1980) Diaconis, P. and Freedman, D. (1980).  De Finetti’s generalizations of exchangeability.  In _Jeffrey _(_1980 _)__. 233–249
  * Diaconis and Freedman (1981) Diaconis, P. and Freedman, D. (1981).  Partial exchangeability and sufficiency.  In _Ghosh and Roy _(_1981 _)__. 205–236.  http://statweb.stanford.edu/~cgates/PERSI/year.html. Also publ. 1982 as technical report https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/diaconis_freedman_PES.pdf
  * Diaconis and Freedman (1986) Diaconis, P. and Freedman, D. (1986).  On the consistency of Bayes estimates.  _Ann. Stat._ 14, 1–26
  * Diaconis and Ylvisaker (1979) Diaconis, P. and Ylvisaker, D. (1979).  Conjugate priors for exponential families.  _Ann. Stat._ 7, 269–281
  * Dupré and Tipler (2009) Dupré, M. J. and Tipler, F. J. (2009).  New axioms for rigorous Bayesian probability.  _Bayesian Anal._ 4, 599–606
  * Edgeworth (1898) Edgeworth, F. Y. (1898).  On the representation of statistics by mathematical formulæ (Part I).  _J. Roy. Stat. Soc._ 61, 670–700.  See also Edgeworth (1899, 1900)
  * Edgeworth (1899) Edgeworth, F. Y. (1899).  On the representation of statistics by mathematical formulæ (Part II & III).  _J. Roy. Stat. Soc._ 62, 125–140, 373–385.  See also Edgeworth (1898, 1900)
  * Edgeworth (1900) Edgeworth, F. Y. (1900).  On the representation of statistics by mathematical formulæ (Supplement).  _J. Roy. Stat. Soc._ 63, 72–81.  See also Edgeworth (1898, 1899)
  * Ellison-Wright and Bullmore (2009) Ellison-Wright, I. and Bullmore, E. (2009).  Meta-analysis of diffusion tensor imaging studies in schizophrenia.  _Schizophrenia Research_ 108, 3–10
  * Ellison-Wright and Bullmore (2010) Ellison-Wright, I. and Bullmore, E. (2010).  Anatomy of bipolar disorder and schizophrenia: A meta-analysis.  _Schizophrenia Research_ 117, 1–12
  * Ferguson (1974) Ferguson, T. S. (1974).  Prior distributions on spaces of probability measures.  _Ann. Stat._ 2, 615–629
  * Fisher (1922) Fisher, R. A. (1922).  On the mathematical foundations of theoretical statistics.  _Phil. Trans. R. Soc. Lond. A_ 222, 309–368.  http://www.stats.org.uk/statistical-inference/Fisher1922.pdf
  * Fraser (1963) Fraser, D. A. S. (1963).  On sufficiency and the exponential family.  _J. Roy. Stat. Soc. B_ 25, 115–123
  * Freedman (1962) Freedman, D. A. (1962).  Invariants under mixing which generalize de Finetti’s theorem.  _Ann. Math. Stat._ 33, 916–923
  * Friston and Frith (1995) Friston, K. J. and Frith, C. D. (1995).  Schizophrenia: A disconnection syndrome?  _Clinical Neuroscience_ 3, 89–97
  * Gelman et al. (2014) Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2014).  _Bayesian Data Analysis_.  Texts in statistical science (Boca Raton, USA: Chapman & Hall/CRC), 3 edn.  First publ. 1995
  * Gheiratmand et al. (2017) Gheiratmand, M., Rish, I., Cecchi, G. A., Brown, M. R. G., Greiner, R., Polosecki, P. I., et al. (2017).  Learning stable and predictive network-based patterns of schizophrenia and its clinical symptoms.  _NPJ Schizophr._ 3, 22
  * Ghosh and Roy (1981) Ghosh, J. K. and Roy, J. (eds.) (1981).  _Statistics: Applications and New Directions_ (Calcutta: Indian Statistical Institute)
  * Good (1950) Good, I. J. (1950).  _Probability and the Weighing of Evidence_ (London: Griffin)
  * Good (1956) Good, I. J. (1956).  The surprise index for the multivariate normal distribution.  _Ann. Math. Stat._ 27, 1130–1135.  See also corrections Good (1957b)
  * Good (1957a) Good, I. J. (1957a).  The appropriate mathematical tools for describing and measuring uncertainty.  In _Good _(_1983 _)__ , chap. 16. 173–177.  First publ. 1957
  * Good (1957b) Good, I. J. (1957b).  Corrections to “The surprise index for the multivariate normal distribution”.  _Ann. Math. Stat._ 28, 1055.  See Good (1956)
  * Good (1983) Good, I. J. (1983).  _Good Thinking: The Foundations of Probability and Its Applications_ (Minneapolis, USA: University of Minnesota Press)
  * Goodman (1999) Goodman, S. N. (1999).  Toward evidence-based medical statistics. 1: The $P$ value fallacy. 2: The Bayes factor.  _Ann. Intern. Med._ 130, 995–1013.  http://www.perfendo.org/docs/bayesprobability/5.3_goodmanannintmed99all.pdf. See also comments and reply Davidoff (1999); Sulmasy et al. (2000)
  * Grabner et al. (2006) Grabner, G., Janke, A. L., Budge, M. M., Smith, D., Pruessner, J., and Collins, D. L. (2006).  Symmetric atlasing and model based segmentation: an application to the hippocampus in older adults.  _Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv_ 9, 58–66
  * Greenland (1998) Greenland, S. (1998).  Probability logic and probabilistic induction.  _Epidemiology_ 9, 322–332.  See also Maclure (1998)
  * Gupta and Nagar (2000) Gupta, A. K. and Nagar, D. K. (2000).  _Matrix Variate Distributions_ , vol. 104 of _Monographs and surveys in pure and applied mathematics_ (Boca Raton, USA: Chapman & Hall/CRC)
  * Hailperin (1996) Hailperin, T. (1996).  _Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications_ (London: Associated University Presses)
  * Halpern (1999) Halpern, J. Y. (1999).  Cox’s theorem revisited.  _J. Artif. Intell. Res._ 11, 429–435.  See also Snow (1998)
  * He et al. (2009) He, Y., Wang, J., Wang, L., Chen, Z. J., Yan, C., Yang, H., et al. (2009).  Uncovering intrinsic modular organization of spontaneous brain activity in humans.  _PLOS ONE_ 4, 1–18
  * Hipp (1974) Hipp, C. (1974).  Sufficient statistics and exponential families.  _Ann. Stat._ 2, 1283–1292
  * Ho et al. (2000) Ho, B.-C., Andreasen, N. C., Flaum, M., Nopoulos, P., and Miller, D. (2000).  Untreated initial psychosis: Its relation to quality of life and symptom remission in first-episode schizophrenia.  _Am. J. Psychiatry_ 157, 808–815
  * Horvitz et al. (1986) Horvitz, E. J., Heckerman, D. E., and Langlotz, C. P. (1986).  A framework for comparing alternative formalisms for plausible reasoning.  _Proc. AAAI_ 5, 210–214
  * Hu et al. (2017) Hu, M.-L., Zong, X.-F., Mann, J. J., Zheng, J.-J., Liao, Y.-H., Li, Z.-C., et al. (2017).  A review of the functional and anatomical default mode network in schizophrenia.  _Neuroscience Bulletin_ 33, 73–84
  * Huszár (2017) Huszár, F. (2017).  Everything that works works because it’s Bayesian: Why deep nets generalize?  http://www.inference.vc/everything-that-works-works-because-its-bayesian-2/
  * Jaynes (2003) Jaynes, E. T. (2003).  _Probability Theory: The Logic of Science_ (Cambridge: Cambridge University Press).  Ed. by G. Larry Bretthorst; http://www-biba.inrialpes.fr/Jaynes/prob.html, http://omega.albany.edu:8008/JaynesBook.html, http://omega.albany.edu:8008/JaynesBookPdf.html. First publ. 1994
  * Jeffrey (1980) Jeffrey, R. C. (ed.) (1980).  _Studies in inductive logic and probability. Vol. II_ (Berkeley: University of California Press)
  * Jeffreys (2003) Jeffreys, H. (2003).  _Theory of Probability_ (London: Oxford University Press), 3 edn.  First publ. 1939
  * Jenkinson et al. (2002) Jenkinson, M., Bannister, P., Brady, M., and Smith, S. (2002).  Improved optimization for the robust and accurate linear registration and motion correction of brain images.  _NeuroImage_ 17, 825–841
  * Jenkinson et al. (2012) Jenkinson, M., Beckmann, C. F., Behrens, T. E., Woolrich, M. W., and Smith, S. M. (2012).  Fsl.  _NeuroImage_ 62, 782–790
  * Jenkinson and Smith (2001) Jenkinson, M. and Smith, S. (2001).  A global optimisation method for robust affine registration of brain images, medical image analysis.  _Medical Image Analysis_ 5, 143–156
  * Johnson (1949) Johnson, N. L. (1949).  Systems of frequency curves generated by methods of translation.  _Biometrika_ 36, 149–176
  * Kallenberg (2005) Kallenberg, O. (2005).  _Probabilistic Symmetries and Invariance Principles_.  Probability and its applications (New York: Springer)
  * Kass and Raftery (1995) Kass, R. E. and Raftery, A. E. (1995).  Bayes factors.  _J. Am. Stat. Assoc._ 90, 773–795.  https://www.stat.washington.edu/raftery/Research/PDF/kass1995.pdf; https://www.andrew.cmu.edu/user/kk3n/simplicity/KassRaftery1995.pdf
  * Kingman (1978) Kingman, J. F. C. (1978).  Uses of exchangeability.  _Ann. Prob._ 6, 183–197
  * Koch and Spizzichino (1982) Koch, G. and Spizzichino, F. (eds.) (1982).  _Exchangeability in Probability and Statistics_ (Amsterdam: North-Holland)
  * Kolmogorov (1942) Kolmogorov, A. N. (1942).  Definition of center of dispersion and measure of accuracy to form a finite number of observations [Sur l’éstimation statistique des paramètres de la loi de Gauss].  _Izv. Akad. Nauk SSSR Ser. Mat._ 6, 3–32.  In Russian
  * Kook et al. (2017) Kook, J. H., Guindani, M., Zhang, L., and Vannucci, M. (2017).  _NPBayes-fMRI_ : Non-parametric Bayesian general linear models for single- and multi-subject fMRI data.  _Stat. Biosci._ 2017, 1–19
  * Koopman (1936) Koopman, B. O. (1936).  On distributions admitting a sufficient statistic.  _Trans. Am. Math. Soc._ 39, 399–409
  * Koopman (1940a) Koopman, B. O. (1940a).  The axioms and algebra of intuitive probability.  _Ann. Math._ 41, 269–292
  * Koopman (1940b) Koopman, B. O. (1940b).  The bases of probability.  _Bull. Am. Math. Soc._ 46, 763–774.  Repr. in Kyburg and Smokler (1980), pp. 159–172
  * Koopman (1941) Koopman, B. O. (1941).  Intuitive probabilities and sequences.  _Ann. Math._ 42, 169–187
  * Kotz et al. (2006) Kotz, S., Balakrishnan, N., Read, C., Vidakovic, B., and Johnson, N. L. (eds.) (2006).  _Encyclopedia of Statistical Sciences_ (Hoboken, USA: Wiley), 2 edn.  First publ. 1982
  * Kotz and Nadarajah (2004) Kotz, S. and Nadarajah, S. (2004).  _Multivariate $t$ Distributions and Their Applications_ (Cambridge: Cambridge University Press)
  * Kurtz and Swartz (2004) Kurtz, D. S. and Swartz, C. W. (2004).  _Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane_ , vol. 9 of _Series in real analysis_ (Singapore: World Scientific)
  * Kyburg and Smokler (1980) Kyburg, H. E., Jr. and Smokler, H. E. (eds.) (1980).  _Studies in Subjective Probability_ (Huntington, USA: Robert E. Krieger), 2 edn.  First publ. 1964
  * Lamoreaux and Armstrong (1998) Lamoreaux, J. and Armstrong, G. (1998).  The fundamental theorem of calculus for gauge integrals.  _Math. Mag._ 71, 208–212
  * Lauritzen (1988) Lauritzen, S. L. (1988).  _Extremal Families and Systems of Sufficient Statistics_ , vol. 49 of _Lecture notes in statistics_ (Berlin: Springer).  First publ. 1982
  * Levine and Tribus (1979) Levine, R. D. and Tribus, M. (eds.) (1979).  _The Maximum Entropy Formalism: A Conference Held at the Massachusetts Institute of Technology on May 2–4, 1978_ (Cambridge, USA: MIT Press)
  * Lindley (2008) Lindley, D. V. (2008).  _Introduction to Probability and Statistics from a Bayesian Viewpoint. Part 2: Inference_ (Cambridge: Cambridge University Press), reprint edn.  First publ. 1965
  * Lindley and Phillips (1976) Lindley, D. V. and Phillips, L. D. (1976).  Inference for a Bernoulli process (a Bayesian view).  _American Statistician_ 30, 112–119
  * Lindquist (2008) Lindquist, M. A. (2008).  The statistical analysis of fMRI data.  _Stat. Sci._ 23, 439–464
  * MacKay (1992a) MacKay, D. J. C. (1992a).  Bayesian interpolation.  _Neural Comp._ 4, 415–447.  http://www.inference.phy.cam.ac.uk/mackay/PhD.html
  * MacKay (1992b) MacKay, D. J. C. (1992b).  _Bayesian Methods for Adaptive Models_.  Ph.D. thesis, California Institute of Technology, Pasadena, USA.  http://www.inference.phy.cam.ac.uk/mackay/PhD.html
  * MacKay (1992c) MacKay, D. J. C. (1992c).  The evidence framework applied to classification networks.  _Neural Comp._ 4, 720–736.  http://www.inference.phy.cam.ac.uk/mackay/PhD.html
  * MacKay (1992d) MacKay, D. J. C. (1992d).  Information-based objective functions for active data selection.  _Neural Comp._ 4, 590–604.  http://www.inference.phy.cam.ac.uk/mackay/PhD.html
  * MacKay (1992e) MacKay, D. J. C. (1992e).  A practical bayesian framework for backpropagation networks.  _Neural Comp._ 4, 448–472.  http://www.inference.phy.cam.ac.uk/mackay/PhD.html
  * MacKay (2003) MacKay, D. J. C. (2003).  _Information Theory, Inference, and Learning Algorithms_ (Cambridge: Cambridge University Press).  http://www.inference.phy.cam.ac.uk/mackay/itila/. First publ. 1995
  * Maclure (1998) Maclure, M. (1998).  How to change your mind.  _Epidemiology_ 9, 233.  See Greenland (1998)
  * Marrelec and Fransson (2011) Marrelec, G. and Fransson, P. (2011).  Assessing the influence of different ROI selection strategies on functional connectivity analyses of fMRI data acquired during steady-state conditions.  _PLoS One_ 6, e14788
  * McCullagh (2002) McCullagh, P. (2002).  What is a statistical model?  _Ann. Stat._ 30, 1225–1267.  http://www.stat.uchicago.edu/~pmcc/publications.html. See also the following discussion and rejoinder Besag et al. (2002)
  * McKenzie (2014) McKenzie, K. J. (2014).  How does untreated psychosis lead to neurological damage?  _Can. J. Psychiatry_ 59, 511–512
  * Mead (1965) Mead, R. (1965).  A generalised logit-normal distribution.  _Biometrics_ 21, 721–732
  * Michael et al. (2010) Michael, A. M., Baum, S. A., White, T., Demirci, O., Andreasen, N. C., Segall, J. M., et al. (2010).  Does function follow form?: Methods to fuse structural and functional brain images show decreased linkage in schizophrenia.  _NeuroImage_ 49, 2626–2637
  * Minka (2001) Minka, T. (2001).  _Inferring a Gaussian distribution_.  Tech. rep., MIT media Lab, Cambridge, USA.  http://research.microsoft.com/en-us/um/people/minka/papers/. First publ. 1998
  * Mosteller and Wallace (1963) Mosteller, F. and Wallace, D. L. (1963).  Inference in an authorship problem: A comparative study of discrimination methods applied to the authorship of the disputed _federalist_ papers.  _J. Am. Stat. Assoc._ 58, 275–309.  https://www.stat.cmu.edu/Exams/mosteller.pdf
  * Murphy (2007) Murphy, K. P. (2007).  Conjugate Bayesian analysis of the Gaussian distribution.  http://thaines.com/content/misc/gaussian_conjugate_prior_cheat_sheet.pdf
  * Murphy (2012) Murphy, K. P. (2012).  _Machine Learning: A Probabilistic Perspective_.  Adaptive computation and machine learning series (Cambridge, USA: MIT Press)
  * Neyman (1935) Neyman, J. (1935).  Su un teorema concernente le cosiddette statistiche sufficienti.  _Giorn. Ist. Ital. Att._ VI, 320–334
  * Nielsen et al. (2016) Nielsen, S. F. V., Madsen, K. H., Røge, R., Schmidt, M. N., and Mørup, M. (2016).  Nonparametric modeling of dynamic functional connectivity in fMRI data.  arXiv:1601.00496
  * Paris (2006) Paris, J. B. (2006).  _The Uncertain Reasoner’s Companion: A Mathematical Perspective_.  No. 39 in Cambridge tracts in theoretical computer science (Cambridge: Cambridge University Press), reprint edn.  Concerning Cox’s theorem see also Snow (1998)
  * Pitman (1936) Pitman, E. J. G. (1936).  Sufficient statistics and intrinsic accuracy.  _Math. Proc. Camb. Phil. Soc._ 32, 567–579
  * Pólya (1949) Pólya, G. (1949).  Preliminary remarks on a logic of plausible inference.  _Dialectica_ 3, 28–35
  * Pólya (1968) Pólya, G. (1968).  _Mathematics and Plausible Reasoning: Vol. II: Patterns of Plausible Inference_ (Princeton: Princeton University Press), 2 edn.  First publ. 1954
  * Porta Mana (2017) Porta Mana, P. G. L. (2017).  Model comparison and Bayes factors: what is a model?  In preparation
  * Porta Mana et al. (2018) Porta Mana, P. G. L., Bachmann, C., and Morrison, A. (2018).  Inferring health conditions from fMRI-graph data.  Open Science Framework doi:10.17605/osf.io/r2huz
  * Raiffa and Schlaifer (2000) Raiffa, H. and Schlaifer, R. (2000).  _Applied Statistical Decision Theory_.  Wiley Classics Library (New York: Wiley), reprint edn.  First publ. 1961
  * Silva and Calhoun (2014) Silva, F. R., Castro, E., Gupta, C. N. , Cetin, M. , Arbabshirani, M., Potluru, V. K., and Plis, S. M. and Calhoun, V. D. (2014).  The tenth annual MLSP competition: Schizophrenia classification challenge.  _2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP),21–24_
  * Shenton et al. (2010) Shenton, M. E., Whitford, T. J., and Kubicki, M. (2010).  Structural neuroimaging in schizophrenia from methods to insights to treatments.  _Dialogues in Clinical Neuroscience_ 12, 317–332
  * Shergill et al. (2000) Shergill, S., Brammer, M., Williams, S., Murray, R., and McGuire, P. (2000).  Mapping auditory hallucinations in schizophrenia using functional magnetic resonance imaging.  _Archives of General Psychiatry_ 57, 1033–1038
  * Smith (2002) Smith, S. M. (2002).  Fast robust automated brain extraction.  _Hum. Brain Mapp._ 17, 143–155
  * Smith et al. (2005) Smith, S. M., Jenkinson, M., Woolrich, M. W., Beckmann, C. F., Behrens, T. E., Johansen-Berg, H., et al. (2005).  Advances in functional and structural mr image analysis and implementation as fsl.  _NeuroImage_ 23, S208–219
  * Smith et al. (2011) Smith, S. M., Miller, K. L., Salimi-Khorshidi, G., Webster, M., Beckmann, C. F., Nichols, T. E., et al. (2011).  Network modelling methods for fMRI.  _NeuroImage_ 54, 875–891
  * Snow (1998) Snow, P. (1998).  On the correctness and reasonableness of Cox’s theorem for finite domains.  _Comput. Intell._ 14, 452–459
  * Sox et al. (2013) Sox, H. C., Higgins, M. C., and Owens, D. K. (2013).  _Medical Decision Making_ (New York: Wiley), 2 edn.  First publ. 1988
  * Stone (1974) Stone, M. (1974).  Cross-validatory choice and assessment of statistical predictions.  _J. Roy. Stat. Soc. B_ 36, 111–133.  See also discussion Barnard et al. (1974)
  * Sulmasy et al. (2000) Sulmasy, D. P., Morgan, T., Caubet, J.-F., and Goodman, S. (2000).  Toward evidence-based statistics [comments and response].  _Ann. Intern. Med._ 132, 507–508.  See Goodman (1999); Davidoff (1999)
  * Swartz (2001) Swartz, C. (2001).  _Introduction to Gauge Integrals_ (Singapore: World Scientific)
  * Terenin and Draper (2017) Terenin, A. and Draper, D. (2017).  Cox’s theorem and the Jaynesian interpretation of probability.  arXiv:1507.06597. First publ. 2015
  * iso (2006) iso (2006).  _ISO 3534-1:2006: Statistics – Vocabulary and symbols – Part 1: General statistical terms and terms used in probability_.  International Organization for Standardization, Geneva
  * iso (2009) iso (2009).  _ISO 80000:2009: Quantities and units_.  International Organization for Standardization, Geneva.  First publ. 1993
  * Tiao and Zellner (1964) Tiao, G. C. and Zellner, A. (1964).  On the Bayesian estimation of multivariate regression.  _J. Roy. Stat. Soc. B_ 26, 277–285
  * Venkataraman et al. (2012) Venkataraman, A., Whitford, T. J., Westin, C.-F., Golland, P., and Kubicki, M. (2012).  Whole brain resting state functional connectivity abnormalities in schizophrenia.  _Schizophr. Res._ 139, 7–12