# A Priori Determination of the Pretest Probability

Jacques Balayla MD, MPH 111To whom correspondence should be addressed: Dr.
Jacques Balayla MD, MPH, CIP, FRCSC. Obstetrician-Gynecologist. Quilligan
Scholar. e-mail<EMAIL_ADDRESS>Lady Davis Institute (LDI), Jewish
General Hospital, McGill University, Montreal, Quebec, Canada

Abstract. In this manuscript, we present various proposed methods estimate the
prevalence of disease, a critical factor in the interpretation of screening
tests. To address the limitations of these approaches, which revolve primarily
around their a posteriori nature, we introduce a novel method to estimate the
pretest probability of disease, a priori, utilizing the Logit function from
the logistic regression model. This approach is a modification of McGee’s
heuristic, originally designed for estimating the posttest probability of
disease. In a patient presenting with $n_{\theta}$ signs or symptoms, the
minimal bound of the pretest probability, $\phi$, can be approximated by:

$\phi\approx\frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right]}$

where $ln$ is the natural logarithm, and $\kappa_{\theta}$ is the likelihood
ratio associated with the sign or symptom in question.

##  1 Background

Evidence-based medicine relies on a proper understanding of epidemiological
principles and biostatistical data retrieved from well-conducted studies,
which then inform the most appropriate clinical management of patients [1].
Pretest probability is a term used in medical and diagnostic contexts to
describe the likelihood or probability of a particular condition or disease
being present before any diagnostic tests or investigations are conducted [2].
In Bayesian terms, it represents the prior probability for a binary
classification system, such as a screening intervention that classifies
individuals in one of two categories, that is, as either “sick” or “not sick”
[3]. The pretest probability, henceforth referred to as $\phi$, represents the
initial estimate of the probability that a patient has a certain condition
based on any signs, symptoms, and risk factors that prompt the clinician to
consider specific tests in light of a potential diagnosis [4]. The pretest
probability is critical because it significantly influences the interpretation
of screening and diagnostic test results [5]. In medical decision-making, the
accuracy of a screening test is often expressed in terms of the sensitivity
and specificity of a screening tool [6]. Sensitivity, or true positive rate,
refers to the ability of a test to correctly identify individuals with the
disease, while specificity, or true negative rate, is the ability to correctly
identify individuals without the disease [7]. The relationship between pretest
probability, sensitivity, and specificity can be understood through Bayes’
theorem, which describes how the pretest probability of disease is updated
based on new evidence provided by the test results [8]. In the screening
context, the posttest probability, or the probability of having the disease
after testing positive, is known as the positive predictive value, henceforth
termed $\rho(\phi)$ [4]. Unfortunately, evidence suggests that clinicians have
difficulty interpreting Bayes’ theorem and its influence on several of the
accuracy parameters of binary classification systems [9]. Therefore,
understanding and considering the pretest probability is a critical and
important step towards understanding diagnostic test results accurately and
making informed clinical decisions.

##  2 Pretest Probability

As alluded to before, accounting for the pretest probability of disease is an
essential prerequisite to evaluate the reliability of screening and diagnostic
assessments in all areas of clinical medicine. Despite its crucial role in the
screening process, there is currently no reliable method to numerically
estimate the pretest probability of disease in a given individual [10].
Several approaches have been proposed – but these are often population-based
and retrospective in the sense that only once the total number of positive
screening results are considered, can an estimate for an average pretest
probability be extrapolated in hindsight [10]. Often, for simplicity’s sake,
we use the prevalence of disease as a proxy statistic, but this approach
overlooks the individualized nature of risk, grouping individuals with
different risk profiles together [11]. The consequences of this approach are
several, and include the dilution of intervention effects, the inability to
account for population changes in disease patterns over time, and the
introduction of selection bias when selecting study participants, amongst
others [12]. Patients with signs and symptoms of disease have a greater
pretest probability than those without, and vice-versa [13]. As a metric,
pretest probability is measured on a continuous spectrum from 0 to 100$\%$ and
its value is proportional to the positive predictive value of screening tests
[14]. Though the interpretative difficulties around these concepts may in fact
be a consequence of the terminology used - which isn’t immediately intuitive,
it also stands to reason that Bayes’ Theorem is fundamentally mathematical and
therefore somewhat abstract [5]. In this manuscript, we present different
proposed methods to estimate the pretest probability of disease - a
cornerstone of the diagnostic process. In order to overcome their limitations,
we propose a method to estimate - a priori \- the pretest probability of
disease using the logit function, as a modification of McGee’s heuristic,
originally developed for the estimation of the posttest probability [15].

###  2.1 Positive Likelihood Ratio ($\kappa$)

The positive likelihood ratio (LR+) for a dichotomous test is defined as the
likelihood of a test result in patients with the disease divided by the
likelihood of the test result in patients without the disease. Otherwise
stated, the positive likelihood ratio $\kappa$ gives the change in the odds of
having a diagnosis in patients with a positive test [16]. For example, a
$\kappa$ close to 1 means that the test result does not change the likelihood
of disease or the outcome of interest appreciably. The more the likelihood
ratio for a positive test is greater than 1, the more likely the disease or
outcome [15]. We can visualise the impact of the likelihood ratio $\kappa$ on
the PPV by using a Fagan nomogram [17]. A Fagan nomogram is a graphical tool
used in medicine and clinical decision-making to calculate and illustrate the
post-test probability of a medical condition based on the results of
diagnostic tests and the pre-test probability of that condition. It’s named
after the British epidemiologist and statistician M. J. Fagan, who developed
this visualization method. The Fagan nomogram consists of a simple chart or
diagram that allows healthcare professionals to estimate how much a diagnostic
test result should change their suspicion of a particular diagnosis [17]. It’s
especially useful when dealing with medical tests that have both false
positive and false negative results, as it helps clinicians interpret the test
results in the context of the patient’s clinical presentation. To use a Fagan
nomogram, you draw a straight line connecting the pre-test probability on one
side to the likelihood ratio (LR) associated with the test result on the other
side. The point where the line intersects the post-test probability scale
represents the updated estimate of the likelihood that the patient has the
condition after considering the test result. That the separation between the
ticks on the pre and post-test probability are not equally spaced suggests
that indeed the relationship between these two variables - as stated in the
screening equation - are not linear. Fagan nomograms are valuable tools
because they provide a visual and intuitive way for healthcare professionals
to integrate diagnostic test results into their clinical decision-making
process. They help clinicians determine how much confidence they should place
in a test result and whether further testing or clinical interventions are
necessary.

##  3 Bayesian fundamentals in the context of screening

Bayes’ theorem provides a mathematical framework to explain how existing
beliefs change in light of new evidence [4]. From Bayes’ theorem, we can
derive the positive predictive value equation of a screening test,
$\rho(\phi)$, defined as the percentage of positive tests which correctly
identify affected individuals. The positive predictive value (PPV) is a
function of a test’s sensitivity and specificity, but also of the pretest
probability:

$\rho(\phi)=\frac{a\phi}{a\phi+(1-b)(1-\phi)}$ (3.1)

where $\rho(\phi)$ = PPV, a = sensitivity, b = specificity and $\phi$ =
prevalence/pretest probability. The PPV, $\rho(\phi)$, is therefore a function
of the pretest probability, $\phi$.

We have previously defined the prevalence threshold as the prevalence level in
the PPV-prevalence curve below which binary classification systems are less
reliable [18]. In technical terms, this point represents the inflection point
of maximum curvature in the aforementioned curve below which the the rate of
change of a tool’s positive predictive value drops at a differential pace
relative to the pretest probability [4], [14]. Below this point, type I errors
- the mistaken rejection of a true null hypothesis, what is colloquially known
as “false positives” - increase [19]. This value, termed $\phi_{e}$, is a
function of a test’s sensitivity $a$ and specificity $b$, and is defined on
the pretest probability axis [20]:

$\phi_{e}=\frac{\sqrt{1-b}}{\sqrt{a}+\sqrt{1-b}}$ (3.2)

The corresponding positive predictive value at this prevalence level is given
by plotting the above equation into the positive predictive value equation
[21], so we obtain:

$\rho(\phi_{e})=\sqrt{\frac{a}{1-b}}\frac{\sqrt{1-b}}{\sqrt{a}+\sqrt{1-b}}=\sqrt{\frac{a}{1-b}}\phi_{e}$
(3.3)

Interestingly, the above expression leads to the well known formulation for
the positive predictive value as a function of prevalence and the positive
likelihood ratio ($\kappa$), defined as the sensitivity $a$ over the
compliment of the specificity $b$ [16]:

$\kappa=\frac{a}{1-b}$ (3.4)

Graphically, an example of a PPV-prevalence curve with some sensitivity $a$
and specificity $b$, reveals the point of maximum curvature where the radius
of curvature $R_{c}$ is at a minimum [22]. The vertical dashed line is the
prevalence threshold (Figure 1).

Figure 1. PPV-pretest probability curve and the prevalence threshold
$\phi_{e}$

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$$\kappa=1/R$$tangent$$R_{c}$$\phi_{e}$$\phi$$\rho(\phi)$$\rho(\phi)$$R_{c}$

Beyond this point, the rate of change of $\rho(\phi)$ is equal to
$\frac{1}{\sqrt{\kappa}}$ (work not shown here), which suggests that for
sufficiently adequate systems, $\rho(\phi_{e})$ is comparable to $\rho(1)$.

###  3.1 Heuristics in establishing pretest probabilities

The main theme of the theory on the prevalence threshold is that the
prevalence threshold, $\phi_{e}$, can act as a benchmark for confidence in
clinical assessments and decision-making, as the predictive values are
comparable by a factor of 1/$\sqrt{\kappa}$ relative to 1 [14]. The critical
question that remains though is, how do we know where an individual lies in
term of risk? How do we therefore know if we have reached the threshold or
not? Indeed, accounting for the pretest probability of disease is an essential
prerequisite to evaluate the reliability of screening and diagnostic
assessments in all areas of clinical medicine. Despite its crucial role in the
screening process, there is currently no reliable method to evaluate the
pretest probability of disease in a given individual. Several approaches have
been proposed – but these are often population-based and retrospective in the
sense that only once the total number of positive screening results are
considered, can an estimate for an average pretest probability be extrapolated
in hindsight. Herein, we review some of these methods.

###  3.2 Qualitative approaches to Bayes’ Theorem

The mathematical complexity of Bayes’ theorem is in fact not insurmountable.
However, for clinicians, it is also not immediately intuitive [23]. Several
attempts to reduce the theorem’s complexity have been proposed, so as to
render it more available for clinical use. Medow and Lucey’s heuristic for
estimating the posttest probability in individuals who undergo screening is a
pragmatic tool which qualitatively categorises an individual’s risk of disease
prior to undergoing testing [5]. In essence, one can first categorise the
pretest probability of disease as very unlikely (less likely than 10$\%$),
unlikely (10–33$\%$), uncertain (34–66$\%$), likely (67–90$\%$) or very likely
(more likely than 90$\%$) (Table 1). They suggest that for disorders that are
very unlikely or very likely, no further testing is needed. On the other hand,
if the prior probability is unlikely, uncertain or likely, a test and a
bayesian-inspired update process is incorporated to obtain a new result [24].
As per the heuristic, a positive result of a good test increases the
probability of the disorder by one likelihood category (eg, from uncertain to
likely) and a negative test decreases the probability by one category (e.g
from unlikely to very unlikely) [5]. Though Medow and Lucey’s heuristic
reduces the mathematical abstractions of Bayes’ Theorem - several potential
issues arise. Indeed, the clinician is called to make subjective assessments
about an a priori risk, without any specific guidance as to how to proceed.
Similarly, such strict cut-off criteria can be problematic when, for example,
a clinician estimates a risk that hovers around two categories. Which category
do we choose? Do we proceed with testing or not? Furthermore, a critical
consideration about the screening process is seemingly overlooked. We know the
accuracy of a classification model, such as that provided by a screening test,
is dependent on several factors, notably: the classification rule or algorithm
used (what is considered the disease state?), the intrinsic characteristics of
the tool used to do the classification (the sensitivity and specificity of the
test), and the relative frequency of the elements being classified (prevalence
of disease/pretest probability) [10]. And yet, Medow and Lucey’s heuristic
does not account for the impact of a test’s sensitivity and specificity.
Instead, they suggest that “using the above intuitive cut-offs, and tests with
sensitivities and specificities between 80$\%$ and 90$\%$ the above procedure
is a good approximation to Bayes’ theorem” [5]. This is problematic, given
that, as clinicians, we should always strive to develop tests whose
sensitivities and specificities approach 100$\%$ so as to reduce Type I and
Type II errors [25], and enhance the screening accuracy for a given pretest
probability. In essence, as screening tests are refined and become more
powerful over time, Medow and Lucey’s heuristic’s utility is poised to
decrease. Van den Ende et al. showcase a logarithmic scale derivation to
propose a similar category-based approach of “clinical power classes” (Table
2) towards the estimation of posttest probabilities [26]. However, neither
method provides a formal, a priori, way to determine the pretest probability
of disease. Indeed, Van den Ende et al. aptly claim: “of course, this model
gives no solution for erroneous estimations of pre-test probability and LRs of
tests, which would have the same deleterious effect in the classical model”
[26]. It is thus critical the we develop methods that can better estimate the
pretest probability of disease.

Table 1. Medow and Lucey’s heuristic for pretest probability

Categorical probability | Numerical probability |  |
---|---|---|---
Very unlikely | Less likely than 10% |  |
Unlikely | Between 10% and 33% |  |
Uncertain | Between 34% and 66% |  |
Likely | Between 67% and 90% |  |
Very likely | More likely than 90% |  |

Table 2. Van der Ende et al. clinical power classes

Table 2.
---
$\kappa$ | $log_{10}\kappa$ | Clinical power class
100 | 2 | Very strong confirmer
33 | 1.5 | Strong confirmer
10 | 1 | Good confirmer
3 | 0.5 | Weak confirmer
1 | 0 | Useless
0.3 | -0.5 | Weak excluder
0.1 | -1 | Good excluder
0.03 | -1.5 | Strong excluder
0.01 | -2 | Very strong excluder

###  3.3 Numerical approaches

The traditional approach in clinical medicine often draws a clear distinction
between information obtained from a medical history, encompassing risk
factors, exposures, and the history of the present illness, and the insights
gleaned from a screening test result. The terms “pretest” and “posttest”
conventionally mark this chronological division based on the information
provided by the test [27]. However, I contend that this distinction is
somewhat artificial, as both sources provide valuable information about the
likelihood of the presence or absence of disease. While a screening test is
typically ordered following indications from the medical history suggesting
potential presence of disease, it is crucial to recognize that a screening
test is not confirmatory [28]. Instead, it acts as an additional piece of
information, akin to introducing another risk factor or symptom that
influences the likelihood of a diagnosis [29]. Traditionally, we have been
trained to consider sensitivity, specificity, and predictive values as
inherent properties of a test. However, there is nothing inherently special
about a test compared to other sources of information. Signs, symptoms, risk
factors, and details from the medical history also possess associated
sensitivity and specificity parameters [30]. This understanding lies at the
core foundation of “clinical diagnoses”, where a diagnosis is made in the
absence of formal testing. Each sign, symptom, exposure, risk factor, or test
result, denoted as $\theta$, contributes to the overall pretest probability
$\phi$. By simple mathematical induction, the more symptoms, the more likely
the presence of disease. As such, the pretest probability can be viewed as the
cumulative effect of individual factors related to $\theta$:

$\phi=\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}....\theta_{i}$

But how does the sum of risk factors translate into probabilities ranging from
0 to 1 given that $a$ and $b$ are also parameters ranging from 0-1? Logically,
each factor must have an associated weight such that their sum yields a value
between 0-1. What interactions, biases and confounders are at play between
different $\theta_{i}$? Is this sum commutative? These are important questions
we must consider when estimating pretest probabilities. In the next sections,
we’ll describe the ways in which pretest probabilities have been previously
numerically estimated.

###  3.4 Richardson’s methods

Richardson et al. [31] list several approaches to estimate the pretest
probability, most of which make use of previously collected data. These
approaches include using disease prevalence data from clinical research
studies, the prevalence data among patients presenting with the symptom/sign
of interest (often derived from practice or administrative databases), the
disease prevalence in the general population as obtained from population
studies, as well as the use of clinical judgement, and clinical prediction
rules/calculators if they exist. Indeed, estimating pretest probabilities
involves a combination of objective data analysis, consideration of relevant
factors, and expert judgement [32]. The specific approach will depend on the
context and the nature of the problem or decision at hand. However, as the
list above suggests, we would need prevalence information to have been
formally collected previously in order to make use of it - unfortunately, this
isn’t always feasible nor is data always readily available. In order to
evaluate individual data points that allow for the estimation of prevalence
using screening test results, we can use Marchevsky, Rogan and Gladen’s
method.

###  3.5 Marchevsky, Rogan and Gladen’s method

Marchevsky, Rogan and Gladen introduced a method for obtaining population
estimates of disease prevalence, based on known sensitivity and specificity
parameters of a screening test, as applied to $n$ individuals [33], [34]. If
an imperfect test is applied to a random sample of $n$ subjects, where $t$
obtain a positive result, we obtain an apparent prevalence $\hat{\phi}$ given
by:

$\hat{\phi}$ = $\frac{t}{n}$ = $a\phi+(1-b)(1-\phi)=(1-b)+J\phi$

By accounting for all positives - even false positives - we say that the
estimate for prevalence is “biased” [35]. This approach requires a large $n$
to be precise, as it estimates the average prevalence of disease and less so
the individual pretest probability of participants. Let us denote $\hat{\phi}$
as the total proportion of positive tests in a population, which is the sum of
true positives and false positives over $n$ and $\phi$ is the prevalence of
disease. Then, for the population studied, $\phi$ is given by:

$\phi=\frac{\hat{\phi}-(1-b)}{a-(1-b)}=\frac{\hat{\phi}-(1-b)}{J}=\frac{\hat{\phi}+b-1}{J}$
(3.5)

Recall that the denominator is the sum of the sensitivity and specificity - 1,
which we have previously defined as Youden’s $J$ statistic. In this case, when
we have a perfect test such that $a=b=1$, we would obtain $\phi$ =
$\hat{\phi}$. Because this measurement is a population estimate whose $a$ and
$b$ parameters are not equal to 1 and by its nature, refers to a sample
proportion of the population, we can apply 95$\%$ confidence intervals to this
proportion estimate $\phi$ using Z-values and assuming a normally distributed
estimate of $\phi$:

$\phi[95\%CI]=\phi\pm 1.96\sqrt{\frac{\phi(1-\phi)}{n}}$ (3.6)

The Rogan–Gladen method is a frequentist approach and, by assuming the
prevalence as a fixed parameter, is based on classical statistical principles.
This estimator shows two important limitations. First, if the apparent
prevalence is lower than the false positive rate $(1-b)$, the Rogan–Gladen
estimate will become negative, which is of course meaningless as a value of
prevalence [36]. Likewise, it is not a given that the test’s $a$ and $b$
parameters are known with certainty. As such, they may be subject to
uncertainty in of in themselves [37]. Rogan-Gladen provide an estimate for the
error in the calculation of prevalence by accounting for changes in the
presumed sensitivity and specificity parameters. However, in order to address
this uncertainty more optimally, we make use of bayesian methods.

###  3.6 Revisiting Bayes’ Theorem

Recall the general statement for Bayes’ theorem [38]:

$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$ (3.7)

where:

$P(A|B)$ is the posterior probability of event $A$ given that event $B$ has
occurred, $P(B|A)$ is the likelihood of event $B$ given that event $A$ has
occurred, $P(A)$ is the prior probability of event $A$, and $P(B)$ is the
probability of event $B$, or marginal probability. Now, the equation above
relays the form of the theorem that applies to point estimates and fixed
parameters. When the point estimates are replaced by variables, these are
interpreted as probabilities and carry uncertainty about their distributions,
so the theorem becomes [35]:

$f(\theta|x)=\frac{f(x|\theta)\cdot f(\theta)}{\int f(x|\theta)\cdot
f(\theta)\,d\theta}\leftrightarrow f(\theta|x)\propto f(x|\theta)\cdot
f(\theta)$ (3.8)

where:

$f(\theta|x)$ is the posterior distribution of the parameter $\theta$ given
the observed data $x$, $f(x|\theta)$ is the likelihood function, describing
the probability of observing the data $x$ given the parameter $\theta$,
$f(\theta)$ is the prior distribution, representing the initial belief or
knowledge about the parameter $\theta$. The denominator is the marginal
distribution and involves integrating over all possible values of $\theta$ to
ensure that the posterior distribution is properly normalized.

In essence, as we have seen repeatedly, the core of Bayesian analysis lies in
[3]:

Posterior $\propto$ prior x likelihood

Put generally, the goal of bayesian statistics is to represent prior
uncertainty about model parameters with a probability distribution and to
update this prior uncertainty with current data to produce a posterior
probability distribution for the parameter that contains less uncertainty [39]
\- thus bringing us closer to the truth .

###  3.7 Modelling an unknown pretest probability

The pretest probability, as the name suggests, is a probability parameter
ranging from 0 to 1. When this probability is unknown, it can be modelled
using a probability distribution function and its conjugate prior [40]. This
transforms the nature of the probability from a fixed parameter in the
binomial distribution to treating the probability as a random variable with a
distribution of its own. One such distribution, often termed the “probability
of probabilities,” is known as the beta distribution [41].

Consider a standard coin. The binomial distribution deals with a probabilistic
problem, answering: given the chance of tossing heads is $p$, what is the
probability of getting $k$ heads in $n$ tosses? This assumes assigning an
exact value to the probability of heads and, consequently, to “not heads” (or
tails). In contrast, the beta distribution approaches the question from a
different perspective, addressing a more statistical problem: given that $k$
heads appeared in $n$ tosses, what would the probability distribution for $p$
originally need to be to yield those results? The beta distribution examines
an outcome and, based on that outcome, infers what the original distribution
of $p$ must have been to obtain those results. It is therefore an excellent
tool to extrapolate the original prior or pretest probability given some
proportion of positive results [42]. As mentioned previously, in classical
statistics, the pretest probability is often regarded as a fixed parameter,
assuming a specific value. However, in bayesian statistics, the pretest
probability can be treated as a random variable with a probability
distribution itself. The beta distribution is well-suited for this purpose, as
it is defined on the interval $[0,1]$ and serves as the conjugate prior for
the binomial distribution [42].

###  3.8 Binomial distribution, beta distribution and conjugate priors

####  3.8.1 Estimation of a prevalence distribution

The binomial distribution is a discrete probability distribution that
describes the number of successes in a fixed number of independent Bernoulli
trials, where each trial has only two possible outcomes. See the the toss of a
coin example above as reference. The outcomes are often denoted as “success”
and “failure” and the probability of success is denoted by $p$. The
probability function of the binomial distribution is given by:

$f(X=t)={n\choose t}p^{t}(1-p)^{n-t}$

For bayesian analyses of binary classifiers such as for screening systems, the
binomial distribution represents the likelihood function and takes the form
[43]:

$f(t|\phi)={n\choose t}\phi^{t}(1-\phi)^{n-t}$

where ${n\choose t}$ is the binomial coefficient, representing the number of
configurations $t$ can be chosen out of $n$, $n$ is the number of trials or
tests undertaken, $t$ is the number of successes or positive tests, $\phi^{t}$
is the probability of observing $t$ positive results given a pretest
probability $\phi$, and $(1-\phi)^{n-t}$ is the probability of observing a
negative result given a pretest probability $\phi$.

On the other hand, the beta distribution is a continuous probability
distribution defined on the interval $[0,1]$. It is a versatile and flexible
distribution commonly employed in bayesian statistics to model random
variables representing proportions or probabilities, such as the pretest
probability or prevalence of disease. The beta distribution is parametrized by
two shape parameters, denoted as $\alpha$ and $\beta$, both of which must be
positive [44]. These parameters influence the shape of the distribution,
determining its skewness and tail behaviour. For bayesian analyses of binary
classifiers such as for screening systems, the beta distribution represents
the conjugate prior function and takes the form:

$f(\phi|\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\cdot\Gamma(\beta)}\cdot\phi^{\alpha-1}\cdot(1-\phi)^{\beta-1},$

where $\phi$ is the random variable, and $\Gamma$ represents the gamma
function.

It thus follows:

$f(\phi|\alpha,\beta)\propto\phi^{\alpha-1}(1-\phi)^{\beta-1}$

Notably, in bayesian inference, the beta distribution serves as the conjugate
prior probability distribution for the Bernoulli, negative binomial, and
geometric distributions as well [45]. This implies that starting with a beta
distribution as a prior and updating it with binomial data will result in a
posterior distribution that is also a beta distribution. Therefore, the beta
distribution, with parameters $(\alpha,\beta)$, proves to be an excellent tool
for estimating pretest probabilities in the screening context, given that it
inherently involves a binomial nature [46]. It follows that during the
modelling phase, we already know the posterior probability will also be a beta
distribution [39]. As a result, after carrying out more experiments, one can
compute the posterior by simply adding the number of successes and failures to
the existing parameters $(\alpha,\beta)$, instead of multiplying the
likelihood with the prior distribution. This significantly simplifies the
computation [47]. The mean and variance of the beta distribution are
contingent on the values of $\alpha$ and $\beta$, allowing practitioners to
tailor the distribution of the data being modeled. In this case, the first
moment or expected mean value of the beta distribution is:

$E[\phi]=\frac{\alpha}{\alpha+\beta}$ (3.9)

The second moment is given by:

$E[\phi^{2}]=\frac{\alpha(\alpha+1)}{(\alpha+\beta)(\alpha+\beta+1)}$ (3.10)

From these two parameters, we can calculate the variance of the beta
distribution:

$Var(\phi)=E[\phi^{2}]-(E[\phi])^{2}=\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$
(3.11)

Therefore, the standard deviation of $B(\phi;\alpha,\beta)$ becomes:

$\sigma(B(\phi;\alpha,\beta))=\sqrt{\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}}$

Following equation 16.4, we thus obtain:

$f(\phi)=f(\phi|t)\propto\phi^{t}(1-\phi)^{n-t}\cdot\phi^{\alpha-1}(1-\phi)^{\beta-1}=\phi^{\alpha^{\prime}-1}(1-\phi)^{\beta^{\prime}-1}$

The updated parameters of the beta distribution become
$\alpha^{\prime}=\alpha+t$ and $\beta^{\prime}=\beta+n-t$. This posterior
distribution provides a probability distribution for the true pretest
probability $\phi$ given the observed number of positive tests [47].
Therefore, the beta distribution is explicitly involved in representing the
prior belief, and it undergoes an update following the observation of test
results in order to narrow its uncertainty.

####  3.8.2 Estimation of prevalence with an imperfect test of known
parameters

Jonathan Baxter [47] proposes a method for estimating prevalence with an
imperfect test having sensitivity and specificity parameters each less than 1.
Assume an imperfect test is applied to a random sample of $n$ subjects, where
$t$ obtain a positive result. We therefore obtain an apparent prevalence
$\hat{\phi}$, as in Marchevsky, Rogan, and Gladen’s method, equal to the
probability that a test applied to a random individual from such a population
yields a positive result.

Once again, $\hat{\phi}$ is given by:

$\hat{\phi}=\frac{t}{n}$

Suppose we know the test’s sensitivity ($a$) and specificity ($b$) parameters.
Denote the population prevalence by $\phi$. The probability $\hat{\phi}$ of a
positive test is the probability of a positive test given the subject has the
disease, plus the probability of a positive test given the subject is disease-
free:

$\hat{\phi}=a\phi+(1-b)(1-\phi)=(1-b)+J\phi$

The probability of retrieving $t$ positive tests out of $n$ subjects
undergoing testing follows a binomial distribution with the probability
parameter $\hat{\phi}$:

$f(t|\phi,n,a,b)=\binom{n}{t}\hat{\phi}^{t}(1-\hat{\phi})^{n-t}$

Following Bayes’ rule, $f(\phi|t,n,a,b)$ is the posterior probability of
$\phi$ given the observed data:

$f(\phi|t,n,a,b)=\frac{f(t|\phi,n,a,b)f(\phi)}{f(t|n,a,b)}$

where $f(\phi)$ is the prior probability of $\phi$, or the prevalence of the
disease, and:

$f(t|n,a,b)=\int_{0}^{1}f(t|\phi,n,a,b)f(\phi)\,d\phi$

If we pick a uniform prior for $\phi$ such that $f(\phi)\propto 1$, and apply
$d\phi=\frac{d\hat{\phi}}{J}$, we obtain:

$\displaystyle f(t|n,a,b)$
$\displaystyle=\frac{1}{J}\binom{n}{t}\int_{1-b}^{a}\hat{\phi}^{t}(1-\hat{\phi})^{n-t}\,d\hat{\phi}$
$\displaystyle=\frac{1}{J}\binom{n}{t}\left[\int_{0}^{a}\hat{\phi}^{t}(1-\hat{\phi})^{n-t}\,d\hat{\phi}-\int_{0}^{1-b}\hat{\phi}^{t}(1-\hat{\phi})^{n-t}\,d\hat{\phi}\right]$
$\displaystyle=\frac{1}{J}\binom{n}{t}[B(a;t+1,n-t+1)-B(1-b;t+1,n-t+1)]$
$\displaystyle=\frac{1}{J}\binom{n}{t}[B(a)-B(1-b)]$

Recall that in Bayesian statistics, a uniform prior (also known as a non-
informative or flat prior) is a type of prior probability distribution that
assigns equal probability to all possible values within a specified range. It
reflects a lack of prior knowledge or bias toward any particular value in that
range, where $f(\phi)\propto 1$ [48]. The absence of $f(\phi)$ beyond that
point means that the likelihood term dominates the posterior, and the specific
form of the prior is no longer explicitly present in the final expression.
Therefore, this yields:

$f(\phi|t,n,a,b)=\left[\frac{J}{B(a)-B(1-b)}\right][(1-b)+\phi(J)]^{t}[1-(1-b)-\phi(J)]^{n-t}$

Or, again, as above:

$f(\phi|t,n,a,b)\propto[(1-b)+\phi(J)]^{t}[b-\phi(J)]^{n-t}$

A summary of the above process can be seen in the flow diagram below (Figure
2):

Uniform prior $f(\phi)$Likelihood function$f(t|\phi,n,a,b)$Posterior
distribution $f(\phi|t,n,a,b)$Bayes’ RuleNormalization

Figure 2. Flow diagram representing the Bayesian process to determine the
average prevalence $\phi$.

###  3.9 Method using a test of unknown parameters

Baxter [47] as well as Larremore et al. [37] propose similar techniques for
scenarios in which the sensitivity and specificity parameters are unknown. If
$a$ and $b$ are unknown, $\phi$ can still be estimated, albeit with reduced
precision, using a Bayesian approach. This requires us to specify a prior
distribution for $\phi$ and informative prior distributions for $a$ and $b$.
The goal is to estimate the population $\phi$, the test’s sensitivity $a$, and
the test’s specificity $b$, as determined by the data collected ($x$) and the
validation data ($v$) used. Recall the relationship at the core of Bayes’
theorem:

$\text{f}(\theta|x)\propto\text{f}(x|\theta)\cdot\text{f}(\theta)$

We can modify the above relationship by accounting for the unknown variables:

$\text{f}(\phi,a,b|x,v)\propto\text{f}(x|\phi,a,b)\cdot\text{f}(v|a,b)$

We break down each equation to obtain the same equation as though the
parameters were known, but add a new term to account for the uncertainty in
the sensitivity and specificity. Suppose the test has been validated with
$t_{b}$ false positives out of $n_{b}$ known negative samples, and $t_{a}$
true positives out of $n_{a}$ known positive samples. Assuming a beta prior on
$b$ with parameters $\alpha_{b}$, $\beta_{b}$, the posterior density on $b$
given the validation data is proportional to a beta density with parameters
$t_{b}+\alpha_{b}$, $n_{b}-t_{b}+\beta_{b}$. Reducing redundancy in the
notation, we write $B_{b}(b)$ for this density and similarly $B_{a}(a)$ for
the corresponding density on $a$. Let $B(1-b+J\phi)$ denote the original
density at $(1-b)+J\phi$ of the beta distribution with parameters $t+1$,
$n-t+1$.

$\text{f}(\phi|t,n,t_{a},n_{a},t_{b},n_{b})\propto\int_{0}^{1}\int_{0}^{a}\frac{1}{B(a)-B(1-b)}B((1-b)+J\phi)\cdot
B_{b}(b)\cdot B_{a}(a)\cdot da\cdot db$ (3.12)

Thus far we have seen Richardson’s list - which estimates prevalence from data
collected in previous studies, and Marchevky, Rogan and Gladen’s method which
utilises the number of positive tests in large cohorts undergoing screening
and surmises a point estimate from the total number of positive results and
the parameters of the test. We likewise saw the use of binomial distributions
and conjugate priors which estimate the prevalence and pretest probability as
a probability distribution instead of as a point estimate, in both cases where
the test’s parameters are either known or unknown. This method too requires a
large number $n$ of tests conducted to best estimate the average prevalence of
disease. While all of these methods are interesting - they share one
fundamental flaw - they are retrospective in nature; that is, they estimate an
average prevalence in a population a posteriori, after data has been
collected. It is therefore of no real use to a clinician who is faced with an
individual patient in whom a pretest probability of disease exists, which may
well vary largely from the average estimated prevalence. Indeed, these methods
are of more particular use in epidemiology and public health domains, but they
are less applicable to the individual patient in a clinical encounter. We
therefore need a technique which estimates - a priori \- the pretest
probability of disease.

###  3.10 Logit function and McGee’s heuristic

Given the differences between odds and probabilities, and the fact that
technically $\kappa$’s measure a change in odds - we need to find a way that
allows the clinician to estimate the change in post-test probability as a
function of the $\kappa$ without having to make cumbersome calculations. In
order to do this, given odds O and probability $p$:

O = $\frac{p}{1-p}$

In passing, by algebraic equivalence, it is also true therefore to say:

ln(O) = ln$\left(\frac{p}{1-p}\right)$

The function ln$\left(\frac{p}{1-p}\right)$ is known as the logit or log-odds
function, and it forms the basis of the logistic regression [15]. However, in
the equations above, we can figure out how probabilities change odds, when in
fact we’re interested in the opposite - how odds and $\kappa$’s change the
probabilities. Re-arranging the equation above we obtain:

O = $\frac{p}{1-p}$ $\leftrightarrow$ p = $\frac{O}{1+O}$

And since the odds are just the exponential of the log-odds, the log-odds can
also be used to obtain probability:

$p$ = $\frac{exp(ln(O))}{1+exp(ln(O))}$ = $\frac{e^{ln(O)}}{1+e^{ln(O)}}$ =
$\frac{1}{1+e^{-ln(O)}}$

We can consider the ln(O) as the independent variable of a plot to assess how
the post-test probability changes accordingly (Figure 3.):

$-15$$-10$$-5$$0$$5$$10$$15$$0$$0.2$$0.4$$0.6$$0.8$$1$$ln(O)$$\rho(\phi)$

Figure 3. The Logit Function.

A probability of 0.5 corresponds to a logit of 0. Negative logit values
indicate probabilities smaller than 0.5, and positive logits indicate
probabilities greater than 0.5. The relationship is symmetrical: logits of
-0.2 and 0.2 correspond to probabilities of 0.45 and 0.55, respectively. Note
that the absolute distance to 0.5 is identical for both probabilities. This
symmetry corresponds well to to the spectrum of the likelihood ratio - where
below 1, the negative predictice value $\Upsilon$ predominates, and above,
$\kappa$ predominates. Indeed, for probabilities of 0.5 the odds equal 1, and
the log-odds equal 0.

In order to provide a heuristic, or shortcut approach, we can study the
geometry of the logit function. In red, a linear approximation of the logit
function is observed which can act as a heuristic to determine the percentage
by which the post-test probability or in this case, the positive predictive
value, $\rho(\phi)$, changes in the presence of a test whose $\kappa$ or
$\Upsilon$ is known. This linear equation of the form y=mx+b is best
approximated between probabilities of 10 and 90 $\%$ and is henceforth
referred to as McGee’s heuristic, after the author who first described it
[15]:

$\Delta\rho(\phi)\approx$ 0.22 $\cdot$ ln($\kappa$)+ 0.5

where $\Delta\rho(\phi)$ represents the difference between the pretest and
post-test probability. The 0.22 is roughly equivalent to 4.54, but we can
round up to 5 for simplicity’s sake. As such, the rate of change $m$ of the
linear approximation is:

Most critically, notice how this change is independent of the pretest
probability - that is, whatever the starting pretest probability is the linear
approximation provides the value it would increase by in the context of a
positive test. Otherwise stated:

$\boxed{\rho(\phi)\approx\phi+\frac{ln(\kappa)}{5}\leftrightarrow{\rho(\phi)-\phi\approx\frac{ln(\kappa)}{5}}}$

With this equation, we can see why a $\kappa$ $>$ 10 is used as a standard for
a “good” test or as having “strong” evidence for the presence of disease. Let
us assume that in an ideal situation, we want to be 100$\%$ certain of the
presence of disease following a positive test, such that $\rho(\phi)$ = 1.
What kind of $\kappa$ should such a test have? We can use the above equation
by inputting $\rho(\phi)$ = 1, to obtain:

1 - $\phi\approx\frac{ln(\kappa)}{5}$

We obtain an equation with two variables, $\phi$ and $\kappa$. The table below
(Table 3.) showcases what $\phi$ should be as $\kappa$ improves so as to
obtain $\rho(\phi)$ = 1:

Table 3. $\phi$ as a function of $\kappa$
---
$\kappa$ | $\frac{ln(\kappa)}{5}$ | $\phi_{1.0}$ | $\phi_{0.5}$
1 | 0.00 | 1.00 | 0.50
2 | 0.15 | 0.85 | 0.35
3 | 0.24 | 0.76 | 0.26
4 | 0.30 | 0.70 | 0.20
5 | 0.35 | 0.65 | 0.15
6 | 0.39 | 0.61 | 0.11
7 | 0.43 | 0.57 | 0.07
8 | 0.46 | 0.54 | 0.04
9 | 0.48 | 0.52 | 0.02
10 | 0.51 | 0.49 | 0.00

Table 3. $\phi$ needed to obtain $\rho(\phi)$ of 1 or 0.50 as a function of
$\kappa$

If we define $\rho(\phi)$ as the probability of disease following a positive
screening test, then by definition, when this probability exceeds 50$\%$, the
individual tested is more likely than not to have the disease in question.
Let’s assume, for argument’s sake, that our prior is entirely uninformative,
implying $\phi$ $\approx$ 0\. While such a scenario is unlikely as no known
condition has prevalence = 0 (except but for say sex-specific conditions, such
as pregnancy in a male or prostate cancer in a female), it nevertheless is
important to contemplate such a situation. In such a scenario, with $\kappa$
$>$ 10, we find that $\rho(\phi)$ $\approx$ 0.50. In other words, solely
considering the test’s performance, a screening test with $\kappa$ $>$ 10
indicates that, in the context of a positive result, it is more likely than
not that the individual indeed has the condition in question. As depicted in
the table, a $\kappa$ value exceeding 10 is the minimum required to ensure
that, irrespective of the initial pretest/prior probability, the test is more
likely than not a true positive, implying that the individual is more likely
than not afflicted with the disease under consideration. This fact is a simple
consequence of the geometry of the logic function and may explain the reason
why $\kappa$ $>$ 10 is considered to reflect strong evidence for the presence
of disease - it moves a patient from less than 50$\%$ risk to beyond 50$\%$.
We can show this is the case algebraically as follows. Taking the limit as
$\phi$ goes to 0, for a $\rho(\phi)$ = 0.5, we obtain:

0.50 - $0\approx\frac{ln(\kappa)}{4.54}$

Isolating $ln(\kappa)$:

$ln(\kappa)\approx{4.54*0.5}$

Isolating $\kappa$ and rounding up:

$\kappa\approx e^{2.27}\approx 9.67\approx 10$

From the screening equation we could determine how each variable influences
the other to attain a $\rho(\phi)$ = 0.50, as follows (Figure 4.):

$2$$4$$6$$8$$10$$0.2$$0.4$$0.6$$0.8$$1$$\kappa$$\phi$$\phi\approx
0.5-\frac{\ln(\kappa)}{5}$

Figure 4. Combination of $\phi$ and $\kappa$ to attain $\rho(\phi)$= 0.50

In essence, for a given $\phi$ value, we determine the corresponding $\kappa$
on the graph, the combination of which yields $\rho(\phi)$= 0.50 - the tipping
point beyond which a positive test is more likely to be a true positive.
Heuristics like these can be very helpful to the clinical decision-making
process the clinician undergoes at the bedside. Of note, this heuristic works
best for pretest probabilities between 10$\%$ and 90 $\%$. As McGee states
clearly in his paper “Although this method is inaccurate for pretest
probabilities less than 10$\%$. or greater than 90$\%$. , this is not a
disadvantage, because these polar extremes of probability indicate diagnostic
certainty for most clinical problems, making it unnecessary to order further
tests (and apply additional LRs)” [15].

###  3.11 Heuristic approximation of a priori pretest probabilities

At the beginning of this section, we alluded to the use of individual risk
factors $\theta_{i}$ to determine the overall a priori risk of disease in an
individual, $\phi$. In essence, the pretest probability can be viewed as the
cumulative effect of individual components $\theta_{i}$ (such as signs,
symptoms, risk factors, etc…) whose sum leads to $\phi$:

$\phi=\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}....\theta_{i}$

In this section, we’ll explore intuitive ways that transform these risk
factors into a probability of disease. In particular, we’ll make use of the
logistic regression and the likelihood ratio. Recall the relationship between
$\kappa$, $\rho(\phi)$ and $\phi$:

$\rho(\phi)=\frac{a\phi}{a\phi+(1-b)(1-\phi)}=\frac{\kappa\phi}{1+({\kappa}-1)\phi}$

In previous work, we used the logit function as dictated by McGee’s heuristic
[15], to evaluate an approximation between $\kappa$, $\rho(\phi)$ and $\phi$
for pretest probability values ranging between 0.10 and 0.90:

$\rho(\phi)\approx\phi+\frac{ln(\kappa)}{5}\leftrightarrow{\phi\approx\rho(\phi)-\frac{ln(\kappa)}{5}}$

Let us denote the difference between $\rho(\phi)$ and $\phi$ as $\Delta p$,
such that:

$\Delta p\approx\frac{ln(\kappa)}{5}$

Bridging the pretest to the posttest probability, $\Delta p$ can be thought of
as the gain in pretest probability following a positive screening test. Now,
since $\kappa$ is a function of the sensitivity $a$ and the specificity $b$,
it follows that it is prevalence independent. We have thus associated a
prevalence-independent measure, as is the likelihood ratio $\kappa$ \- to a
probability of disease. Since $\phi$ and $\rho(\phi)$ are probabilities, they
each can hold values between 0 and 1, representing minimal and maximal bounds
for both $\phi$ and $\rho(\phi)$ can take. Let’s look at both scenarios a
little closer.

####  3.11.1 Maximal bound of $\phi$

On the one hand, intuitively, we know that $\phi$ is maximally bound at 1 - by
definition. That is, if $\phi$ = 1 then given the invariant points on the
probability square plane $S$, no matter the level of $\kappa$ applied,
$\rho(\phi)$ = 1. That said, a theoretical $\phi$ = 1 would obviate the need
for testing in the first place.

$\rho(\phi)=\frac{\kappa(1)}{1+({\kappa}-1)(1)}$ = $\frac{\kappa}{\kappa}=1$

####  3.11.2 Minimal bound of $\phi$

If we assume that the pretest probability $\phi$ is such that when subjected
to a positive screening test $\rho(\phi)$ is maximized to 1, then it would
follow that $\phi$ would increase by:

$\phi\approx 1-\frac{ln(\kappa)}{5}$

Recall that when subject to orthogonal testing - the use of different “tests”
with individual sensitivity and specificity parameters (a thus $\kappa$, too)
- $\rho(\phi)$ requires multiplication of the individual parameters [24]. As
such, the gain in pretest probability $\phi$ is given by:

$\phi\approx
1-\frac{1}{5}{ln\left(\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right)}$

Therefore, given that such equation represents the gain in $\phi$, the the
underlying pretest probability must be its complement, such that:

$\phi\approx
1-\left[1-\frac{1}{5}{ln\left(\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right)}\right]=\frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right]}$

We say that this is a minimum in the sense that for a given $\kappa$, this
represent the minimal pretest probability needed to obtain a $\rho(\phi)$ = 1.
However, it is possible that for the same $\kappa$ a higher $\phi$ would also
yield a $\rho(\phi)$ = 1. We can illustrate graphically (Figure 5.) how these
bounds change as a function of
$\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}$, as follows:

$0$$20$$40$$60$$80$$100$$0$$0.2$$0.4$$0.6$$0.8$$1$$\displaystyle\prod\kappa_{\theta}$$\phi$

Figure 5. Range of minimal and maximal potential values of $\phi$ based on the
presence of signs, symptoms, or risk factors, each with an individual
$\kappa$.

These values are conservative in the sense that they represent the widest
possible intervals for the calculation of $\phi$ given a series of
$\kappa_{\theta}$. Given the heuristic nature of this technique, it is
probably best to err in the side of caution when estimating pretest
probabilities by selecting the minimal bound as the approximation. This way,
we yield the highest possible impact to the $\kappa$ of the test to bridge the
pretest to the posttest probability. Nevertheless, we could calculate an
average $\phi$ as simply the mean value of both functions, with the interval
min-max range, R$(\bar{\phi})$, as follows (Figure 6.):

$\bar{\phi}\approx\frac{1+\frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right]}}{2}$,
R$(\bar{\phi})$ =
$\left[\frac{1}{5}{ln\left(\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right)},1\right]$

$0$$20$$40$$60$$80$$100$$0$$0.2$$0.4$$0.6$$0.8$$1$$\displaystyle\prod\kappa_{\theta}$$\phi$

Figure 6. Range of minimal and maximal potential values of $\phi$ based on the
presence of signs, symptoms, or risk factors, each with an individual
$\kappa$. The blue dashed line represents the mean value of the pretest
probability $\bar{\phi}$ as a function of $\kappa_{\theta}$ and the black
dashed lines represent the range of potential values for $\phi$.

Though it appears like it, the graph above is misleading in that there is no
asymptote to that equation. In other words, for a sufficient product of
$\kappa_{\theta}$, the pretest probability would go beyond 1. This is not a
real problem since that would imply a $\kappa_{\theta}$ of near 150! Recall we
showed earlier that a combined $\kappa$ of 10 would be sufficient to deem a
test a “good confirmer”, so it would be unnecessary to obtain levels
significantly higher than that to rule-in the presence of disease. Similarly,
the premise of this theory is that the prevalence threshold $\phi_{e}$ may
serve as a good alternate benchmark to rule-in disease for adequate-enough
classification systems. In the figure below, one can see that the intersection
of these two functions happens at a $\kappa$ level of $\kappa_{\theta}$
$\approx$ 4.5 (Figure 7).

$0$$20$$40$$60$$80$$100$$0$$0.2$$0.4$$0.6$$0.8$$1$$\displaystyle\prod\kappa_{\theta}$$\phi$

Figure 7. Intersection of $\phi$ and $\phi_{e}$ as a function of
$\kappa_{\theta}$ .

$\kappa_{\theta}$ | $\bar{\phi}$ | Min | Max
---|---|---|---
1 | 0.50 | 0.00 | 1
2 | 0.57 | 0.14 | 1
3 | 0.61 | 0.22 | 1
4 | 0.64 | 0.28 | 1
5 | 0.66 | 0.32 | 1
6 | 0.68 | 0.36 | 1
7 | 0.69 | 0.39 | 1
8 | 0.71 | 0.42 | 1
9 | 0.72 | 0.44 | 1
10 | 0.73 | 0.46 | 1
20 | 0.80 | 0.60 | 1
30 | 0.84 | 0.68 | 1
40 | 0.87 | 0.74 | 1
50 | 0.89 | 0.78 | 1
60 | 0.91 | 0.82 | 1
70 | 0.92 | 0.85 | 1
80 | 0.94 | 0.88 | 1
90 | 0.95 | 0.90 | 1
100 | 0.96 | 0.92 | 1

Table 4. Mean value of $\phi$ and its associated min-max range as a function
of $\displaystyle\prod\kappa_{\theta}$.

Thus, by accounting for the individual $\theta_{\theta}$, or signs and
symptoms, we can multiply their $\kappa_{\theta}$ to approximate $\phi$
through this equation. Then - we can update this probability using Bayes’
theorem with a screening test to obtain a final probability of disease. As
such:

$\rho(\phi)=\frac{\kappa\phi}{1+({\kappa}-1)\phi}\rightarrow\frac{\kappa\frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right]}}{1+({\kappa}-1)\frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right]}}$

In deriving the equation for $\bar{\phi}$ we made the assumption that we
wanted to obtain a perfect positive predictive value, such that $\rho(\phi)$ =
1. In reality, this is likely not necessary. What would the effect of
considering a different threshold for $\rho(\phi)$ be?

$0$$20$$40$$60$$80$$100$$0$$0.2$$0.4$$0.6$$0.8$$1$$\displaystyle\prod\kappa_{\theta}$$\phi$

Figure 8. Range of minimal and maximal potential values of $\phi$ based on the
presence of signs, symptoms, or risk factors, each with an individual
$\kappa$, assuming the ultimate target would be a $\rho(\phi)$ of 1.00 (red),
0.90 (blue), 0.80 (orange), 0.70 (black).

We see thus in Figure 8. that lowering the ultimate $\rho(\phi)$ threshold
narrows the range of $\bar{\phi}$. Since we seek ultimately to err in the side
of caution using this heuristic, we’re better off using $\rho(\phi)$ = 1 as
the most appropriate approach, as it yields the widest range of values, and
thus the lowest bound of $\bar{\phi}$.

###  3.12 The effect of baseline prevalence on $\kappa_{\theta}$

While the product of individual $\kappa_{\theta}$ is a good way to approximate
the pretest probability of disease - a criticism could be made that the
baseline risk - or prevalence of disease - is not accounted for in this
calculation. The question whether $\kappa_{\theta}$ accounts for the
prevalence of disease is to some degree philosophical and academic. But let us
for argument’s sake assume that prevalence is not accounted for in this
calculation. The effect of accounting for baseline prevalence would be simply
to narrow the range of $\bar{\phi}$ by moving the line upwards. Since we err
in the side of caution by taking the widest possible range - this addition
would not modify the estimates for the purposes of this heuristic. Secondly,
in the presence of a high enough $\kappa_{\theta}$, the impact of baseline
prevalence would likely be negligible for the simple reason that if the
prevalence was high enough we would not need to account for $\kappa_{\theta}$
in the first place, as the individual would likely be at a risk level such
that $\epsilon>$ $\phi_{e}$, where $\epsilon$ is the prevalence of disease.

$\phi\approx\frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_{\theta}\right]}+\epsilon$

As we have thus seen, using the logit function and the rules of logistic
regression allowed us to approximate the impact of the likelihood ratios of
individual signs and symptoms. These are then used to approximate - a priori -
the lower bound of the pretest probability of disease. Clinical confirmation
of these relationships should be undertaken prior to clinical implementation.

## References

  * [1] Vivek Subbiah.  The next generation of evidence-based medicine.  Nature medicine, 29(1):49–58, 2023.
  * [2] Anthony K Akobeng.  Understanding diagnostic tests 2: likelihood ratios, pre-and post-test probabilities and their use in clinical practice.  Acta paediatrica, 96(4):487–491, 2007.
  * [3] Rens van de Schoot, Sarah Depaoli, Ruth King, Bianca Kramer, Kaspar Märtens, Mahlet G Tadesse, Marina Vannucci, Andrew Gelman, Duco Veen, Joukje Willemsen, et al.  Bayesian statistics and modelling.  Nature Reviews Methods Primers, 1(1):1, 2021.
  * [4] Jacques Balayla.  Prevalence threshold ($\phi$ e) and the geometry of screening curves.  Plos one, 15(10):e0240215, 2020.
  * [5] Mitchell A Medow and Catherine R Lucey.  A qualitative approach to bayes’ theorem.  Bmj evidence-based medicine, 16(6):163–167, 2011.
  * [6] Pat Croskerry.  Achieving quality in clinical decision making: cognitive strategies and detection of bias.  Academic emergency medicine, 9(11):1184–1204, 2002.
  * [7] Robert Trevethan.  Sensitivity, specificity, and predictive values: foundations, pliabilities, and pitfalls in research and practice.  Frontiers in public health, 5:307, 2017.
  * [8] Martijn JL Bours.  Bayes’ rule in diagnosis.  Journal of Clinical Epidemiology, 131:158–160, 2021.
  * [9] Penny F Whiting, Clare Davenport, Catherine Jameson, Margaret Burke, Jonathan AC Sterne, Chris Hyde, and Yoav Ben-Shlomo.  How well do health professionals interpret diagnostic information? a systematic review.  BMJ open, 5(7), 2015.
  * [10] Nasser M Gayed and David E Kern.  A formula for estimating pretest probability: evaluation and clinical application.  Journal of general internal medicine, 5:292–297, 1990.
  * [11] Thomas Agoritsas, Delphine S Courvoisier, Christophe Combescure, Marie Deom, and Thomas V Perneger.  Does prevalence matter to physicians in estimating post-test probability of disease? a randomized trial.  Journal of general internal medicine, 26:373–378, 2011.
  * [12] Brian R Jackson.  The dangers of false-positive and false-negative test results: false-positive results as a function of pretest probability.  Clinics in laboratory medicine, 28(2):305–319, 2008.
  * [13] DAVID L Sackett and R BRIAN Haynes.  The architecture of diagnostic research.  Bmj, 324(7336):539–541, 2002.
  * [14] Jacques Balayla.  Bayesian updating and sequential testing: Overcoming inferential limitations of screening tests.  BMC medical informatics and decision making, 22(1):1–8, 2022.
  * [15] Steven McGee.  Simplifying likelihood ratios.  Journal of general internal medicine, 17:647–650, 2002.
  * [16] Jonathan J Deeks and Douglas G Altman.  Diagnostic tests 4: likelihood ratios.  Bmj, 329(7458):168–169, 2004.
  * [17] Charles GB Caraguel and Raphaël Vanderstichel.  The two-step fagan’s nomogram: ad hoc interpretation of a diagnostic test result without calculation.  BMJ Evidence-Based Medicine, 18(4):125–128, 2013.
  * [18] Davide Chicco and Giuseppe Jurman.  A statistical comparison between matthews correlation coefficient (mcc), prevalence threshold, and fowlkes–mallows index.  Journal of Biomedical Informatics, page 104426, 2023.
  * [19] Yasin Memari.  Low incidence rate of covid-19 undermines confidence in estimation of the vaccine efficacy.  arXiv preprint arXiv:2101.10005, 2021.
  * [20] Jacques Balayla.  On the formalism of the screening paradox.  Plos one, 16(9):e0256645, 2021.
  * [21] Saeed Safari, Alireza Baratloo, Mohamed Elfil, and Ahmed Said Negida.  Part 2: Positive and negative predictive values of diagnostic tests.  Archives of Academic Emergency Medicine, 3(3):87–88, 2015.
  * [22] Jacques Balayla.  Information threshold, bayesian inference and decision-making.  arXiv preprint arXiv:2206.02266, 2022.
  * [23] Martin H Reed.  Pretest probability: should we care?  Journal of the American College of Radiology, 10(7):486–487, 2013\.
  * [24] Jacques Balayla.  Derivation of generalized equations for the predictive value of sequential screening tests.  arXiv preprint arXiv:2007.13046, 2020.
  * [25] Philip Sedgwick.  Pitfalls of statistical hypothesis testing: type i and type ii errors.  Bmj, 349, 2014.
  * [26] Jef Van den Ende, Zeno Bisoffi, Hugo Van Puymbroek, Patrick Van der Stuyft, Alfons Van Gompel, Anselm Derese, Lutgarde Lynen, Juan Moreira, and Paul Adriaan Jan Janssen.  Bridging the gap between clinical practice and diagnostic clinical epidemiology: pilot experiences with a didactic model based on a logarithmic scale.  Journal of evaluation in clinical practice, 13(3):374–380, 2007\.
  * [27] Milo A Puhan, Johann Steurer, Lucas M Bachmann, and Gerben ter Riet.  A randomized trial of ways to describe test accuracy: the effect on physicians’ post-test probability estimates.  Annals of internal medicine, 143(3):184–189, 2005.
  * [28] Lori A Boardman and Jeffrey F Peipert.  Screening and diagnostic testing.  Clinical obstetrics and gynecology, 41(2):267–274, 1998.
  * [29] Andrew S Dunn, Kanan V Shridharani, Wendy Lou, Jeffrey Bernstein, and Carol R Horowitz.  Physician–patient discussions of controversial cancer screening tests.  American journal of preventive medicine, 20(2):130–134, 2001.
  * [30] Holli A DeVon, Anne Rosenfeld, Alana D Steffen, and Mohamud Daya.  Sensitivity, specificity, and sex differences in symptoms reported on the 13-item acute coronary syndrome checklist.  Journal of the American Heart Association, 3(2):e000586, 2014.
  * [31] W Scott Richardson.  Where do pretest probabilities come from.  BMJ Evidence-Based Medicine, 4(3):68, 1999.
  * [32] M Granger Morgan.  Use (and abuse) of expert elicitation in support of decision making for public policy.  Proceedings of the National academy of Sciences, 111(20):7176–7184, 2014.
  * [33] Walter J Rogan and Beth Gladen.  Estimating prevalence from the results of a screening test.  American journal of epidemiology, 107(1):71–76, 1978.
  * [34] N Marchevsky.  Errors in prevalence estimates in population studies. a practical method for calculating real prevalence.  Zoonosis, 16:98–109, 1974.
  * [35] Peter J Diggle.  Estimating prevalence using an imperfect test.  Epidemiology Research International, 2011, 2011.
  * [36] Patrick Pithua, Nathaniel S Kollias, et al.  Estimated prevalence of caprine paratuberculosis in boer goat herds in missouri, usa.  Veterinary medicine international, 2012, 2012.
  * [37] Daniel B Larremore, Bailey K Fosdick, Sam Zhang, and Yonatan H Grad.  Jointly modeling prevalence, sensitivity and specificity for optimal sample allocation.  bioRxiv, pages 2020–05, 2020.
  * [38] Karel GM Moons, Gerrit-Anne van Es, Jaap W Deckers, Dik JF Habbema, and Diederick E Grobbee.  Limitations of sensitivity, specificity, likelihood ratio, and bayes’ theorem in assessing diagnostic probabilities: a clinical example.  Epidemiology, 8(1):12–17, 1997.
  * [39] Scott M Lynch.  Basics of bayesian statistics.  In Introduction to applied bayesian statistics and estimation for social scientists, pages 47–75. Springer, 2007.
  * [40] Giulio D’Agostini.  Bayesian inference in processing experimental data: principles and basic applications.  Reports on Progress in Physics, 66(9):1383, 2003.
  * [41] Keegan E Hines.  A primer on bayesian inference for biophysical systems.  Biophysical journal, 108(9):2103–2113, 2015.
  * [42] Tony Hsiu-Hsi Chen, Amy Ming-Fang Yen, Jean Ching-Yuan Fann, Paula Gordon, Sam Li-Sheng Chen, Sherry Yueh-Hsia Chiu, Chen-Yang Hsu, King-Jen Chang, Won-Chul Lee, Khay Guan Yeoh, et al.  Clarifying the debate on population-based screening for breast cancer with mammography: a systematic review of randomized controlled trials on mammography with bayesian meta-analysis and causal model.  Medicine, 96(3), 2017.
  * [43] Claes Enøe, Marios P Georgiadis, and Wesley O Johnson.  Estimation of sensitivity and specificity of diagnostic tests and disease prevalence when the true disease state is unknown.  Preventive veterinary medicine, 45(1-2):61–81, 2000.
  * [44] Ricardo A Olea.  On the use of the beta distribution in probabilistic resource assessments.  Natural resources research, 20:377–388, 2011.
  * [45] Edson Zangiacomi Martinez and Jorge Alberto Achcar.  Trends in epidemiology in the 21st century: time to adopt bayesian methods.  Cadernos de Saúde Pública, 30:703–714, 2014.
  * [46] Angus R Cameron and F Chris Baldock.  A new probability formula for surveys to substantiate freedom from disease.  Preventive veterinary medicine, 34(1):1–17, 1998.
  * [47] Jonathan Baxter.  Bayesian beta-binomial prevalence estimation using an imperfect test.  arXiv preprint arXiv:2009.05446, 2020.
  * [48] Stefan Van Dongen.  Prior specification in bayesian statistics: three cautionary tales.  Journal of theoretical biology, 242(1):90–100, 2006.


##  Appendix C General case of regularisation: $g$ is inhomogeneous

In this section, we briefly describe the method for solving semi-blind inverse
problems in which the regularisation term $g$ is in-homogeneous, making the
normalisation constant (5) computationally and analytically intractable due to
the high dimensional integral on ${\mathbb{R}}^{d}$. In the Bayesian
framework, several works have been done to address this difficulty by
replacing (5) with an approximation such as variational approximations [14],
pseudo-likelihood approximations [55] and Monte Carlo approximations [71]. In
a manner akin to [71], we approximate the normalisation constant by driving a
proximal Markov kernel $(\bar{X}_{k})_{k\in{\mathbb{N}}}$ associated with the
MYULA algorithm and targeting the prior distribution $p(x|\theta)$. The Markov
kernel starting from $\bar{X}_{0}\in{\mathbb{R}}^{d}$ is given by the
following recursion,

(63)
$\bar{X}_{k+1}=(1-\frac{\gamma^{\prime}}{\lambda^{\prime}})\bar{X}_{k}+\frac{\gamma^{\prime}}{\lambda}\text{prox}^{\lambda^{\prime}}_{\theta
g}(\bar{X}_{k})+\sqrt{2\gamma^{\prime}}Z_{k+1},$

where $\gamma^{\prime}>0$ and $\lambda^{\prime}>0$ are the step-size and
smoothing parameters respectively.
From Algorithm 1, we obtain Algorithm 2 with two Markov kernels
$(X_{k})_{k\in{\mathbb{N}}}$, $(\bar{X}_{k})_{k\in{\mathbb{N}}}$ targeting the
posterior $p(x|y,\theta,\alpha,\sigma^{2})$ and prior $p(x|\theta)$
distribution respectively.

1: Initialization: $\\{\theta_{0},\alpha_{0},\sigma^{2}_{0},X_{0}^{0}\\}$,
$\Theta_{\theta},\Theta_{\alpha},\Theta_{\sigma^{2}}$, kernel parameters
$\gamma,\lambda$, iteration $N$.

2: for $n=0:N-1$ do

3: if $n>0$ then

4: set $X_{0}^{n}=X_{m_{n}-1}^{n-1}$

5: set $\bar{X}_{0}^{n}=\bar{X}_{m_{n}-1}^{n-1}$

6: end if

7: for $k=0:m_{n}-1$ do

8:
$X_{k+1}^{n}=(1-\dfrac{\gamma}{\lambda})X_{k}-\gamma\nabla_{x}f^{y}_{\alpha_{n},\sigma_{n}^{2}}(X_{k})+\dfrac{\gamma}{\lambda}\text{prox}^{\lambda}_{\theta_{n}g}(X_{k})+\sqrt{2\gamma}Z_{k+1}$

9:
$\bar{X}_{k+1}^{n}=(1-\dfrac{\gamma^{\prime}}{\lambda^{\prime}})\bar{X}_{k}+\dfrac{\gamma^{\prime}}{\lambda^{\prime}}\text{prox}^{\lambda^{\prime}}_{\theta_{n}g}(\bar{X}_{k})+\sqrt{2\gamma^{\prime}}Z_{k+1}$

10: end for

11:
$\theta_{n+1}=\Pi_{\Theta_{\theta}}\left[\theta_{n}+\frac{\delta_{n+1}}{m_{n}}\sum_{k=1}^{m_{n}}\left\\{g(\bar{X}_{k}^{n})-g(X_{k}^{n})\right\\}\right]$

12:
$\alpha_{n+1}=\Pi_{\Theta_{\alpha}}\left[\alpha_{n}-\frac{\delta_{n+1}}{m_{n}}\sum_{k=1}^{m_{n}}\nabla_{\alpha}f^{y}_{\alpha_{n},\sigma_{n}^{2}}(X_{k}^{n})\right]$

13:
$\sigma^{2}_{n+1}=\Pi_{\Theta_{\sigma^{2}}}\left[\sigma^{2}_{n}-\frac{\delta_{n+1}}{m_{n}}\left(\sum_{k=1}^{m_{n}}\nabla_{\sigma^{2}}f^{y}_{\alpha_{n},\sigma_{n}^{2}}(X_{k}^{n})+\dfrac{d}{2\sigma_{n}^{2}}\right)\right]$

14: end for

15: $\bar{\theta}_{N}$, $\bar{\alpha}_{N}$ and $\bar{\sigma}^{2}_{N}$ are
evaluated using (25), (26) and (27) respectively.

16:
$\bar{x}_{MAP}=\operatorname*{argmax}_{x\in{\mathbb{R}}^{d}}p(x|y,\bar{\theta},\bar{\alpha},\bar{\sigma}^{2})$

Algorithm 2 SAPG Algorithm

##  Appendix D Robustness of the method to hyper parameters

This section extends Section 3.6 with more details on setting the parameters.
Additionally, we illustrate the robustness of the proposed method w.r.t.
setting the parameters of Algorithm 1.

#####  $\Theta_{\alpha},\Theta_{\theta}$ and $\Theta_{\sigma^{2}}$

The admissible sets $\Theta_{\theta}$ and $\Theta_{\alpha}$ for $\theta$ and
$\alpha$ are chosen base on some a priori knowledge on these parameters. For
the Gaussian observational model, where
$\alpha=\left(\alpha_{h},\alpha_{v}\right)$ with $\alpha_{h}$ and $\alpha_{v}$
representing the horizontal and vertical bandwidths of the Gaussian kernel,
respectively, we consider their inverse for numerical stability within the
method. Consequently, we establish
$\Theta_{\alpha_{h}}=\Theta_{\alpha_{v}}=[0.1,1]$. The reason for setting
these inverse bandwidths below $1$ is that a Gaussian blur kernel with
$\alpha_{h}>1$ and $\alpha_{v}>1$ essentially approximates the identity blur.
This implies that the kernel will not significantly degrade the image. We
adopted a similar approach to define $\Theta_{\alpha}$ for the Laplace and
Moffat observational models.

Regarding the regularisation parameter $\theta$, we observe that when
$\theta>1$, the resulting image tends to be overly smoothed. On the contrary,
when $\theta$ is close to $0$, the degree of regularisation introduced to
constrain the solution is minimal. Given these observations, we establish
$\Theta_{\theta}=[10^{-3},1]$ in all of our experiments. Finally, to define
the admissible set $\Theta_{\sigma^{2}}$, we assume that the actual value of
BSNR falls within the range of $15$ dB and $45$ dB. This information on the
BSNR value allows us to calculate $\sigma^{2}_{min}$ and $\sigma^{2}_{max}$ by
rearranging (40). More precisely, we use the approximation
$\|y-Hx\|_{2}^{2}\approx d\sigma^{2}$ which holds when $d=\textrm{dim}(x)$ is
large, and obtain the following.

(64)
$\text{BSNR}(y,x)=-10\log_{10}\left(\frac{d\sigma^{2}}{||Hx||^{2}}\right)\,,$

and therefore to achieve BSNR values of approximately $15$ dB and $45$ dB,
respectively we set

$\sigma^{2}_{min}=\dfrac{||Hx||^{2}_{2}}{\sqrt{d10^{15/10}}}\,,\quad\text{and
}\quad\sigma^{2}_{max}=\dfrac{||Hx||^{2}_{2}}{\sqrt{d10^{45/10}}}\,.$

Recall that these values are merely used to bound the sequence of iterates,
hence the approximation involved in their calculation does not directly affect
the accuracy of our estimates.

##### Initialisation $\alpha_{0}$, $\theta_{0}$ and $\sigma^{2}_{0}$

To demonstrate the robustness of the proposed method w.r.t. the initial values
$\theta_{0},\alpha_{0}$ and $\sigma_{0}^{2}$, we focus on the Gaussian model,
which serves as a representative case, to examine the effect of initialising
$\alpha_{0}=(\alpha_{h,0},\alpha_{v,0})$. Figure 16 $(a)$ and $(b)$ depict the
evolution of the iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ and
$\left(\alpha_{v,n}\right)_{n\in{\mathbb{N}}}$, respectively, with various
initial values of $\alpha_{h,0}$ and $\alpha_{v,0}$. It can be seen that all
sequences exhibit convergence and ultimately stabilise at the same solution.
It should be noted that similar results and conclusions can be derived by
considering both $\theta$ and $\sigma^{2}$. Consequently, this demonstrates
the robustness of the proposed method with respect to the initial parameters
$\alpha_{0},\theta_{0}$ and $\sigma_{0}^{2}$.

$(a)$ Iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ | $(b)$ Iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$
---|---
Figure 16: Gaussian experiment under a noise regime achieving a BSNR value of
$30$ dB. (a) Evolution of iterates
$\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ for different initialisation
$\alpha_{h,0}$. (b) Evolution of the sequence of iterates
$\left(\alpha_{v,n}\right)_{n\in{\mathbb{N}}}$ for different initialisation
$\alpha_{v,0}$.

##### Lipschitz constant $L_{f}$

To assess the robustness of the proposed method w.r.t. the Lipschitz constant,
we assume that we have incorrectly estimated $L_{f}$ by different orders of
magnitude, from $0.1L_{f}$ to $100L_{f}$. Figure 17 (a) and (b) depict the
evolution of the iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ and
$\left(\alpha_{v,n}\right)_{n\in{\mathbb{N}}}$. It can be seen that all
sequences exhibit convergence and ultimately stabilise at the same solution.
This results indicate that the proposed method exhibits reasonable robustness
w.r.t. the estimation of the Lipschitz constant.

$(a)$ Iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ | $(b)$ Iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$
---|---
Figure 17: Gaussian experiment under a noise regime achieving a BSNR value of
$30$ dB. (a) Evolution of the iterates
$\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ for different Lipschitz
constants $L_{f}$. (b) Evolution of the iterates
$\left(\alpha_{v,n}\right)_{n\in{\mathbb{N}}}$ for different Lipschitz
constants $L_{f}$.

##### Step size: $\delta_{n}$

Lastly, we evaluate the robustness of the proposed method w.r.t. the step size
$c_{0}*\delta_{n}$ with $c_{0}>0$. Figure 18 $(a)$ and $(b)$ depict the
evolution of the sequence of iterates
$\left(\alpha_{h,n}\right)_{n\in\mathbb{N}}$ and
$\left(\alpha_{v,n}\right)_{n\in\mathbb{N}}$ for the Gaussian model, with
noise variance set to achieve a BSNR value of $30$ dB. It can be seen that
both the excessively large and very small step sizes significantly impact the
computational efficiency of the method. Consequently, it is recommended to
carefully set the step size to enhance the computational efficiency of the
proposed Algorithm 1.

$(a)$ Iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ | $(b)$ Iterates $\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$
---|---
Figure 18: Gaussian experiment under a noise regime achieving a BSNR value of
$30$ dB. (a) Evolution of the iterates
$\left(\alpha_{h,n}\right)_{n\in{\mathbb{N}}}$ for different step sizes
$\delta_{n}$. (b) Evolution of the iterates
$\left(\alpha_{v,n}\right)_{n\in{\mathbb{N}}}$ for different step sizes
$\delta_{n}$.

##  Appendix E Additional Experiments

###  E.1 Model evaluation for $\theta$ and $\sigma^{2}$

We recall that each test image is associated with a distinct true noise
variance $\sigma^{\star 2}$, and the optimal regularisation parameter
$\theta^{\star}$. To evaluate the precision of the proposed method in
calibrating $\sigma^{2}$ and $\theta$, we consider 8 different test images.
Figure 19 illustrates the estimated values $\bar{\sigma}^{2}$ and
$\bar{theta}$ along with the true values $\sigma^{\star 2}$ and the optimal
values $\theta^{\star}$. It can be seen that our proposed method is robust in
calibrating $\sigma^{2}$ and $\theta$ from observation $y$.

|  |  |
---|---|---|---

|  |  |
---|---|---|---

$(a)$$\bar{\theta}$ and $\theta^{*}$ (20 dB) | $(b)$$\bar{\sigma}^{2}$ and $\sigma^{2*}$ (20 dB) | $(c)$$\bar{\theta}$ and $\theta^{*}$ (30 dB) | $(d)$$\bar{\sigma}^{2}$ and $\sigma^{2*}$ (30 dB)
---|---|---|---

Figure 19: Summary of the experiments involving the Gaussian model (first
row), the Laplace model (second row), and the Moffat model (last row), each
conducted under two noise regimes achieving a BSNR value of $20$ dB and $30$
dB, respectively. $(a)$ and $(b)$ depict the optimal value $\theta^{\star}$
alongside the estimated value $\bar{\theta}$ . $(b)$ and $(d)$ depict the
optimal value $\sigma^{2^{\star}}$ together with the estimated value
$\bar{\sigma}^{2}$.

###  E.2 Non-blind deblurring results

This section presents additional results to illustrate the effectiveness of
our proposed method when the noise variance is set to achieve a BSNR value of
$20$ dB. Figure 20-$(a)$ depicts the true image along with the true blur
kernel positioned in the upper right corner. Figure 20-$(b)$ shows the blurred
image along with its corresponding PSNR value, and finally Figure 20-$(c)$
shows the MAP estimate, including the estimated blur kernel in the upper right
corner, along with the associated PSNR value.

$(a)$ Ground truth | $(b)$ Blurred (20.23 dB) | $(c)$$x_{MAP}$ (27.84 dB)
---|---|---

$(a)$ Ground truth | $(b)$ Blurred (20.25 dB) | $(c)$$x_{MAP}$ (27.94 dB)
---|---|---

$(a)$ Ground truth | $(b)$ Blurred (20.24 dB) | $(c)$$x_{MAP}$ (28.0 dB)
---|---|---

Figure 20: Qualitative results for the noise variance set to achieve a BSNR
value of $20$ dB. From the top raw to the bottom raw, the reconstruction
results obtained with Gaussian, Laplace, and Moffat blur kernels are
presented, respectively. $(a)$ True test image man with the true blur kernel
in the top right corner. $(b)$ Observation $y$. $(c)$ MAP estimate with the
estimated blur kernel in the top right corner.