# Artificial Broadcasts as Galactic Populations: I. A Point Process Formalism
for Extraterrestrial Intelligences and Their Broadcasts

Brian C. Lacki Breakthrough Listen, Department of Astronomy, University of
California Berkeley, Berkeley CA 94720

###### Abstract

Artificial broadcasts from extraterrestrial intelligences (ETIs) are a
hypothetical class of celestial phenomena. Unlike known astrophysical objects,
the societies that generate them may be able to replicate on galactic scales
through interstellar travel. Different galaxies could thus have drastically
different populations, with abundance variations of many orders of magnitude.
I present a probabilistic formalism to treat this shared history, in which
societies and their broadcasts are described by distributions over basic
properties like lifespan and energy released. The framework contains a
hierarchy of objects related by a tree structure. Discrete societies, the
sources of broadcasts, are organized into potentially interstellar
“metasocieties.” The population of each type of object is represented by a
random point process in an abstract parameter hyperspace, a “haystack.” When a
selection like an observation draws a sample, the point process is thinned.
Given assumptions of interchangeability and independence, observables are
modeled with compound Poisson random variables. I present an example of how
selection bias can favor sampling longer-lived objects. I rederive the Drake
Equation for societies in the limit of no expansion. When interstellar
replication is present, however, the mean number of detected broadcasts can
depend quadratically on stellar mass, suggesting a search strategy favoring
large galaxies.

Search for extraterrestrial intelligence — Technosignatures — Astronomical
techniques — Spatial point processes — Poisson distribution

##  1 Introduction

The Search for Extraterrestrial Intelligence (SETI; Tarter 2001; Worden et al.
2017) rests on the premise that the technosignatures of extraterrestrial
intelligences (ETIs) are astrophysical phenomena, arising in and shaped by a
cosmic environment and detectable over cosmic distances. ETIs are not ruptures
in the laws of physics but consequences of them, forming a cosmic population,
however rare they actually are. The properties of this population are
constrained by physical limitations on technology and unknown internal
factors.

Negative SETI results, although leaving plenty of room for ETIs, are at least
strong enough to eliminate many possible classes of broadcasts as the main
contributor to the Milky Way’s luminosity. Narrowband radio beacons with
isotropic luminosities around $10^{12}\text{--}10^{{26}}\ \textrm{W}$ cannot
possibly make up a significant fraction of its current radio emission
(Enriquez et al. 2017; Price et al. 2020; Tremblay & Tingay 2020; Wlodarczyk-
Sroka et al. 2020; Gajjar et al. 2021; see Paper II). Optical SETI constraints
also are sensitive to energy outputs far smaller than the Galaxy’s stellar
luminosity (Horowitz & Sagan, 1993; Tellis & Marcy, 2017; Maire et al., 2019).
Very little work has been done in SETI at higher energies like X-rays (Corbet,
1997; Hippke & Forgan, 2017), or alternate messengers like neutrinos
(Subotowicz, 1979; Learned et al., 1994; Hippke, 2018). But our ability to
interpret the cosmos as natural implicitly suggests that artificial broadcasts
are not the main emission process in other wave bands (admittedly, not a
certainty as per Ćirković 2018a).

If we were looking for conventional sources around stars, we would conclude
that the brightest examples of the source class are rare in all galaxies as
they should more or less just trace stellar mass, with additional biases from
things like metallicity and age. Only exotic environments like globular
clusters and galactic nuclei might have much higher abundances than seen in
the field, as for close binary star systems, stellar mergers, and their
products (Bailyn 1995; Pooley et al. 2003; Muno et al. 2005; for ETIs, see Di
Stefano & Ray 2016).

Unlike all known astrophysical phenomena, however, ETIs potentially
_replicate_.111Life might also replicate on interstellar scales via panspermia
(e.g., Napier, 2004). Interstellar travel can allow intelligent beings who
evolved on a single world to spread to a large fraction of a galaxy in well
less than a billion years if such propagation is practical (Hart, 1975;
Tipler, 1980; Jones, 1981; Zackrisson et al., 2015; Carroll-Nellenback et al.,
2019). Widespread replication presents great opportunities for SETI, as it
vastly increases the technosignature profile of a galaxy (Kuiper & Morris,
1977; Wright et al., 2014). We have no evidence that this has happened in our
Galaxy, a statement known as the Fermi Paradox (Brin 1983; Wright et al. 2014;
Webb 2015; Ćirković 2018b; Forgan 2019; Lingam & Loeb 2021).

However, this does not rule out it happening in other galaxies. If ETIs with
interstellar travel are rare but not impossible, we could end up with a
situation where most galaxies (perhaps including our own) have none, but a few
have millions of inhabited worlds. Galaxies that look alike to us could have
divergent technosignature populations due to unobservable historical factors
(perhaps set by the first ETIs as in Scheffer 1994 and Hair 2011). The
menagerie of societies in galaxies populated by starfarers did not all arise
independently, but have a shared origin in perhaps a single world, even if the
resulting population is extremely diverse. This shared history can shape the
entire population of descendant ETIs, and so the group might be treated as a
whole – neither a simple collection of noninteracting societies nor a generic
galaxy solely described by astrophysical properties. I call these
intermediate-level groupings _metasocieties_.222Compare analogous notions of
_metapopulations_ and _metacommunities_ in ecology, both describing situations
in which organisms are clustered in many small-scale habitats that influence
each other through dispersal – members of a single species for metapopulations
(Hanski, 1998), and sets of multiple interacting species for metacommunities
(Leibold et al., 2004). Metapopulation ecology bears a direct analogy with
several of the dynamics of interstellar migration and metasocieties – for
example, the survival of a metapopulation depends on quality and number of
habitat patches (see Carroll-Nellenback et al., 2019). The analogy between
interstellar migration and metapopulations is explicitly noted in Lingam &
Loeb (2021), who argue it implies that not all worlds will be inhabited at any
given time. Therefore, the underlying parameters of the technosignature
distribution in each galaxy are themselves the result of a stochastic process,
relating to the properties of the underlying metasociety. As a result,
constraints on ETIs in one galaxy could mean little for the presence or
absence of ETIs in another.333Neglecting the possibility of intergalactic
travel, which could lead to cosmological “bubbles” within which ETIs may
synchronize (Kardashev, 1985; Armstrong & Sandberg, 2013; Olson, 2016). Even
then, different “domains” of the Universe could have varied technosignatures.
SETI has set the first limits on technosignatures in other galaxies (e.g.,
Horowitz & Sagan, 1993; Shostak et al., 1996; Annis, 1999; Griffith et al.,
2015; Gray & Mooley, 2017; Garrett & Siemion, 2023), but these limits still
lag far behind those in the Milky Way.

Since ETIs could be biased toward specific galaxies where interstellar travel
has flourished, this suggests a treatment of galactic ETI populations and
their technosignatures is in order. We can use probability theory to calculate
the expected emission from all the ETIs covered by an observation. Developing
this population approach is the focus of this paper and its sequels.

###  1.1 The problem of notation

The tables in this paper list the important variables. There are many kinds of
objects with their own populations and derived constraints (Figure 1). Each of
these levels has its own associated parameters, which leads to a great
proliferation of variables. Additionally, many of them already have often
conflicting symbols in the literature. This can easily lead to a highly
overloaded namespace: for example, many different types of quantity may lay
claim to $N$ – number of stars, number of societies, number of broadcasts,
number of polarizations emitted, number of pointings, number of photons
counted, and so on. Subscripts and superscripts help, but risk confusion when
two completely different types of variables share the same symbol (e.g., $L$
for luminosity and longevity in the Drake equation), and lead to clutter when
combined with the selection notation developed later. I use different fonts to
help distinguish between different types of variables, as listed in Table 1.
The chosen notation does not always match with their usage in other fields.

Table 1: Summary of fonts used

Font | Case | Examples | Use | Frame
---|---|---|---|---
Blackboard bold | Upper |  $\mathbb{R}$, $\mathbb{M}$, $\mathbb{V}$ |  CCertain sets and operations |  $\cdots$
Hebrew |  $\cdots$ |  $\aleph(\mathrm{K})$, $\beth(\mathrm{B})$ |  OObject type operations |  $\cdots$
Roman | Any |  $\mathrm{W}$, , |  Mathematical functions and probability distributions |  $\cdots$
| Lower |  $\mathrm{j}$, $\mathrm{m}$, $\mathrm{b}$ |  IIndices for realized objects |  $\cdots$
| Upper |  $\mathrm{J}$, $\mathrm{G}$, $\mathrm{MW}$ |  LLabels for types of objects, random objects, and certain named objects |  $\cdots$
San-serif | Lower | , $\mathsf{o}$, $\mathsf{x}$, $\mathsf{t}$ |  LLabels for windows |  $\cdots$
| Upper |  $\mathsf{B}_{\mathsf{o}}$, $\mathsf{\Pi}_{\mathsf{x}}$, $\mathsf{S}_{\mathsf{y}}^{\mathsf{x}}$ |  VVariables defining a selection window | Source
Italics | Any |  $X$, $t$, $f(x)$, $k_{B}$, $H_{0}$ |  Generic variables and functions; physical and cosmological constants |  $\cdots$
| Lower |  $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$, $\boldsymbol{\mathnormal{w}}_{\mathrm{u}}$, $\mathnormal{\vartheta}_{\mathrm{m}}$, $\mathring{\mathnormal{\epsilon}}_{\mathrm{b}}$ |  Quantities describing intrinsic properties of individual objects | Source
| Upper |  $\mathnormal{\Psi}_{\mathrm{k}}^{\mathrm{J}}$, $\mathnormal{M}_{\mathrm{g}}^{\star}$, $\mathnormal{\Sigma}_{\mathsf{x}}^{\mathrm{M}}$, $\mathring{\mathnormal{L}}_{\nu;\mathrm{J}}^{\mathrm{B}}$ |  Quantities describing samples of objects | Source
|  LLower |  ${d}_{\mathrm{b}}$, ${z}$ |  Distances, redshifts, dilutions, and transmittances |  $\cdots$
FFraktur | Lower |  $\mathfrak{m}_{\mathsf{x}|\mathrm{b}}$, $\mathfrak{h}_{\mathsf{o}|\mathrm{B}}$ |  FFluence, flux, or observables for a single object or background | Observer
| Upper |  $\mathfrak{M}_{\mathsf{x}}$, $\mathfrak{a}_{\upmu|\mathfrak{n}}$ |  FFluence, flux, or observables from a sample, or in total | Observer
CCalligraphic | Upper |  $\mathcal{A}$, $\mathcal{N}_{A}$ |  Quantities describing instrument | Observer

Note. — Some characters, mainly the Greek letters, have no counterparts in the
fonts used. For these, the standard italic or upright character is used. A ⊕
superscript attached to a source-frame variable changes it to an observer-
frame variable.

Figure 1: Overview sketch illustrating the tree structure of objects in this
series. A selection draws samples of galaxies, metasocieties, societies, and
broadcasts. The galaxy $\mathrm{g}_{1}$ is in the “classic” SETI scenario with
isolated worlds, each developing its own metasociety, while $\mathrm{g}_{i}$
has a single galaxy-spanning metasociety, and $\mathrm{g}_{2}$ and
$\mathrm{g}_{n}$ are uninhabited (additional objects are represented by the
ellipses). Here, an observation (${\mathsf{o}}$) of a galaxy
($\mathrm{g}_{i}$) defines a more restrictive selection
$(\mathsf{o},\mathrm{g}_{i})$ that picks a subsample of objects, highlighted
in bright blue. The windowed emission
$\mathfrak{M}_{\mathsf{o},\mathrm{g}_{i}}^{\mathrm{B}}$ from the broadcast
sample $\mathnormal{\Sigma}_{\mathsf{o},\mathrm{g}_{i}}^{\mathrm{B}}$ is mixed
with background ($\mathfrak{m}_{\mathsf{o}|\mathfrak{n}}$) into an observed
quantity, $\mathfrak{M}_{\mathsf{o}}$. The selection also picks the ancestors
of $\mathrm{g}_{i}$ (dark blue). Some societies and broadcasts in
$\mathrm{g}_{i}$ are excluded because they lie outside the ${\mathsf{o}}$
window (in a different field or at a different time, for example). The objects
and emission sampled by another selection, $(\infty,{\mathrm{c}_{n_{1}}})$,
are highlighted in red (dark red for ancestors of ${\mathrm{c}_{n_{1}}}$).

###  1.2 Outline of paper

I begin the paper with a review of point process theory (Section 2). The
construction of selections and distributions is detailed in Section 3. The
nature of randomness in ETIs and broadcasts is discussed in Section 4. A
discussion of the modeling of the universe, galaxies, and their stellar
populations (Section 5) is followed by the treatment of the interplay of ETI
metasocieties and societies (Section 6). Next is consideration of the
properties of individual broadcasts (Section 7), and the results are assembled
into calculations of broadcast populations and their total emission (Section
8). Sections 9 and 10 present the box and chord models of broadcasts for
calculating observables. A brief discussion of noisy measurements comes in
section 11. Section 12 demonstrates the formalism by considering how a
galaxy’s stellar mass affects the number of broadcasts in the face of sporadic
interstellar travel. After the conclusion (section 13), appendices present
additional details and derivations.

##  2 Background on point processes

Point processes have a rich theoretical background. The concepts throughout
this section are informed by Kingman (1993); Daley & Vere-Jones (2003, 2008);
Baddeley (2007); Chiu et al. (2013); Haenggi (2013); Last & Penrose (2017).
The reader may consult them for further details. Certain technicalities are
given in footnotes for interested readers.

###  2.1 Random variables

Probability is defined over a sample space of all outcomes, each of which is a
distinguishable configuration of all variables. Because outcomes are so
particular, it is practical to group them into events defined by a shared
quality. An event’s probability $\mathnormal{P}(\mathnormal{E})$ is summed (or
integrated) over all outcomes in the event $\mathnormal{E}$ (Wasserman, 2004).
Outcomes can be summarized by a random variable, which assumes one value per
outcome called a variate. Formally speaking, a random variable is defined as a
function mapping each outcome onto a set, conventionally a real number
(Klenke, 2020). Let’s say we are interested in the positions of stars in the
Milky Way. Every single possible configuration of stars is a distinct outcome.
Those outcomes with two stars within one parsec of the Earth form an event;
the number of stars within one parsec of the Earth is a random variable, which
depends on what outcome is realized.

Distributions describe the relative probability of different variates, among
them the cumulative distribution function (CDF)
$F[X](x)\equiv\mathnormal{P}(X\leq x)$ and probability density function (PDF)
$\uppsi[X](x)\equiv dF[X](x)/dx$ for real-valued variables (Wasserman,
2004)444The PDF is conventionally written as $f_{X}$, I use to avoid confusion
with other uses like the factors in the Drake equation. (refer to Table 2 for
the statistical notation used in this series).

Random variables are very general, and can have completely arbitrary values.
Among the simplest is the indicator variable: $\mathbb{I}[\mathnormal{E}]$ is
$1$ if event $\mathnormal{E}$ happens and $0$ if it does not. Random variables
can have a deterministic value regardless of outcome, like always being $0$,
with a degenerate distribution. Thus, random variables _per se_ do not imply a
lack of rhyme or reason.

Random variables are independent if we learn nothing new about the value of
one if we know the value of the other. Often, however, random variables depend
on an outside factor, without influencing each other. Conditional independence
of $X$ and $Y$ means that, if we know the value of a third random variable
$Z$, knowing what $X$ is tells us nothing more about the value of $Y$. It
neither requires nor implies independence.555sSpecifically, it means
$F[X,Y|Z](x,y|z)=F[X|Z](x|z)\cdot F[Y|Z](y|z)$ (Wasserman, 2004; Klenke,
2020). Conditionalization is very important in this series because it can
isolate the shared influences between random variables. The conditional mean
$\left\langle X|Y\right\rangle$ and variance $\mathbb{V}\left[X|Y\right]$ come
up frequently. The law of total expectation states

$\left\langle X\right\rangle=\left\langle\left\langle
X|Y\right\rangle\right\rangle,$ (1)

and the law of total variance is

$\mathbb{V}\left[X\right]=\mathbb{V}\left[\left\langle
X|Y\right\rangle\right]+\left\langle\mathbb{V}\left[X|Y\right]\right\rangle$
(2)

(Brillinger, 1969; Wasserman, 2004; Bas, 2019).

###  2.2 Point processes

An astrophysical object may be described by a tuple
$\boldsymbol{\mathnormal{w}}$ of quantitative parameters. Each possible tuple
is a point in the set of all tuples, the state space $\mathnormal{W}$. Even
objects that are extended in an observable space are represented as points,
their internal structure following from the point’s location – a core method
in stochastic geometry. As the common metaphor goes, our observations trawl
through that space to find the “needles in the haystack” (Zwicky, 1957;
Harwit, 1981; Tarter, 1984; Djorgovski et al., 2013; Wright et al.,
2018).666In this series, these state spaces are those describing physical
properties of objects, not the “observable parameter spaces” of Harwit 1981
and Djorgovski et al. 2013 describing observational capabilities.

A population is represented as a point process. Basically, a _point process_
is some random set $\mathnormal{\Sigma}$ of points on a state space
$\mathnormal{W}$.777EEach bounded region has only a finite number of points
(almost surely), which is no issue for realistic populations. A point may be
sampled multiple times, requiring some way to encode multiplicity. However, I
assume the point processes are “simple,” each point included only once with
probability $1$ (Baddeley, 2007; Daley & Vere-Jones, 2003). A random variable
$\mathnormal{N}(\mathnormal{A})=|\mathnormal{\Sigma}\cap\mathnormal{A}|$
counts the number of points appearing in each region
$\mathnormal{A}\subset\mathnormal{W}$.888FFormally, each region must be a
Borel set (Daley & Vere-Jones, 2003). Any rigorously defined subset of
$\mathbb{R}^{n}$ prone to be encountered in astronomy is Borel: all continuous
curves and surfaces, including fractals, whether unbounded or closed; their
interiors and exteriors; Cantor dusts; all open sets; all closed sets; and
countable unions of these, among others.

A core property of each point process is the mean number of points999BBoth
$\mathnormal{N}(\mathnormal{A})$ and
$\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle$ are measures. One
consequence is that they are additive when taking the (countable) union of
disjoint subsets. in each region $\mathnormal{A}$. The means
$\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle$ are described by a
density distribution $\mathnormal{\Psi}$ known as the intensity:

$\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle=\int_{\mathnormal{A}}\mathnormal{\Psi}(\boldsymbol{\mathnormal{w}})d\boldsymbol{\mathnormal{w}}.$
(3)

The intensity does not have to be a smooth function; it can include Dirac
distributions to indicate exact locations where some points are found. Points
can also fall along lines or surfaces in the space – just like how the
properties of galaxies and stars can be idealized to fall along lines or
surfaces in graphs describing their properties, like the main sequence or the
Tully-Fisher relation.

Often, we wish to calculate the sum of random variables associated with each
point $\boldsymbol{\mathnormal{w}}$ in the point process
$\mathnormal{\Sigma}$. Campbell’s formula is a very general result that
relates it to the distribution:

$\left\langle\sum_{\boldsymbol{\mathnormal{w}}\in\mathnormal{\Sigma}\cap\mathnormal{A}}f(\boldsymbol{\mathnormal{w}})\right\rangle=\int_{\mathnormal{A}}f(\boldsymbol{\mathnormal{w}})\mathnormal{\Psi}(\boldsymbol{\mathnormal{w}})d\boldsymbol{\mathnormal{w}}.$
(4)

There are many ways to manipulate point processes. The _superposition_ of
several given point processes includes all the points from each of them in a
new point process, with the $\mathnormal{\Psi}(\boldsymbol{\mathnormal{w}})$
summing together.101010As long as the collection of superposed point processes
is countable, there can even be infinitely many of them. _Thinning_ a point
process keeps some of the points based on their locations in the space.
Independent thinnings keep or remove a point solely based on its own location
and not on any other point’s. We can also use a point’s position to _map_ it
to another location in the space.

Finally, we can _mark_ the points in a point process, attaching a random
variable with additional information. If the marks are drawn from set
$\mathbb{K}$, a marked point process is equivalent to a point process on
$\mathnormal{W}\times\mathbb{K}$. Furthermore, if the mark
$\mathnormal{\kappa}$ of a single point is drawn from the distribution
$\uppsi[\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}]$, independent of the
number of other points and their locations, then the intensity on this new
space is
$\mathnormal{\Psi}{(\boldsymbol{\mathnormal{w}})}\times\uppsi[\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}]$.
That is why the “haystack” concept works – although objects are located in
physical spacetime, their parameters serve as “marks,” which can then be
interpreted as additional dimensions in an abstract space.111111The reverse
operation – treating one or more of the dimensions as marks for a point
process on a lower-dimensional space – is only allowed if the resulting
projected process does not have infinite
$\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle$ for any bounded
$\mathnormal{A}$ (Baddeley, 2007), but happens whenever we simplify
distributions by marginalizing over a parameter.

###  2.3 Poisson point processes

Poisson point processes are special point processes with many useful
properties (Kingman, 1993).121212It is common to use “Poisson process” to
refer to the specific case counting the number of hits along a nonnegative
real line. A “compound Poisson process” then measures the accumulated sum of
jumps occurring at each hit (e.g., Ross, 1996; Wasserman, 2004; Embrechts et
al., 2013). They have seen use in modeling populations of ETIs (Glade et al.,
2012; Kipping, 2021), fast radio bursts (Lawrence et al., 2017), and galaxies
(Neyman & Scott, 1952; Martínez & Saar, 2002). In a Poisson point process,
each of the $\mathnormal{N}(\mathnormal{A})$ has a Poisson distribution:

$\mathnormal{P}(\mathnormal{N}(\mathnormal{A})=n)=e^{-\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle}\frac{\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle^{n}}{n!}.$
(5)

Additionally, the number of points in nonoverlapping regions are independent
random variables. A Poisson point process is completely specified by its
intensity.131313AAdditionally, a Poisson point process has
$\left\langle\mathnormal{N}(\\{\boldsymbol{\mathnormal{w}}\\})\right\rangle=0$
for any single point $\boldsymbol{\mathnormal{w}}$ (there are no “atoms” in
the intensity; Kingman 1993). It is fine for lower-dimensional curves and
surfaces to have $\left\langle\mathnormal{N}(\mathnormal{A})\right\rangle>0$,
however (Daley & Vere-Jones, 2003). The Poissonian character of the point
processes is unaffected by superposition, independent thinning, mapping, and
displacing points by a random offset. The resulting intensities are what one
naively expects – added together for superposition, reduced by a position-
dependent fraction in independent thinning, preserved in the image of a subset
for mapping, and blurred for displacement – subject to certain technical
conditions (Kingman, 1993). A marked Poisson process with points on
$\mathnormal{W}$ and marks from $\mathbb{K}$ is equivalent to a new Poisson
point process on $\mathnormal{W}\times\mathbb{K}$, as long as the marks are
mutually independent. Other point processes converge to Poissonian when
repeated randomizing operations are applied to them, most notably displacing
their points independently and randomly (Daley & Vere-Jones, 2008).

These results formalize common sense notions in a very general way. Often in
astronomy, we model a field of objects with a Poisson process, like counting
stars in a patch of sky or transients observed by a radio telescope. These
results show that we can add information, like the colors and magnitudes of
stars, and still use the Poisson distribution. Poisson statistics also apply
when we change the parameterization (e.g., frequency to wavelength), combine
or filter out subclasses of objects (like early- and late-type stars), or have
scatter (as from time delays or errors). There only has to be a determined
intensity (Section 2.5).

###  2.4 Compound Poisson variables and point processes

A compound Poisson random variable has the form

$\mathnormal{K}=\sum_{\mathrm{J}=1}^{\mathnormal{N}}{\mathnormal{\kappa}_{\mathrm{J}}},$
(6)

where the ${\mathnormal{\kappa}_{\mathrm{J}}}$ are independent and identically
distributed (i.i.d.), and also independent of
$\mathnormal{N}\sim\mathrm{Pois}(\left\langle\mathnormal{N}\right\rangle)$
(e.g., Adelson 1966; Barbour & Chryssaphinou 2001; Karlis & Xekalaki 2005;
Geringer-Sameth et al. 2015). If $\mathnormal{N}=0$, then $\mathnormal{K}=0$.
The distribution of $\mathnormal{K}$ has a compound form as the weighted sum
of the distribution for each fixed $\mathnormal{N}$:

$\mathnormal{P}(\mathnormal{K}\leq
x)=\sum_{n=0}^{\infty}\mathnormal{P}(\mathnormal{N}=n|\left\langle\mathnormal{N}\right\rangle)\mathnormal{P}({\mathnormal{K}\leq
x|\mathnormal{N}=n}).$ (7)

Although this distribution often has no closed form expression, its Fourier
transform, the characteristic function, is

$\upphi[\mathnormal{K}](\tilde{x})={\left\langle
e^{i\mathnormal{K}\tilde{x}}\right\rangle=}\exp[\left\langle\mathnormal{N}\right\rangle(\upphi[\mathnormal{\kappa}_{\mathrm{J}}](\tilde{x})-1)],$
(8)

where $\upphi[\mathnormal{\kappa}_{\mathrm{J}}]$ is the characteristic
function of $\mathnormal{\kappa}_{\mathrm{J}}$ (Kemp, 1967; Daley & Vere-
Jones, 2003). An inverse Fourier transform then gives a numerical probability
distribution (Geringer-Sameth et al., 2015). Equation 8 appears in $P(D)$
analyses, which implicitly use the compound Poisson distribution (Scheuer,
1957). Both the mean and variance have simple expressions:

$\displaystyle\left\langle\mathnormal{K}\right\rangle$
$\displaystyle=\left\langle\mathnormal{N}\right\rangle\left\langle\mathnormal{\kappa}_{\mathrm{J}}\right\rangle$
$\displaystyle\mathbb{V}\left[\mathnormal{K}\right]$
$\displaystyle=\left\langle\mathnormal{N}\right\rangle\left\langle\mathnormal{\kappa}_{\mathrm{J}}^{2}\right\rangle$
(9)

(Adelson, 1966; Bas, 2019).

Whenever we have a Poisson point process $\mathnormal{\Sigma}$, with each
point marked by i.i.d. nonnegative real numbers, the compound Poisson point
process describes the sum of the marks for different subsets
$\mathnormal{K}(\mathnormal{A})$ (Daley & Vere-Jones, 2003; Last & Penrose,
2017). Each $\mathnormal{K}(\mathnormal{A})$ is a compound Poisson random
variable. In fact, even the ${\mathnormal{\kappa}_{\mathrm{J}}}$ themselves
can be compound Poisson random variables, representing hierarchical clusters
of events, as long as they are i.i.d. and independent of
$\mathnormal{N}(\mathnormal{A})$ on each level.

What if the marks are independent but their distribution varies with
$\boldsymbol{\mathnormal{w}}$? Although the sum is no longer compound Poisson,
Campbell’s formula for the mean (equation 4) still applies, and a second
Campbell’s theorem gives us

$\mathbb{V}\left[\mathnormal{K}(\mathnormal{A})\right]=\int_{\mathnormal{A}}\left\langle\mathnormal{\kappa}(\boldsymbol{\mathnormal{w}})^{2}\right\rangle\mathnormal{\Psi}(\boldsymbol{\mathnormal{w}})d\boldsymbol{\mathnormal{w}}$
(10)

if $\left\langle\mathnormal{K}(\mathnormal{A})\right\rangle$ exists (e.g.,
Kingman, 1993). This equation is valid only for Poisson point processes,
failing when the points are dependent or $\mathnormal{N}(\mathnormal{A})$ is
fixed.

Table 2: Summary of statistical notation used Notation | Explanation
---|---
Variables
$X$ |  Generic random variable
$\mathnormal{N}$ |  Generic nonnegative integer random variable
$\mathbb{I}[\mathnormal{E}]$ |  IIndicator random variable; $1$ if event $\mathnormal{E}$ happens and $0$ otherwise
$X|Y$ |  Random variable $X$, with outcomes limited by value of random variable $Y$
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ |  RArbitrary singleton random variable describing object ${\mathrm{J}}$; quantity filtered by window ${\mathsf{x}}$
$\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}$ |  SArbitrary aggregate random variable, summing $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ from $\mathrm{J}$-type objects selected by $(\mathsf{y},\mathrm{K})$
$|S|$ |  Cardinality of set $S$, when $S$ is nonnumeric
Probability distributions and operations
$X\sim\mathrm{Q}(\alpha)$ |  $X$ has $\mathrm{Q}$ distribution with parameter $\alpha$
$P(\mathnormal{E})$ |  Probability of some event $\mathnormal{E}$
$\mathbb{O}[X]$ |  AAny operation on the distribution of the random variable $X$: can stand for event probability, cumulative distribution function (CDF), probability density function (PDF), characteristic function, mean, variance, standard deviation, and order statistics including maxima and minima
$\mathbb{O}\left[X_{\mathrm{J}}\right]_{\mathsf{x},\mathrm{K}}$ |  EEnsemble selection-relative distribution operation on $X_{\mathrm{J}}$ for a random object ${\mathrm{J}}$ drawn by $(\mathsf{x},{\mathrm{K}})$ (Section 3.4)
$\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{z}|\mathsf{y},\mathrm{K}}$ |  MMultiwindow distribution operation on $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ that lets $\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}$ range over all the ${\mathsf{x}_{i}}$ windows in ${\mathsf{z}}$ (Section 3.5).
${\mathnormal{P}_{\mathsf{x},\mathrm{K}}}(\mathnormal{E}_{\mathrm{J}})$ |  EEnsemble probability of an event related to random object ${\mathrm{J}}$ selected by $(\mathsf{x},{\mathrm{K}})$
$F[X](x)$ |  CCDF of $X$, $P(X\leq x)$
$F[X|Y](x|y)$ |  CConditional CDF of $X$ conditionalized on $Y$, $P(X\leq x|Y\leq y)$
$F[X,Y](x,y)$ |  JJoint CDF of $X$ and $Y$, $P(X\leq x\ \cap\ Y\leq y)$. Can be extended to arbitrarily many variables.
$\uppsi[X](x)$ |  PDF of $X$, $dF[X](x)/dx$. Conditional and joint PDFs substitute for $F$ in the CDF notation.
$\upphi[X](\tilde{x})$ |  Characteristic function of $X$
${\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}^{[\mathsf{y},\mathrm{K}]}}$ |  RRegularization (trimming) of $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ to exclude values unlikely to occur in $\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}$
Means and variances
$\left\langle X\right\rangle$ |  Mean (expectation value) of random variable $X$; simple mean for variables describing an object
$\mathbb{V}\left[X\right]$ |  Variance of random variable $X$ ($\left\langle X^{2}\right\rangle-\left\langle X\right\rangle^{2}$); simple variance for variables describing an object
Order statistics
$\mathbb{M}\left[X\right]$ |  Median of random variable $X$
$X_{(j)}$ |  Order statistic: $j$th smallest in set of random variables $X_{k}$; distinct from unsorted $X_{j}$
$X_{(1)}$ |  Minimum of the set of random variables $X_{k}$
$X_{(N)}$ |  Maximum of the set of $N$ random variables $X_{k}$, where $N$ itself is a random variable

###  2.5 Other point processes

Actual counts may not be Poissonian. The binomial point process is what we get
when the total number of points in a Poisson point process is fixed. The
positions of the points are mutually independent, although the numbers of
points in each $\mathnormal{A}\in\mathnormal{W}$ are dependent.

While the Poisson point process is a mixture of binomial point processes with
random $\mathnormal{N}$, the Cox point process is a mixture of Poisson point
processes where the intensity distribution $\mathnormal{\Psi}$ itself is
random. Only one intensity field is ever realized for the point process.
Looking at it, we would only “see” that realized intensity, and it would “look
like” a Poisson point process, no matter how much we sampled it. Cox point
processes are thus _nonergodic_. Only if we had a family of identical Cox
point processes could we see the true range of variability in them. Cox point
processes have found use in modeling ecological populations when the
configuration of offspring is hypothesized to cluster around parents (e.g.,
Wiegand & Moloney, 2013). The populations of societies and broadcasts in
galaxies are regarded as Cox point process later in this paper as well.

Even more general are point processes where the random points interact. At the
deterministic limit are lattices, with points “repelling” one another. Another
example, the Gibbs point process, has a global energy function penalizing some
configurations of points (e.g., forbidding those with points too close
together), but practical results are difficult to come by. They remain outside
of this work’s scope.

In this work, objects of a single type usually appear randomly and do not
interact (with some exceptions described in Section 6.3). Poisson point
processes are natural when we have objects appearing independently according
to a well-defined background rate. Several kinds of object populations behave
this way (see discussion in Section 4), if their immediate host has specified
properties, including broadcasts in a society. But sometimes we wish to
consider the population hosted by a randomized collection of hosts. The
intensity then is itself random, and the point process is Cox. The locations
of stars in a specific galaxy are a Poisson point process (e.g., Tonry &
Schneider, 1988), with the brightness profile serving as intensity. However,
the star counts of “a galaxy” in an of itself is not defined, because the
brightness profile itself varies with galactic mass, type, and so on – it is
more accurately described as a function that returns a Poisson point process
given the galactic parameters. The star counts of a galaxy cluster are a Cox
point process, since the gross properties of the cluster do not necessarily
tell us about the characteristics of the member galaxies, only their
_distribution_. The star counts are a superposition of Poisson processes for
all possible configurations of galaxies in the cluster. Broadcasts in galaxies
are one example of a Cox point process in this paper because they are
clustered into random societies.

##  3 Defining selections and samples

We do not observe the properties of ETI broadcast populations directly, only
measure limited samples of the entire population. Measurements are treated as
random variables that depend on, but are not determined by, the properties of
the sampled broadcasts. This introduces two forms of variance: the sampling
variance due to the randomness of which objects are included, and noise
variance introduced by microscopic fluctuations in observables for that sample
(Figure 2). Describing how the sampling is modeled, and how it affects random
variables, is the focus of this section.

Figure 2: ETI broadcasts of a galaxy over its history form a population
described by distributions (left). Surveys, and observations within the
surveys, select subsets of the positions, times, and frequencies (and
polarizations, not shown) spanned by this population, drawing samples treated
as random sets (middle). In turn, measurement yields observable quantities,
which are themselves random variables with probability distributions dependent
on the sample drawn by the observation (right). Both the sampling itself and
the measurement conditionalized on the observed sample contribute variance to
the final quantity.

###  3.1 Basic structure of populations

####  3.1.1 The tree and the haystacks

A theory of the population of ETI technosignatures encompasses the habitat
that hosts the ETIs, the ETIs themselves, and their technosignatures. In this
formalism, there are two kinds of structures relating them.

First is a multilevel _tree_ , assigning each object to a host. The ancestors
of an object are its host, the hosts of the host, and so on, with the parent
being the immediate node one level closer to the root. Likewise, each object
can host a subpopulation of descendants, with the children being those that
are one level deeper.

Five “levels” are considered in this series, each for a different type of
object. The root is the _universe_ (type $\mathrm{U}$), containing all other
objects. On the next level down are _galaxies_ (type $\mathrm{G}$). These
contain _metasocieties_ (type $\mathrm{M}$), a collection of societies with a
common origin or influence, which may be localized or extended. Within them,
_(communicative) societies_ of ETIs (type $\mathrm{C}$) may send out
_broadcasts_ (type $\mathrm{B}$). The objects in each level form a population
of objects (Figure 1).

The levels are related by type operations. The ancestor-type
$\aleph(\mathrm{J})$ returns the set of object types that are ancestors of
${\mathrm{J}}$, with the parent-type operation $\aleph_{1}(\mathrm{J})$
returning the type one level up. The descendant-type $\daleth(\mathrm{J})$
gives the set of object types that are lower on the tree, regardless of
whether any such descendants exist, and the type-of operation
$\beth({\mathrm{J}})$ (or $\beth(\mathrm{j})$) which returns an object’s type.
Thus, for the metasocietal type $\mathrm{M}$,
$\aleph(\mathrm{M})=\\{\mathrm{U},\mathrm{G}\\}$,
$\aleph_{1}(\mathrm{M})=\mathrm{G}$, $\beth(\mathrm{M})=\mathrm{M}$, and
$\daleth(\mathrm{M})=\\{\mathrm{C},\mathrm{B}\\}$.

The other ingredient is a series of state spaces for objects, a collection of
_haystacks_ (see Wright et al., 2018). Every type of object $\mathrm{J}$ has a
haystack $\mathnormal{W}^{\mathrm{J}}$ (Figure 3). A single object
${\mathrm{J}}$ is described fully by a corresponding tuple
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$ in its haystack. Each tuple is a
Cartesian product of parameters describing the object. These can include basic
quantities like the lifespan of the object, its location, energetics, and so
on.

Included among the intrinsic properties of the object are parameters
describing the statistical properties of its internal child population. So,
for every object ${\mathrm{k}}$ of type $\mathrm{K}$ with
$\mathrm{J}\in\daleth(\mathrm{K})$, $\boldsymbol{\mathnormal{w}}_{\mathrm{k}}$
specifies the _distribution_ of the descendants on the corresponding haystack
$\mathnormal{W}^{\mathrm{J}}$, but generally not which
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$ are included in the actual
subpopulation.141414This framework echoes others developed in the past
literature across science, without directly following them. In the Neyman-
Scott process, clusters of galaxies are represented by random points in real
space, and associated with each is a collection of galaxies represented by
points randomly distributed by some distribution. The clusters themselves can
be clustered this way, to arbitrarily high order (Neyman & Scott, 1952, 1958;
Martínez & Saar, 2002; Haenggi, 2013). Also related are multiplicative
population chains, modeling the growth of population between generations
(Moyal, 1962a, b; Daley & Vere-Jones, 2003). Unlike those formalisms, however,
the underlying type of state space changes between “generations” because each
level is a new type of object.

####  3.1.2 Models

A model describes the populations within some object of interest. It picks
some node ${\mathrm{K}}$ in the tree, whether the universe or a single host
that we are studying, to serve as the root of a subtree. The model specifies
enough parameters that the descendant populations of that node are
statistically characterized. The model can be identified with the tuple
$\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$ describing this subtree root object
${\mathrm{K}}$. Thus, the model is essentially a generalized random variable.
The actual population of objects that exists is a realization of the model.

Only one set of model parameters actually describes the model’s root. We may
know some of these parameters – the stellar mass of a model galaxy, or the
cosmological parameters of the universe – but others, particularly those
describing ETI subpopulations, are unknown. However, the random nature of the
model root is different than the randomness of the descendant populations. The
distributions of objects on lower levels have a frequentist character. The
subpopulation of $\mathrm{J}$ objects within a host ${\mathrm{K}}$ is a single
trial; if we had an infinite number of trials, with an ensemble of hosts all
with the same $\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$, the mean density of
$\mathrm{J}$ objects in the haystack really would converge to
$\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}$. We use
$\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$ to predict the properties of the
descendant subpopulation.

In contrast, there is only one root object, as is clear when it is the
universe as a whole, and cannot repeat “trials.” Yet we are uncertain about
the true values of $\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$, so we have to
consider many possible models and compare the kinds of populations that result
with reality. To the extent there is any “distribution” for the root object,
the probabilities involved are Bayesian, describing our level of belief in
each hypothetical model; the goal is inference, not prediction. The results of
SETI surveys are phrased as statements about likelihood, the consistency of
each model with the data. The classic rate-versus-luminosity plots of SETI
(e.g., Enriquez et al., 2017) are an example, describing the consistency with
an empirical null result.

####  3.1.3 Random and realized objects

Random variables describe the observables of objects, but sometimes we need a
general result – the object itself is a “random variable” of sorts. These are
the random objects, in opposition to realized objects. With a random object,
we mean “for anything of this type,” but for a realized object, we mean “
_this_ (possibly hypothetical) object.” Samples draw random objects from the
haystack, where the object’s tuple itself is the random variable.

Each dimension of a haystack is a coordinate variable; realized objects have
specific values for each coordinate, while random objects leave them as free
parameters.151515TThe interpretation of the basic parameters as coordinates
and not random variables is implicit in their direct use in the distributions
(e.g.,
$d\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}\right\rangle/d\mathnormal{\tau}_{\mathrm{J}}$).
The same is true of deterministic combinations of the basic parameters (like
broadcast isotropic luminosity from total energy release and lifespan), which
can be regarded as using a different coordinate system. However, these
intrinsic quantities also could be interpreted as random variables with
degenerate distributions, as in section 3.3.1. The notation for these basic
parameters is the same as for the random quantities that mark each notation,
however, reflecting this ambiguity. If we wished to be absolutely rigorous, we
would make a distinction between the basic parameter as coordinate variable
and a random variable “wrapper,” but I ignore the distinction to avoid further
clutter. Every quantity associated with a realized object that is not a
deterministic combination of these coordinate parameters is a random variable,
with a distribution that is fixed by their values. With random objects, these
quantities are not formally random variables, as we cannot specify a single
distribution without the object’s properties being known. Instead, they
actually are _functions_ that yield random variables when the tuple describing
a realized object is fed into them as input. The number of photons intercepted
from random broadcast ${\mathrm{B}}$ during window ${\mathsf{x}}$,
${\mathfrak{q}_{\mathsf{o}|\mathrm{B}}}$, is an example of one of these random
functions, and should in all strictness be written as
${\mathfrak{q}_{\mathsf{o}|\mathrm{B}}(\boldsymbol{\mathnormal{w}}_{\mathrm{B}})}$.
When $\boldsymbol{\mathnormal{w}}_{\mathrm{B}}$ is assigned a single value
$\boldsymbol{\mathnormal{w}}_{\mathrm{b}}$, then
${\mathfrak{q}_{\mathsf{o}|\mathrm{b}}\equiv\mathfrak{q}_{\mathsf{o}|\mathrm{B}}(\boldsymbol{\mathnormal{w}}_{\mathrm{b}})}$
is a _bona fide_ random variable with a single distribution. In practice, the
random functions may be written as if they are variables to avoid cluttering
notation, but the dependence is still implicitly there.

The notation makes this distinction in the way it indexes random variables. An
uppercase index designating a type of object also refers to a random object of
that type. When the index is a lowercase letter, it denotes a realized object.
Thus, for a random variable $X$ and a $\mathrm{J}$-type object ${\mathrm{j}}$
described by a parameter tuple $\boldsymbol{\mathnormal{w}}_{\mathrm{j}}$,

$X_{\mathrm{j}}=X_{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{j}}).$
(11)

This distinction is very general, applying even to distributions, point
processes, and the selection-relative operations – so
$\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
is the distribution of $\mathrm{J}$-type objects in a random $\mathrm{K}$-type
host, while
$\mathnormal{\Psi}_{\mathrm{k}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$
is the distribution in a realized host ${\mathrm{k}}$.

Most of the results that follow apply to both random and realized objects, and
can be interconverted between the two by substituting notation and using the
appropriate dependence on the random object’s tuple:

$\displaystyle\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
$\displaystyle\rightarrow\mathnormal{\Psi}_{\mathrm{k}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}=\boldsymbol{\mathnormal{w}}_{\mathrm{k}})$
$\displaystyle\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}(\mathnormal{\kappa})$
$\displaystyle\rightarrow\begin{aligned}
&\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{k}}(\mathnormal{\kappa})\\\
=~{}&\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}(\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}};\boldsymbol{\mathnormal{w}}_{\mathrm{K}}=\boldsymbol{\mathnormal{w}}_{\mathrm{k}})\end{aligned}$
$\displaystyle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$
$\displaystyle\rightarrow\mathnormal{\kappa}_{\mathsf{x}|\mathrm{j}}=\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}=\boldsymbol{\mathnormal{w}}_{\mathrm{j}})$
$\displaystyle\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
$\displaystyle\rightarrow\mathnormal{K}_{\mathsf{x},\mathrm{k}}^{\mathrm{J}}=\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}}=\boldsymbol{\mathnormal{w}}_{\mathrm{k}}).$
(12)

####  3.1.4 Populations as point processes

The population of each kind $\mathrm{J}$ of objects is described by a point
process on its haystack $\mathnormal{W}^{\mathrm{J}}$ (Figure 3). Furthermore,
a point process exists for every possible host. The point processes are
equivalently described by a random set on the haystack space,
$\mathnormal{\Sigma}^{\mathrm{J}}$, or by its resultant collection of
$\mathnormal{N}^{\mathrm{J}}{(\mathnormal{A})}$ (section 2.2). Only one of the
possible sets is actually drawn, the realized sample or sample variate
$\mathnormal{S}^{\mathrm{J}}$.

Figure 3: Example illustration of the various haystacks for (left to right)
the universe, galaxies, metasocieties, societies, and broadcasts, depicted
here as having only two dimensions. Each host, chosen along one branch of
Figure 1, has its own point process for each type of haystack. The realized
samples illustrated here represent one possible population, each object a
single point. Those objects that are selected by $(\mathsf{x},\mathrm{g}_{i})$
(blue region) or $(\infty,\mathrm{c}_{n_{1}})$ (red outline) are colored blue
and red, respectively. In some cases, an object falling in the ${\mathsf{x}}$
window is missed because its ancestors on the tree (its hosts) are not
selected (black dots in blue regions).

Let ${{\mathrm{K}}}$ be the parent of $\mathrm{J}$-type children. These
children form a random set
${\mathnormal{\Sigma}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$
on $\mathnormal{W}^{\mathrm{J}}$, the $\mathrm{J}$-haystack.161616OOr, to be
pedantic about it,
$\mathnormal{\Sigma}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
is a random function that yields a random set (point process) based on
$\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$; for the realized host
${\mathrm{k}}$,
$\mathnormal{\Sigma}_{\mathrm{k}}^{\mathrm{J}}\equiv\mathnormal{\Sigma}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}}=\boldsymbol{\mathnormal{w}}_{\mathrm{k}})$
is a random set (section 3.1.3). So the broadcasts produced by a realized
society ${\mathrm{c}}$ are characterized by a point process
$\mathnormal{\Sigma}_{\mathrm{c}}^{\mathrm{B}}$ on the broadcast haystack
$\mathnormal{W}^{\mathrm{B}}$, for instance. Of course, the children of an
object are constrained by the properties of the parent. Descendant objects
cannot be born before their ancestors, for example. As a result, the region of
the haystack where children objects may be found is restricted.

Each point process has an intensity distribution

${\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}|{\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})=\frac{d\left\langle{\mathnormal{N}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}\right\rangle}{d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}),$
(13)

where ${\mathnormal{N}_{\mathrm{K}}^{\mathrm{J}}}$ is the number of
$\mathrm{J}$-type objects descended from the random host
${\mathrm{K}}$.171717TThe haystack is multidimensional, and the tuples are
more or less vectors with many components. Even some of the parameters
themselves are multidimensional (e.g., spatial position). The number of
dimensions depends on the type of object and model, however; for an
$n$-dimensional variable $\boldsymbol{x}$, $d/d\boldsymbol{x}$ should be
understood to be the $n$th order derivative, and $d\boldsymbol{x}$ as a
hypervolume element. This equation applies generally, but it serves as a
definition for the distribution of child objects in a parent. Because the
characteristics of the parent determines the statistical properties of the
child subpopulation, the value of the intensity function at each point
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$ in the $\mathrm{J}$-haystack is
specified by the model and ${\boldsymbol{\mathnormal{w}}_{\mathrm{K}}}$. Note
that these mean numbers refer to the population of objects over the entire
history of the parent, _not_ the typical number of objects that are active at
any one time (contrast the $N$ in the Drake equation). Finally, the
distribution is an average over an ensemble of all possible realizations of
the subpopulations in ${{\mathrm{K}}}$.

Of course, we are not just interested in the subpopulation in the parent, but
also in the higher-order ancestors. We often wish to know the total number of
broadcasts in an entire galaxy, not just one society, for instance. These
populations are just the superposition for the point processes of descendant
hosts – for an $\mathrm{L}$-type random host with
$\mathrm{L}\in\aleph(\mathrm{K})$ and $\mathrm{K}\in\aleph(\mathrm{J})$,

$\displaystyle{\mathnormal{\Sigma}_{\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$
$\displaystyle=\bigcup_{{\boldsymbol{\mathnormal{w}}_{\mathrm{K}}\in\mathnormal{\Sigma}_{\mathrm{L}}^{\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}}\mathnormal{\Sigma}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
$\displaystyle{\mathnormal{N}_{\mathrm{L}}^{\mathrm{J}}}(\mathnormal{A}{|\boldsymbol{\mathnormal{w}}_{\mathrm{L}}})$
$\displaystyle=\sum_{{i}=1}^{{\mathnormal{N}_{\mathrm{L}}^{\mathrm{K}}}(\mathnormal{A}{|\boldsymbol{\mathnormal{w}}_{\mathrm{L}}})}{\mathnormal{N}_{\mathrm{K}_{i}}^{\mathrm{J}}(\mathnormal{A}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}_{i}}})$
(14)

for each region $\mathnormal{A}\subset\mathnormal{W}^{\mathrm{J}}$. The number
of $\mathrm{K}$-type subancestors is itself a random variable, and
${\mathnormal{\Sigma}_{\mathrm{L}}^{\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$
is random too. Now, the mean value of
$\mathnormal{N}_{\mathrm{L}}^{\mathrm{J}}(\mathnormal{A}|\boldsymbol{\mathnormal{w}}_{\mathrm{L}})$
is closely related to
$\mathnormal{\Psi}_{\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{L}})=d\left\langle\mathnormal{N}_{\mathrm{L}}^{\mathrm{J}}(\mathnormal{W}^{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{L}})\right\rangle/d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$
through a derivative. The distribution is thus also a weighted sum over the
possible $\mathrm{K}$ hosts. The intensity for the total $\mathrm{J}$-object
population is

${\mathnormal{\Psi}_{\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}=\int_{\mathnormal{W}^{\mathrm{K}}}\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}){\mathnormal{\Psi}_{\mathrm{L}}^{\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}}|\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}d\boldsymbol{\mathnormal{w}}_{\mathrm{K}}.$
(15)

This follows essentially from applying Campbell’s formula (equation 4),
because
$\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
is a random variable at each $\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$. This
way, we can build up the distribution functions up from the parent all the way
to the universe, given a model (see Moyal, 1962a). The distribution of objects
in their parent-type hosts is fundamental. Given these, we can derive all the
other distributions for objects in ancestors, informing us about the
populations we expect to be contained within each host.

What if the “host” object type is not ancestral, though? The $\mathrm{J}$-type
population in a $\mathrm{J}$-type object is the self-population, namely,
itself:
${\mathnormal{S}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\mathnormal{\Sigma}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\\{\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\\}}$.
It is a Dirac point process, picking the object with certainty:

$\mathnormal{\Psi}_{\mathrm{J}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\updelta(\boldsymbol{\mathnormal{w}}-\boldsymbol{\mathnormal{w}}_{\mathrm{J}}).$
(16)

What about the point process of ancestors “contained” within an object? It
helps to distinguish between what ancestors an object _could have_ had, and
what ancestors an object _actually_ has. Only the latter, the realized sample,
is used, and it only contains one point for each level. When ${\mathrm{k}}$ is
a $\mathrm{K}$-type ancestor of ${\mathrm{j}}$,
$\mathnormal{S}_{\mathrm{j}}^{\mathrm{K}}=\mathnormal{S}_{\mathrm{k}}^{\mathrm{K}}=\\{\boldsymbol{\mathnormal{w}}_{\mathrm{k}}\\}$.

###  3.2 Selections

Any realistic observation program cannot observe the entire realized
population of objects throughout the universe, much less all possible objects
in the haystack. An observation, survey, or other selection instead draws a
sample of them based on position, time, frequency, and other constraints, as
well as which ancestors they have (Figure 2). A _selection_ makes these cuts.
Each selection $(\mathsf{x},{\mathrm{J}})$ is the combination of a _window_
${\mathsf{x}}$ and a _host_ object ${{\mathrm{J}}}$. Notation for windows is
listed in Table 3.

Table 3: Notation for selection windows  Notation | Explanation
---|---
$(\mathsf{x},{\mathrm{J}})$ |  TThe selection picking objects and emission using window $\mathsf{x}$ from a population hosted by object ${{\mathrm{J}}}$
Useful windows
${\mathsf{x}}$, ${\mathsf{y}}$, ${\mathsf{z}}$ |  Denote generic windows; can be substituted with any other unless otherwise indicated.
|  Picks objects and emission covered by in one electromagnetic field mode
(see Paper II)
${\mathsf{o}}$ |  Picks objects and emission covered by an observation
|  Picks objects and emission covered by a single beam or resolution element
with a fixed sky position in a series of observations
|  Picks objects and emission covered by a “tuning,” a series of observations
covering a fixed frequency range (e.g., in a single filter in optical)
|  Picks objects and emission covered by a pointing, a series of observations
over a fixed sky field
${\mathsf{s}}$ |  Picks objects and emission covered by a survey
|  FFunction returning a window that picks all objects and emission coincident
with a fixed value of an arbitrary basic variable $\rho$
${\mathsf{t}}$ |  Picks all objects and emission active at one instant as viewed from Earth (i.e., along our past light cone); a function of $t$
${\mathsf{h}}$ |  Picks all objects and emission in past history, within our past light cone; a function of $t$
|  Picks all objects and emission at one source-frame frequency, regardless of
time; a function of $\nu$
${\mathsf{t}\cap\upnu}$ |  Picks all objects active at one instant as viewed from Earth at one source-frame frequency; a function of $t$ and $\nu$
${\infty}$ |  AALL window: picks all objects and emission, regardless of parameters
Quantities describing window definition
$\mathsf{T}_{\mathsf{x}}$ |  Duration of window ${\mathsf{x}}$
$\mathsf{B}_{\mathsf{x}}$ |  Bandwidth covered by ${\mathsf{x}}$
$\mathsf{\Theta}_{\mathsf{x}}$ |  Central time of ${\mathsf{x}}$
$\mathsf{\Upsilon}_{\mathsf{x}}$ |  Central frequency of ${\mathsf{x}}$
$\mathsf{\Delta}_{\mathsf{x}}$ |  Drift rate of ${\mathsf{x}}$ window; used for describing dedrifted observations
$\mathsf{\Pi}_{\mathsf{x}}$ |  Set of polarizations covered by ${\mathsf{x}}$
$|\mathsf{\Pi}_{\mathsf{x}}|$ |  Number of independent polarizations covered by ${\mathsf{x}}$
$\mathsf{\Omega}_{\mathsf{x}}$ |  Sky field covered by ${\mathsf{x}}$
$\mathsf{V}_{\mathsf{x}}$ |  Spatial volume covered by ${\mathsf{x}}$
Window manipulation
$\mathsf{S}_{\mathsf{y}}^{\mathsf{x}}$ |  SSet of windows of type ${\mathsf{x}}$ that are fused into ${\mathsf{y}}$
$\mathsf{N}_{\mathsf{y}}^{\mathsf{x}}$ |  $|\mathsf{S}_{\mathsf{y}}^{\mathsf{x}}|$, number of component windows of type ${\mathsf{x}}$ making up ${\mathsf{y}}$
$\mathsf{x}\cup\mathsf{y}$ |  FFused selection, inclusive of both ${\mathsf{x}}$ and ${\mathsf{y}}$
$\mathsf{x}\cap\mathsf{y}$ |  Joint selection, intersection of ${\mathsf{x}}$ and ${\mathsf{y}}$
$\mathsf{x}{:}\rho$, $\mathrm{j}{:}\rho$ |  Projection of ${\mathsf{x}}$ or ${\mathrm{j}}$ into basic variable $\rho$; picks all objects and emission that have a $\rho$ value in the range covered by ${\mathsf{x}}$ or ${\mathrm{j}}$

####  3.2.1 Windows

A window ${\mathsf{x}}$ selects things that pass cuts on variables like
location, lifespan, or frequencies covered. They include the reach of a survey
or observation, but the cuts can also be more abstract, picking all broadcasts
that are active at an arbitrary frequency, for example. A window is thus a
kind of selection defined on the haystack of each object type.

Each window acts to thin a point process according to position in the
haystack. A window ${\mathsf{x}}$ has two functions. First, it includes a
filter on observable quantities, generally a bounded region over some basic
variables so that any object that “touches” it is selected (see Section
3.3.1). Second, a window restricts object selections: formally it includes a
collection of probability functions
$\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$
for each type of object $\mathrm{J}$, giving the probability that the window
selects an object with tuple $\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$.
Basically, $\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}$ gives the completeness of
the window selection for an object with
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. All selected objects fall within a
subset of the haystack,181818Windows do not have to be contiguous, but that is
assumed in the box and chord models.

$\mathnormal{W}_{\mathsf{x}}^{\mathrm{J}}=\\{\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{W}^{\mathrm{J}}:\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})>0\\}.$
(17)

Whether an object ${\mathrm{J}}$ is selected by the window depends solely on
$\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$.
It does not otherwise matter whether an object is detectable or
distinguishable from a background. For example, the leakage from a cell phone
call in a distant galaxy would be selected by a survey window if the survey
looks at that galaxy at the right frequency when it happens, unless the window
also imposes a luminosity or fluence cut.

A few fundamental variables – position, time, and polarization – define an
arena in which things are situated (Table 4). Windows and objects may cover
extended regions in this space (Figure 4), and their arrangement is a problem
of stochastic geometry. A basic kind of window is one that picks objects along
a fixed value of one of these variables. For each variable $\rho$, a function
$\uprho{(\rho)}$ returns a window that intersects with a given fixed value of
it. The most commonly used one is the time window function,
${\mathsf{t}}{(t)}$, selecting every object that exists during a particular
moment $t$ in time as viewed from Earth.

Table 4: Notation for the basic general variables  Notation | Explanation
---|---
$t$ | Time
$\nu$ | Frequency
$p$ | Polarization
$d$ | Distance
$\boldsymbol{r}$ | Position in space
$\boldsymbol{\theta}$ | Location on receiver sky
$\rho$ | Arbitrary basic variable

In this work, an observation ${\mathsf{o}}$ is the lowest level window for
which independent data are analyzed: each distinct combination of angular
resolution element, channel, and time yields a separate measurement and can
count as an observation. Other useful windows are the survey ${\mathsf{s}}$, a
complete collection of observations; and ${\mathsf{h}}{(t)}$, picking all
objects in past history to the present as viewed from Earth. Still more are
listed in Table 3.191919Although not used in this work, windows can be random.
These would be a point process on their own “haystack.” In fact, Wright et al.
(2018) basically describe a survey window haystack (see also Djorgovski et
al., 2013).

A collection $\mathsf{S}_{\mathsf{y}}^{\mathsf{x}}$ of windows can be fused to
form another,

$\mathnormal{W}_{\mathsf{y}}^{\mathrm{J}}=\bigcup_{\mathsf{x}_{i}\in\mathsf{S}_{\mathsf{y}}^{\mathsf{x}}}\mathnormal{W}_{\mathsf{x}_{i}}^{\mathrm{J}},$
(18)

and if their selections are independent,

$\mathsf{P}_{\mathsf{y}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=1-\prod_{\mathsf{x}_{i}\in\mathsf{S}_{\mathsf{y}}^{\mathsf{x}}}(1-\mathsf{P}_{\mathsf{x}_{i}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})).$
(19)

The number of these subwindows is $\mathsf{N}_{\mathsf{y}}^{\mathsf{x}}$. For
example, a survey (${\mathsf{s}}$) is divided into
$\mathsf{N}_{\mathsf{s}}^{\mathsf{o}}$ observation windows (generically
labeled ${\mathsf{o}_{i}}$) from the set
$\mathsf{S}_{\mathsf{s}}^{\mathsf{o}}$. The window resulting from fusing two
subwindows ${\mathsf{x}}$ and ${\mathsf{y}}$ in this way can be written
$\mathsf{x}\cup\mathsf{y}$. A joint window $\mathsf{x}\cap\mathsf{y}$ instead
picks an event only if it is selected by both subwindows.

Figure 4: Illustration of how different selection windows could manifest in a
spectrogram. window ${\mathsf{x}}$ (shaded blue) can be projected onto time
($\mathsf{x}{:}t$) and frequency ($\mathsf{x}{:}\nu$). A second window
${\mathsf{y}}$ (shaded red) can be fused with ${\mathsf{x}}$ (violet outline),
or a joint selection can be made (gray shading). Also shown are windows
picking a specific time (green line), frequency (gold line), or both (black
dot).

The last operation on windows is projection, working on a window or host and a
fundamental variable $\rho$ like time or frequency. The projection
$\mathsf{x}{:}\rho$ (or ${\mathrm{K}}{:}\rho$) makes a selection according to
the range of $\rho$ covered by ${\mathsf{x}}$ or ${\mathrm{K}}$ (Figure 4).
The projection of a window is given by

$\mathnormal{W}_{\mathsf{x}{:}\rho}^{\mathrm{J}}=\bigcup_{\rho:\mathnormal{W}_{\mathsf{x}}^{\mathrm{J}}\cap\mathnormal{W}^{\mathrm{J}}(\rho)\neq\varnothing}\mathnormal{W}^{\mathrm{J}}(\rho)$
(20)

and the projection of an object is

${\mathnormal{W}_{\mathrm{K}{:}\rho}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})=\bigcup_{\rho:\boldsymbol{\mathnormal{w}}_{\mathrm{K}}\in\mathnormal{W}^{\mathrm{K}}(\rho)}}\mathnormal{W}^{\mathrm{J}}(\rho)$
(21)

with
$\mathsf{P}_{\mathsf{x}{:}\rho}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}};\rho)={\max[}\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}({\boldsymbol{\mathnormal{w}}})\mathsf{P}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}};\rho){|\boldsymbol{\mathnormal{w}},}\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{W}^{\mathrm{J}}(\rho){]}$
and
${\mathsf{P}_{\mathrm{K}{:}\rho}^{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}};\rho{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})=\mathbb{I}[\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in{\mathnormal{W}_{\mathrm{K}{:}\rho}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}]$.
Projection is a versatile operation, letting us make selections according to a
single shared quantity. For example,
$\mathring{\mathnormal{\epsilon}}_{\mathsf{x}{:}\nu|\mathrm{b}}$ is the
effective isotropic energy release of broadcast ${\mathrm{b}}$ at _any_ time
or polarization within the bandwidth of ${\mathsf{x}}$;
$\mathnormal{N}_{\mathrm{c}{:}t,\mathrm{g}}^{\mathrm{C}}$ is the number of
communicative societies in galaxy ${\mathrm{g}}$ that are ever active during
the lifespan of the specific society ${\mathrm{c}}$. Projection is
particularly useful when describing broadcasts that can run in and out of a
window, like in the chord model later on (Appendix C).

By default, this series uses the ALL window ($\infty$) selecting every event
in a population when no other window is specified or implied.

####  3.2.2 Hosts

Specifying a host ${{\mathrm{K}}}$ in a selection restricts the population to
its subpopulation, as defined by its point process: it draws $\mathrm{J}$-type
objects from
${\mathnormal{\Sigma}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$.
The host-based selection of objects is determined solely by their position on
the tree – the object must be the direct ancestor or descendant of the host,
or be the host itself. Objects are indexed, and the indices serve to denote
their associated host selection as well. Most variables in this series use a
host, but when none is otherwise specified, the cosmic host ${\mathrm{U}}$ is
assumed.

####  3.2.3 Samples and selection-relative intensities

A full selection $(\mathsf{x},{\mathrm{L}})$, as a combination of a window and
a host, selects $\mathrm{J}$ objects from the ${\mathsf{x}}$-thinning of the
point process
${\mathnormal{\Sigma}_{\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$
over the set $\mathnormal{W}_{\mathsf{x}}^{\mathrm{J}}$. The resulting point
process,
${\mathnormal{\Sigma}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$,
is the random sample drawn by $(\mathsf{x},{\mathrm{L}})$. An object
${{\mathrm{J}}}$ is selected by $(\mathsf{x},{\mathrm{L}})$
(${\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{\Sigma}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$)
if and only if:

  * •

It is within the random sample
$\mathnormal{\Sigma}_{\mathsf{x}}^{\mathrm{J}}$; that is, its parameter tuple
${\boldsymbol{\mathnormal{w}}_{\mathrm{J}}}$ is within
$\mathnormal{W}_{\mathsf{x}}^{\mathrm{J}}$ and selected by the
${\mathsf{x}}$-thinning.

  * •

It is a member of the subpopulation of its host
(${\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{\Sigma}_{\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$),
picked by the host selection ${{\mathrm{L}}}$.

  * •

For each ancestor ${{\mathrm{K}}}$ of ${{\mathrm{J}}}$ up to and including
${\mathrm{L}}$,
${\boldsymbol{\mathnormal{w}}_{\mathrm{K}}\in\mathnormal{\Sigma}_{\mathsf{x},\mathrm{L}}^{\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}$;
so that each of these ancestors is _also_ selected by the window.

This final condition is a somewhat tricky one, but an object can be missed
because its ancestor was not sampled. Suppose we defined a window
${\mathsf{x}}$ that picked all galaxies and metasocieties in a patch of the
sky, but only societies that are older than a million years old, and all their
broadcasts. A radio broadcast from a society that is only one thousand years
old would not be selected by $(\mathsf{x},{\mathrm{U}})$ even though
${\mathsf{P}_{\mathsf{x}}^{\mathrm{B}}(\boldsymbol{\mathnormal{w}}_{\mathrm{B}})=1}$
for all ${\boldsymbol{\mathnormal{w}}_{\mathrm{B}}}$, because its host society
is excluded.

If ${\mathrm{L}}$ is a host for objects of type
$\mathrm{K}\in\aleph(\mathrm{J})$, the intensity relative to a selection
$(\mathsf{x},{\mathrm{L}})$ is

${\mathnormal{\Psi}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{L}}})=\mathsf{P}_{\mathsf{x}}^{\mathrm{L}}({\boldsymbol{\mathnormal{w}}_{\mathrm{L}}})\int_{\mathnormal{W}^{\mathrm{K}}}{\mathnormal{\Psi}_{\mathrm{L}}^{\mathrm{K}}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{L}}})\mathnormal{\Psi}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})d\boldsymbol{\mathnormal{w}}_{\mathrm{K}},$
(22)

with

$\mathnormal{\Psi}_{\mathsf{x},\mathrm{J}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}})\mathnormal{\Psi}_{\mathrm{J}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}})\updelta(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}-\boldsymbol{\mathnormal{w}}).$
(23)

Integrating gives us the expected number of $\mathrm{J}$-type objects sampled
by the selection:

${\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}\right\rangle\equiv}\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}\right\rangle=\int_{\mathnormal{W}^{\mathrm{J}}}{\mathnormal{\Psi}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{L}}})d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}.$
(24)

The process of selection often introduces biases in object properties, so
${\mathnormal{\Psi}_{\mathsf{x},\mathrm{L}}^{\mathrm{J}}}$ is not usually
proportional to ${\mathnormal{\Psi}_{\mathrm{L}}^{\mathrm{J}}}$. Generally,
objects that are “larger” are overrepresented in a sample (see section 3.8).

####  3.2.4 Simplifying distributions through marginalization, rates, and
abundances

The full intensities often involve many parameters that are of no interest in
a given problem. The distribution is simplified by marginalizing these
irrelevant quantities, treating them as marks to be ignored (Section 2.2). If
a parameter tuple $\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$ can be divided
into components that we are interested in,
$\boldsymbol{\mathnormal{w}}^{\ast}_{\mathrm{J}}$, and those that we are not,
$\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}$, with
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}=\boldsymbol{\mathnormal{w}}^{\ast}_{\mathrm{J}}\times\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}$,
then

${\mathnormal{\Psi}^{\mathrm{J}\ast}_{\mathsf{x},\mathrm{K}}}(\boldsymbol{\mathnormal{w}}^{\ast}_{\mathrm{J}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})=\int{\mathnormal{\Psi}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}^{\ast}_{\mathrm{J}}\times\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})d\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}.$
(25)

Of course, this equation applies to
${\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}}$ itself, which uses the
$\infty$ window.

The most common of these marginalized intensities is the rate that objects
form, a function over time, and possibly other fundamental variables like
frequency. When $\mathnormal{\vartheta}_{\mathrm{J}}$ is the time an object is
born, the rate is

${\breve{\mathnormal{\Gamma}}_{\mathrm{K}}^{\mathrm{J}}}(t,\nu{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})=\frac{d\left\langle{\mathnormal{N}_{\upnu,\mathrm{K}}^{\mathrm{J}}}(\nu{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})\right\rangle}{d\mathnormal{\vartheta}_{\mathrm{J}}}(\mathnormal{\vartheta}_{\mathrm{J}}=t),$
(26)

where
${\mathnormal{N}_{\upnu,\mathrm{K}}^{\mathrm{J}}}(\nu{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})$
is the number of objects hosted by ${\mathrm{K}}$ and selected by a window
picking all objects that cover the frequency $\nu$. Broadcasts have limited
bandwidth, but other types of objects (including societies, metasocieties, and
galaxies) are not confined by frequency and have
${\mathnormal{N}_{\upnu,\mathrm{K}}^{\mathrm{J}}}(\nu{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})={\mathnormal{N}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$.
If the number of selected objects is proportional to the number of stars
(e.g., more stars means more inhabited worlds), a convenient variable is the
stellar rate,

${\mathnormal{\Gamma}_{\mathrm{K}}^{\mathrm{J}}}(t,\nu{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})={\breve{\mathnormal{\Gamma}}_{\mathrm{K}}^{\mathrm{J}}(t,\nu|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})/\left\langle\mathnormal{N}_{\mathsf{t},\mathrm{K}}^{\star}(t,\boldsymbol{\mathnormal{w}}_{\mathrm{K}})\right\rangle},$
(27)

where
${\mathnormal{N}_{\mathsf{t},\mathrm{K}}^{\star}}(t{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})$
is the number of stars extant at time $t$ within the host ${\mathrm{K}}$.

If ${\mathnormal{\Psi}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}}$ is marginalized
over all parameters, then the resulting zero-dimensional distribution simply
gives
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}}\right\rangle$
as per Campbell’s formula (equation 4). This itself can be regarded as a
parameter, with
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}}\right\rangle$
defined for some well-characterized window ${\mathsf{x}}$ scaling any other
distribution. For objects that trace stars, the most useful of these scaling
parameters is the stellar abundance

${\mathnormal{\Xi}_{\mathrm{K}}^{\mathrm{J}}}(t,\nu{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})=\left\langle{\mathnormal{N}_{\mathsf{t}\cap\upnu,\mathrm{K}}^{\mathrm{J}}}(t,\nu{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})\right\rangle/\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{K}}^{\star}}(t{,\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})\right\rangle,$
(28)

with $\mathsf{t}\cap\upnu$ picking all objects active at both time $t$ and,
for broadcasts, frequency $\nu$. It is the mean number of objects per star
that “cover” a point on a spectrogram.

###  3.3 Random variables

####  3.3.1 Singleton variables

A _singleton variable_ $\mathnormal{\kappa}_{\mathrm{J}}$ is a random variable
that describes a single object ${\mathrm{J}}$.202020TTechnically, for a random
object ${\mathrm{J}}$,
$\mathnormal{\kappa}_{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$
is a random function that outputs a random variable from the input
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$ (Section 3.1.3). The same caveat
applies to aggregate “variables” with random hosts. I assume that it is
independent of all other objects, depending only on
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. Examples include position of an
object, luminosity, flux received at Earth, and total number of photons
collected in a telescope from that object in one pixel of a detector. The
value of a singleton variable is not always determined by
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$, only its distribution is. The
number of photons counted even from a perfectly coherent laser beacon has
detector and shot noise, for example.

The basic notation for singleton variables is
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$, where $\mathnormal{\kappa}$ may
be replaced with another symbol, usually a lowercase letter. In some cases, a
random variable depends on how long we are observing, in what frequencies, at
what polarizations, as is true for emission measured. The _quantity window_
${\mathsf{x}}$ sets the bounds of integration for $\mathnormal{\kappa}$
(Figure 5). Generally, an object with no activity or emission within this
window is not selected by ${\mathsf{x}}$,

$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}=0~{}{\Rightarrow}~{}\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\notin\mathnormal{W}_{\mathsf{x}}^{\mathrm{J}}{,}$
(29)

although an object can be excluded for other reasons. When no quantity window
is specified, the ALL window implicitly is used
($\mathnormal{\kappa}_{\mathrm{J}}=\mathnormal{\kappa}_{\infty|\mathrm{J}}$).
The basic parameters for each object can be interpreted as special singleton
variables that naturally use the ALL window; they are intrinsic properties,
unaffected by observation.

A calculation with singleton variables over a population
${\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$
is performed by marking each point with the value of
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}$.
As noted in Section 2.2, this is equivalent to considering a point process in
$\mathnormal{W}^{\mathrm{J}}\times\mathbb{R}$ for a real-valued random
variable. As long as the marks are independent of any and all other points in
the process, the intensity of this new process is

$\frac{d^{2}\left\langle{\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle}{d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}d\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}}(\boldsymbol{\mathnormal{w}},\mathnormal{\kappa}{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}})={\mathnormal{\Psi}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}}(\boldsymbol{\mathnormal{w}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}}){\cdot}\uppsi[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}](\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}).$
(30)

####  3.3.2 Aggregate variables

An _aggregate variable_ $\mathnormal{K}$ is the sum of the singleton variables
for a sample of objects drawn from a host ${\mathrm{K}}$:

${\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}\equiv{\sum_{\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}}\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})},$
(31)

where $\mathnormal{K}$ may be replaced with another symbol, usually an
uppercase letter. In addition to the quantity window ${\mathsf{x}}$, a
potentially distinct _object window_ is combined with the host ${\mathrm{K}}$
to define a selection that picks out which of the objects included in a sum.
Each ${\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}$ is
treated using the marked point process for the corresponding
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$.

To simplify notation, the quantity window may be omitted to indicate that it
is the same as the object window:
$\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}\equiv\mathnormal{K}_{\mathsf{x}|\mathsf{x},\mathrm{K}}^{\mathrm{J}}$
(Table 5). Applying equation 29 allows us to “expand” or “contract” the object
window:

${{\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}=\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}~{}\text{if}~{}(\forall\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{W}_{\mathsf{x}}^{\mathrm{J}})~{}\mathsf{P}_{\mathsf{x}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})=\mathsf{P}_{\mathsf{y}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}.$
(32)

Aggregate and singleton variables can be interconverted,

${\mathnormal{\kappa}_{\mathsf{y}|\mathrm{K}}\leftrightarrow\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}},$
(33)

where the object window of a singleton variable is always identical to its
quantity window. These simply reflect different ways of viewing the quantity:
the aggregate variable interprets it as a sum over a population, while the
singleton variable interprets it as an intrinsic property of the host. A
common example in extragalactic astronomy is the mass of a galaxy’s stars –
whether it is viewed as the collective property of a stellar population or a
property of a single galaxy, it is still the same quantity. This lets us build
high-level aggregates recursively, like the total emission from all broadcasts
in all societies in a galaxy.

Number variables can be interpreted as a special case of aggregate variables:

${\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}={\sum_{\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}}1=|{\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}|.$
(34)

###  3.4 Selection-relative probabilities, means, and variances: The effects
of selection bias

A distribution operation ${\mathbb{O}}$ takes a variable and returns a
function or number derived from its probability distribution – it needs a
random variable, and cannot stand on its own. These operations include the
CDF, PDF, mean, and variance. In this section, I consider how these operations
are affected by selection biases.

####  3.4.1 Selection-relative probabilities

Let us say we want to know the probability of an event
$\mathnormal{E}_{\mathrm{J}}$ describing some property of
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$. Now, this probability is
associated with a random object ${\mathrm{J}}$ and thus is actually a random
function of $\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. Our general assumption
has been that the probability distribution of any random variable
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ is fully determined by
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$, automatically implying
independence between other such variables and the number of objects. This is
the simple probability of the event,
$\mathnormal{P}(\mathnormal{E}_{\mathrm{J}})\equiv\mathnormal{P}(\mathnormal{E}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$.

In turn, the probability $\mathnormal{P}(\mathnormal{E}_{\mathrm{J}})$ is also
a function of $\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. However, any
distribution for the $\mathrm{J}$ objects themselves allows us to generate a
mixture distribution for $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$, and
then we can evaluate the probability of $\mathnormal{E}_{\mathrm{J}}$ for this
mixture. This includes distributions modified by selection windows. When the
value of a random variable $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ for a
single object is independent of all other objects, the selection-relative
probability is thus defined

${\mathnormal{P}_{\mathsf{y},\mathrm{K}}}(\mathnormal{E}_{\mathrm{J}})\equiv\frac{1}{\left\langle{\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle}\int_{\mathnormal{W}_{\mathsf{y}}^{\mathrm{J}}}{\mathnormal{\Psi}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}\mathnormal{P}(\mathnormal{E}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}.$
(35)

This is the probability that $\mathnormal{E}_{\mathrm{J}}$ occurs for a
$\mathrm{J}$-type object fairly drawn from the random sample
$\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$.
It is an ensemble probability, averaging over all possible selections, not
just for a realized sample. It reflects the bias of a selection.

An archetypal event is $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ being
within some specified range. The previous equation immediately yields a
selection-relative CDF, with
${F\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}(x)\equiv\mathnormal{P}_{\mathsf{y},\mathrm{K}}(\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\leq
x)}$. Differentiating then gives the selection-relative PDF,

${\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}}({\mathnormal{\kappa}})\equiv\\\
\frac{1}{\left\langle{\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle}\int_{\mathnormal{W}_{\mathsf{y}}^{\mathrm{J}}}{\mathnormal{\Psi}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}\uppsi[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}]({\mathnormal{\kappa}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})d\boldsymbol{\mathnormal{w}}_{\mathrm{J}},$
(36)

where $\uppsi[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}]$ is the (unbiased)
simple probability distribution for
$\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ determined by
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. An analogous equation can be
written for aggregate variables.

Note that the objects sampled by a selection are random. Thus, selection-
relative probabilities and operations are used on things describing random
objects, although the selection itself may have a realized host.212121IIf for
some reason a selection-relative operation were applied to a variable
describing a realized object, the selection would drop out, and it would
become a simple operation described in the next subsection (e.g.,
$\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{j}}\right]_{\mathsf{y},\mathrm{K}}=\uppsi[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{j}}]$).

####  3.4.2 A notational convention for simple and selection-relative
operations

The simple probabilities are selection-relative probabilities where the self-
point process is being used. This rule extends to all distribution operations
on the probability:

$\displaystyle\mathbb{O}[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}]$
$\displaystyle=\mathbb{O}[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}}]=\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{x},\mathrm{J}}$
$\displaystyle\mathbb{O}[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}]$
$\displaystyle=\mathbb{O}[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}]=\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}$
(37)

including mean, variance, and order statistics.

A similar simplification is used in this series for windows – a selection-
relative distribution operation “missing” its window inherits it from the
variable’s object window (which is the quantity window too for singleton
variables):

${\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathrm{L}}}\equiv{\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{x},\mathrm{L}}}~{}\text{and}~{}{\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathrm{L}}}\equiv{\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{y},\mathrm{L}}},$
(38)

including mean and variance (Table 5). Contrast with the convention for
variables, which stand on their own and have no other variable to draw a
window from. This convention is motivated by practicality: in most windows we
use, the vast majority of objects do not contribute to aggregate quantities
(equation 29) and do not concern us. When we want to know the mean broadcast
fluence over an observation, we do not want to include those that were only
visible during the Jurassic, for instance.222222If we _do_ want to include
these objects, we apply the ALL window operator,
${\mathbb{O}\left[X\right]_{\infty,\mathrm{K}}}$ (compare Figure 5). Remember,
if the variable itself does not specify a window, it uses the ALL ($\infty$)
window (Section 3.2.1), which the mean then inherits
(${\mathbb{O}\left[\mathnormal{\kappa}_{\mathrm{J}}\right]_{\mathrm{K}}}={\mathbb{O}\left[\mathnormal{\kappa}_{\infty|\mathrm{J}}\right]_{\infty,\mathrm{K}}}$).
This applies to the intrinsic quantities describing objects like their
lifespan and position.

####  3.4.3 Simple means for a single object

The most straightforward averaging operation is the simple mean. Say we have a
singleton quantity $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$ associated
with a random object of type $\mathrm{J}$ with a parameter tuple
$\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. Its simple mean is:

$\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle\equiv\left\langle{\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\right\rangle=\int{\mathnormal{\kappa}}\cdot\uppsi[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}](\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})d\mathnormal{\kappa}.$
(39)

This is a function of $\boldsymbol{\mathnormal{w}}_{\mathrm{J}}$. The simple
mean of an aggregate variable is, with the help of equation 33,

$\displaystyle\left\langle\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right\rangle$
$\displaystyle\equiv\left\langle\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})\middle|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}\right\rangle$
$\displaystyle=\int{\mathnormal{K}}\cdot\uppsi[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}](\mathnormal{K}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})d\mathnormal{K}.$
(40)

Recall that $\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$ does _not_ determine
the subpopulation of ${\mathrm{K}}$’s descendants, only its statistical
properties. The mean is therefore an average over all realizations of this
subpopulation, conditionalized on the properties of the host. This sampling
variance within the host is regarded as “noise.”

The average is found by applying Campbell’s formula (equation 4) to the
intensity of the marked point process:

$\left\langle{\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle=\int_{\mathnormal{W}_{\mathsf{y}}^{\mathrm{J}}}\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle{\mathnormal{\Psi}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}.$
(41)

The integral for
$\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle$ in
equation 39 is just a special case of this, summing over
$\mathnormal{\Sigma}_{\mathrm{J}}^{\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}$.

####  3.4.4 Selection-relative means

But what if we want to average, not just over the “noise” in a random
variable, but over different possible “draws” of its associated object? This
is what a selection-relative mean does (Figure 5). For a selection
$(\mathsf{z},{\mathrm{L}})$, the selection-relative mean is

$\displaystyle{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{z},\mathrm{L}}}$
$\displaystyle\equiv\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in{\mathnormal{\Sigma}_{\mathsf{z},\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}\right\rangle$
$\displaystyle{\left\langle\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right\rangle_{\mathsf{z},\mathrm{L}}}$
$\displaystyle\equiv\left\langle\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}\middle|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}\in{\mathnormal{\Sigma}_{\mathsf{z},\mathrm{L}}^{\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}\right\rangle.$
(42)

Means with respect to realized hosts have analogous definitions. Unlike the
simple means, the selection-relative means averages over all the different
possible parameter tuples for the object being described by the random
variable, weighted by the probability of it being a part of the sample drawn
by the selection. It is an _ensemble_ average, the mean that we would get if
we repeated the selection again and again.

To give a concrete example, suppose we had a society ${\mathrm{c}}$ in a
galaxy ${\mathrm{g}}$, and we observe it over a window ${\mathsf{o}}$. The
total energy received from all broadcasts in that society over ${\mathsf{o}}$
is $\mathfrak{E}_{\mathsf{o},\mathrm{c}}^{\mathrm{B}}$. Then
$\left\langle\mathfrak{E}_{\mathsf{o},\mathrm{c}}^{\mathrm{B}}\right\rangle$
is the mean value of the energy we receive from the broadcasts in that
society, or more precisely, the average for an ensemble of societies each with
an identical parameter tuple $\boldsymbol{\mathnormal{w}}_{\mathrm{c}}$.
However,
$\left\langle\mathfrak{E}_{\mathsf{o},\mathrm{C}}^{\mathrm{B}}\right\rangle_{\mathrm{g}}$
is the mean value over all the possible societies in the galaxy, roughly the
value of $\mathfrak{E}_{\mathsf{o},\mathrm{C}}^{\mathrm{B}}$ expected from a
typical society.

Figure 5: Comparison of different quantity windows and selections applied to a
variable and an averaging operation. Here we have a cluster of
$\mathrm{J}$-type spherical objects in a host ${\mathrm{k}}$, observed in the
sky field as pictured at upper right. A survey ${\mathsf{s}}$ covers this
field with two observations, ${\mathsf{o}_{\mathsf{A}}}$ and
${\mathsf{o}_{\mathsf{B}}}$, each an aperture. Any object that touches a
window is selected by it. The subplots depict different multiwindow selection-
relative means, with simplified (above) and full (below) notation. The
underlying variable $\mathnormal{\kappa}$ is a quantity that is integrated
over its quantity window (e.g., starlight from a galaxy that falls within the
window), with the amount falling within its bounds indicated by the shading.
The means average this quantity over only some of the objects, the ones
outlined by rings. The multiwindow means on the bottom average over several
observations; ${\mathsf{o}_{i}}$ is a placeholder for
${\mathsf{o}_{\mathsf{A}}}$ and ${\mathsf{o}_{\mathsf{B}}}$. Note however that
the means are actually ensemble averages, over all possible configurations of
the objects, whereas only one realized configuration is shown. Other
operations like the variance and PDF also have multiwindow selection-relative
versions analogous to the means shown here.

The selection-relative mean is found using the selection-relative (biased) PDF
(equation 36). Assuming all the variables are mutually independent, equation
39 gives us

${\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{z},\mathrm{L}}}=\frac{1}{\left\langle{\mathnormal{N}_{\mathsf{z},\mathrm{L}}^{\mathrm{J}}}\right\rangle}\int_{\mathnormal{W}_{\mathsf{z}}^{\mathrm{J}}}\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle{\mathnormal{\Psi}_{\mathsf{z},\mathrm{L}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{L}})}d\boldsymbol{\mathnormal{w}}_{\mathrm{J}},$
(43)

with the selection-relative mean of an aggregate variable found with an
analogous integral (equation 33). A consequence of Campbell’s formula is that

$\left\langle{\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle=\left\langle{\sum_{\boldsymbol{\mathnormal{w}}_{\mathrm{J}}\in\mathnormal{\Sigma}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}}\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}\right\rangle=\left\langle{\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}}.$
(44)

Once completed, the averages may themselves be averaged, most naturally when
the inner average’s selection host is a random object.232323IIf the
operation’s host is a realized object ${\mathrm{k}}$, then the selection-
relative mean (or variance) has a fixed value determined by
$\boldsymbol{\mathnormal{w}}_{\mathrm{k}}$ (i.e., it is a degenerate random
variable); whereas, if the operation’s host is random object ${\mathrm{K}}$,
then it is a function that yields a single value for each
$\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$, or equivalently, the degenerate
random variable that always attains that value. As a random variable, this
variable inherits the object window and host from the averaging operator
itself: $\left\langle
X_{\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}\rightarrow{\mathnormal{\kappa}^{\prime}_{\mathsf{y}|\mathrm{K}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$.

Note that, generally,

${\left\langle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}\right\rangle_{\mathsf{z},\mathrm{L}}\neq\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{z},\mathrm{L}}},$
(45)

and similarly for aggregate variables. That is because the left-hand side
averages over subselections before averaging over the outermost selection.
Some subselections may have more $\mathrm{J}$-type objects than others, but
these are all weighted equally on the left-handed side, despite their
different population sizes. By evenly sampling the $\mathrm{K}$-type host
objects, the left-hand side fails to evenly sample the $\mathrm{J}$-type
objects like the right-hand side. This effect is related to Simpson’s Paradox
(Haigh, 2013). However, it _is_ true that

${\left\langle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle\right\rangle_{\mathsf{z},\mathrm{L}}=\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{z},\mathrm{L}}},$
(46)

because the inner mean’s selection necessarily has a population size of $1$ –
this is just the law of total expectation and an expression of Campbell’s
formula for marked point processes.

####  3.4.5 Simple and selection-relative variances

The simple variance of a variable is found using the distribution of that
variable that comes from its tuple. Selection-relative variances then average
over a distribution of random objects. Their definition follows from that of a
general conditional variance, $\mathbb{V}\left[X|Y\right]=\left\langle
X^{2}|Y\right\rangle-\left\langle X|Y\right\rangle^{2}$ (Wasserman, 2004; Bas,
2019). Thus ${\mathbb{V}\left[X\right]_{\mathsf{x},\mathrm{K}}=\left\langle
X^{2}\right\rangle_{\mathsf{x},\mathrm{K}}-\left\langle
X\right\rangle_{\mathsf{x},\mathrm{K}}^{2}}$.

The law of total variance is an important tool in evaluating both types of
variances. By first conditionalizing on
$\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$,
we derive the simple variance of
$\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$:

${\mathbb{V}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]=\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}^{2}\mathbb{V}\left[\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]+\left\langle\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right\rangle\mathbb{V}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}},}$
(47)

as long as all of the variables are independent. When
${\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$
for a fixed $\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$ is also Poisson, the
variance reduces to

${\left\langle\mathnormal{N}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}^{2}\right\rangle_{\mathsf{y},\mathrm{K}}=\int_{\mathnormal{W}^{\mathrm{J}}}\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}^{2}\right\rangle\mathnormal{\Psi}_{\mathsf{y},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})d\boldsymbol{\mathnormal{w}}_{\mathrm{J}}},$

equivalent to Campbell’s second formula (equation 10). The law also gives us,
for selection-relative variances,

${\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right]_{\mathsf{z},\mathrm{L}}=\mathbb{V}\left[\left\langle{\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right\rangle\right]_{\mathsf{z},\mathrm{L}}+\left\langle\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}}\right]\right\rangle_{\mathsf{z},\mathrm{L}}.}$
(48)

###  3.5 Multiwindow operations

An observing program is usually a collection of many observations: a survey
${\mathsf{s}}$ might take several pointings i, each in turn involving
thousands of distinct observations ${\mathsf{o}_{j}}$. The problem is that the
statistical properties of a sample may depend on which subwindow we are
considering. Differences between subwindows also introduce still more
variance. When mapping a spiral galaxy, for instance, the stars are far more
dense in the core than the outer disk, increasing the variance in star counts
because the underlying density changes. We need to be able to ask questions
like “What is the expected number of stars per observation averaged over all
observations in the survey?” and “What’s the maximum broadcast fluence during
an observation among all the observations we do?”

Just as the selection-relative operations have an object window that picks
which objects are included in the operation, the multiwindow operations have a
quantity window, which defines which subwindows are considered. The most
general operation is written
$\mathbb{O}\left[X\right]_{\mathsf{z}|\mathsf{y},\mathrm{K}}$, where
${\mathsf{z}}$ is its object window, and $(\mathsf{y},\mathrm{K})$ is its
selection.

For our purposes, there are a fixed number of subwindows with specified
properties. I define the PDF as a mixture of the PDFs for the individual
windows:

$\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{z}|\mathsf{y},\mathrm{K}}(\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})\equiv\frac{1}{\mathsf{N}_{\mathsf{z}}^{\mathsf{x}}}\sum_{i=1}^{\mathsf{N}_{\mathsf{z}}^{\mathsf{x}}}\uppsi\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}(\mathnormal{\kappa}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}}).$
(49)

The general multiwindow mean is

$\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{z}|\mathsf{y},\mathrm{K}}\equiv\left\langle\left\langle\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}|\mathsf{x}_{i}\in\mathsf{S}_{\mathsf{z}}^{\mathsf{x}}\right\rangle.$
(50)

We simply take the mean over the given selection for each object window, then
take the average over the object windows (Figure 5). The general multiwindow
variance is, by the law of total variance,

$\mathbb{V}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{z}|\mathsf{y},\mathrm{K}}\equiv\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}^{2}\right\rangle_{\mathsf{z}|\mathsf{y},\mathrm{K}}-\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{z}|\mathsf{y},\mathrm{K}}^{2}\\\
=\left\langle\mathbb{V}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}|\mathsf{x}_{i}\in\mathsf{S}_{\mathsf{z}}^{\mathsf{x}}\right\rangle+\mathbb{V}\left[\left\langle\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}|\mathsf{x}_{i}\in\mathsf{S}_{\mathsf{z}}^{\mathsf{x}}\right].$
(51)

The second term in the final equality is the variance due to the changes in
the population between ${\mathsf{x}_{i}}$ windows. Finally the general
multiwindow maximum is

$\mathrm{max}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{z}|\mathsf{y},\mathrm{K}}\equiv\mathrm{max}[\mathrm{max}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}|\mathsf{x}_{i}\in\mathsf{S}_{\mathsf{z}}^{\mathsf{x}}],$
(52)

and likewise for the minimum. Thus, the mean energy fluence from a broadcast
during a typical observation in a survey ${\mathsf{s}}$ among the broadcasts
the survey selects from a galaxy ${\mathrm{g}}$ is written
$\left\langle\mathfrak{h}_{\mathsf{o}_{i}|\mathrm{B}}\right\rangle_{\mathsf{s}|\mathsf{s},\mathrm{g}}$,
for instance.

The populations in different windows are not generally independent. Adjacent
windows can cover the same objects: the sidelobes of one pointing can
intersect the primary beam of another for instance. Sometimes the objects
themselves are extended, crossing into different windows, as when a frequency-
drifting narrowband broadcast cuts across many channels. The greatest effect
of this cross-correlation is on the maxima and minima because they reduce the
effective number of independent samples. If the subwindows are very broad, the
variance can be reduced as well, because the same objects are covered in all
pointings, averaging out each measurement.

When working with multiwindow operations, there is a distinction between
general unspecified windows and specific windows, similar to that between
random objects and realized objects. We can assign a unique label to a unique
window, like ${\mathsf{o}_{1}}$ or ${\mathsf{o}_{2}}$, and apply it to a
variable. This fixes a unique subwindow; the operation’s object window becomes
irrelevant because the subwindow is one of a kind. In other cases, we really
do want to consider the full gamut of subwindows in a collection, and to
emphasize this, we might use a subwindow label like ${\mathsf{o}_{i}}$, where
$i$ is a free variable. Most often, I assume that the statistical properties
of the population do not change appreciably from window to window. Then the
multiwindow selection-relative means and variances simply collapse to the
selection-relative means and variances, and the multiwindow maxima are a
simple application of extreme value theory (Appendix A). Only then can we
elide the distinction between unspecified and specified windows: any
observation is just an observation ${\mathsf{o}}$.

The multiwindow operations come with their own notational simplifications. The
quantity window can be inherited from the operation’s selection but not the
variable, while an operation labeled _solely_ by a window uses it as a
quantity window:

$\displaystyle{\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}}$
$\displaystyle\equiv{\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y}|\mathsf{y},\mathrm{K}}}$
$\displaystyle{\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y}}}$
$\displaystyle\equiv{\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y}|\mathsf{x}_{i},\mathrm{J}}}$
(53)

Table 5 summarizes all of the notation rules, and Figure 5 provides an
illustrated example. As before, the object window of an operation is inherited
from the variable, even when a separate quantity window is specified:

$\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y}|\mathrm{K}}\equiv\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}_{i}|\mathrm{J}}\right]_{\mathsf{y}|\mathsf{x}_{i},\mathrm{K}}.$
(54)

Table 5: Simplifications in notation for variables and operations

Singleton variable | Aggregate variable
---|---
Short | Full | Short | Full
Variables
$\mathnormal{\kappa}_{\mathrm{J}}$ | $\mathnormal{\kappa}_{\infty|\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})$ | $\mathnormal{K}_{\mathrm{K}}^{\mathrm{J}}$ | $\mathnormal{K}_{\infty|\infty,\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
|  | $\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}$ | $\mathnormal{K}_{\mathsf{x}|\mathsf{x},\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
|  | $\mathnormal{K}_{\mathsf{x}|\mathrm{K}}^{\mathrm{J}}$ | $\mathnormal{K}_{\mathsf{x}|\infty,\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
|  | $\mathnormal{K}_{\mathsf{x}}^{\mathrm{J}}$ | $\mathnormal{K}_{\mathsf{x}|\mathsf{x},\mathrm{U}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{U}})$
|  | $\mathnormal{K}_{\mathsf{x}|\mathsf{y}}^{\mathrm{J}}$ | $\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{U}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{U}})$
Operations on variables
$\mathbb{O}[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}]$ | $\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{x}|\mathsf{x},\mathrm{J}}$ | $\mathbb{O}[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}]$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{y}|\mathsf{y},\mathrm{K}}$
$\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathrm{K}}$ | $\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{x}|\mathsf{x},\mathrm{K}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathrm{L}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{y}|\mathsf{y},\mathrm{L}}$
$\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y}}$ | $\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y}|\mathsf{x},\mathrm{J}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{z}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{z}|\mathsf{y},\mathrm{K}}$
$\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{K}}$ | $\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y}|\mathsf{y},\mathrm{K}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{z},\mathrm{L}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{z}|\mathsf{z},\mathrm{L}}$
$\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y}|\mathrm{K}}$ | $\mathbb{O}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y}|\mathsf{x},\mathrm{K}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{z}|\mathrm{L}}$ | $\mathbb{O}\left[\mathnormal{K}_{\mathsf{x}|\mathsf{y},\mathrm{K}}^{\mathrm{J}}\right]_{\mathsf{z}|\mathsf{y},\mathrm{L}}$

Note. — The simplifications are applied from the inside out. Thus,
$\left\langle\mathnormal{\kappa}_{\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}\to\left\langle\mathnormal{\kappa}_{\infty|\mathrm{J}}\right\rangle_{\mathsf{y},\mathrm{K}}\to\left\langle\mathnormal{\kappa}_{\infty|\mathrm{J}}\right\rangle_{\mathsf{y}|\mathsf{y},\mathrm{K}}$.
When the host of a selection-relative operation (including a general
multiwindow operation) is a random object, the result should be understood as
a function of the host’s properties.

###  3.6 Regularization of variables with broad distributions

Like other astrophysical entities, ETIs and technosignatures may be described
by quantities with heavy-tailed distributions. The radiated energy is widely
thought to be a possible example (Drake et al., 1973); others may include the
lifespan of societies or transmitters, or the number of societies per
metasociety. Plausible distributions like shallow power laws can have
variances and means dominated by very rare events with extreme properties.
These objects may be too uncommon to detect in any practical selection,
however, and thus throw off estimates for the properties of a typical sample.

When needed, this series estimates mean and variance by trimming values of a
random variable that are far outside the usual range in a typical sample. This
regularization is done by imposing cutoff values $\mathnormal{\kappa}^{L}$ and
$\mathnormal{\kappa}^{H}$ based on extreme value statistics (Appendix A).
Starting from a random variable $\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}$
and trimming with respect to a selection $(\mathsf{y},{\mathrm{K}})$, the
regularized variable
${{\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}^{[\mathsf{y},\mathrm{K}]}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}};\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$
has a truncated distribution,

$\uppsi[{{\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}^{[\mathsf{y},\mathrm{K}]}}}](\mathnormal{\kappa})\equiv\displaystyle\frac{\uppsi[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}](\mathnormal{\kappa})\cdot\mathbb{I}[\mathnormal{\kappa}^{L}\leq\mathnormal{\kappa}\leq\mathnormal{\kappa}^{H}]}{{\mathnormal{P}_{\mathsf{y},\mathrm{K}}}(\mathnormal{\kappa}^{L}\leq\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\leq\mathnormal{\kappa}^{H})}.$
(55)

Equation 55 also regularizes aggregate variables, as they can be treated as
just a special case of singleton variables (equation 33). Because aggregate
variables are sums, the calculations are more complicated. If the individual
contributions have a narrow enough distribution that their full range is
expected to be well-sampled in each aggregate, any large fluctuations in the
sum must be due to fluctuations in the number of objects included:
$\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J};[\mathsf{y},\mathrm{L}]}\approx\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J};[\mathsf{y},\mathrm{L}]}\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{K}}$.
These fluctuations may be relatively small and the regularization has
essentially no effect. If the individual contributions have a very broad
distribution – when we are calculating the aggregate luminosity of objects
with a shallow power-law distribution, for example – the sum is almost
entirely dominated by the largest contribution in the sample and
$\mathnormal{K}_{\mathsf{x},\mathrm{K}}^{\mathrm{J};[\mathsf{y},\mathrm{L}]}\approx\mathrm{max}\left[\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right]_{\mathsf{y},\mathrm{L}}$.

The cutoff is a statistical quantity, however, and any actual sample may
include extreme outliers. Thus this regularization will underestimate the
sample mean and variance some of the time, possibly by a large degree in rare
instances.

###  3.7 The discreteness criterion

In order to make a detection of a broadcast in a survey ${\mathsf{s}}$, at
least one must be sampled during a program242424The converse is not always
true – a sampled broadcast may be too faint to detect, or confused with many
others., and a null result should occur whenever zero broadcasts are sampled.
It is possible our observation was too short, too narrowband, or had too small
a field to intercept rare broadcasts even when they are extremely bright. This
happens outside of SETI too: a supernova or a quasar episode can boost a
galaxy’s luminosity far above its typical value for a short time. If a
galaxy’s measured luminosity is fainter than a supernova or a quasar, that
just means one was not active during the observation, not that they never are
found there. Thus, a null result is compatible with any models with a high
enough probability of zero broadcasts being detected.

The discreteness criterion states that we should only expect a detection in
survey ${\mathsf{s}}$ with models having

$\mathnormal{P}(\mathnormal{N}_{\mathsf{s}}^{\mathrm{B}}=0)\leq{\bar{\mathnormal{P}}}$
(56)

for a suitably conservative false negative threshold $\bar{\mathnormal{P}}$
near $0$. Only models that fulfill this criterion are likely to be constrained
by observations. To order of magnitude, we expect a nonempty sample only if
the mean number of objects is $\gtrsim 1$.252525This follows from Markov’s
inequality, which gives
$\mathnormal{P}({\mathnormal{N}_{\mathsf{s}}^{\mathrm{J}}}\geq
1)\leq\left\langle{\mathnormal{N}_{\mathsf{s}}^{\mathrm{J}}}\right\rangle$ for
the nonnegative ${\mathnormal{N}_{\mathsf{s}}^{\mathrm{J}}}$ (Wasserman,
2004). However, no broadcasts will be detected if the sample is missing any of
the ancestor nodes on the “tree” – societies, metasocieties, or galaxies.
Equation 56 implies the often-simpler formula

${\left\langle\mathnormal{N}_{\mathsf{s}}^{\mathrm{K}}\right\rangle}\geq
1-{\bar{\mathnormal{P}}}\approx
1~{}{\text{if}~{}\mathrm{K}\in\aleph(\mathrm{B})~{}\text{or}~{}\mathrm{K}=\mathrm{B}.}$
(57)

###  3.8 An example: lifespan bias in time-limited selection

Long-lived objects are disproportionately represented in a field once a
population has reached equilibrium. This phenomenon serves as an example of
how to make selection-relative calculations.

Suppose we have objects of type $\mathrm{J}$ in a host ${\mathrm{K}}$. How is
the probability distribution of object lifespan affected by a time-limited
selection? To start, let us suppose that each such object forms at time
$\mathnormal{\vartheta}_{\mathrm{J}}$ and exists for duration
$\mathnormal{\tau}_{\mathrm{J}}$ without interruption. The distribution
function for these objects has the form

${\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}=\frac{d^{3}\left\langle{\mathnormal{N}_{\mathrm{K}}^{\mathrm{J}}}\right\rangle}{d\mathnormal{\tau}_{\mathrm{J}}d\mathnormal{\vartheta}_{\mathrm{J}}d\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$
(58)

where $\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}$ stands for a tuple
of other, irrelevant parameters to marginalize over. Now consider a window
${\mathsf{x}}$ that selects all objects in ${\mathrm{K}}$ active anytime
during the contiguous time interval
$[\mathsf{\Theta}_{\mathsf{x}},\mathsf{\Theta}_{\mathsf{x}}+\mathsf{T}_{\mathsf{x}}]$.
Clearly, an object in this population is selected if it either formed before
$\mathsf{\Theta}_{\mathsf{x}}$ and lives long enough to overlap the window
($\mathsf{\Theta}_{\mathsf{x}}-\mathnormal{\tau}_{\mathrm{J}}\leq\mathnormal{\vartheta}_{\mathrm{J}}<\mathsf{\Theta}_{\mathsf{x}}$)
or if it forms within the window
($\mathsf{\Theta}_{\mathsf{x}}\leq\mathnormal{\vartheta}_{\mathrm{J}}\leq\mathsf{\Theta}_{\mathsf{x}}+\mathsf{T}_{\mathsf{x}})$.
Thus, if the objects form at a constant rate
${\breve{\mathnormal{\Gamma}}_{\mathrm{K}}^{\mathrm{J}}}$ (equation 27),

$\displaystyle\frac{d\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}}\right\rangle}{d\mathnormal{\tau}_{\mathrm{J}}}{(\mathnormal{\tau}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$
$\displaystyle=\int_{\mathsf{\Theta}_{\mathsf{x}}-{\mathnormal{\tau}}}^{\mathsf{\Theta}_{\mathsf{x}}+\mathsf{T}_{\mathsf{x}}}\int{\mathnormal{\Psi}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{J}}|\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}d\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{J}}d\mathnormal{\vartheta}_{\mathrm{J}}$
$\displaystyle=({\mathnormal{\tau}}+\mathsf{T}_{\mathsf{x}}){\cdot\breve{\mathnormal{\Gamma}}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})\cdot}{\uppsi\left[\mathnormal{\tau}_{\mathrm{J}}\right]_{\mathrm{K}}(\mathnormal{\tau})}$
(59)

from equation 24, where
${\uppsi\left[\mathnormal{\tau}_{\mathrm{J}}\right]_{\mathrm{K}}}$ is the
unbiased lifespan distribution. It follows immediately that
${\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}\right\rangle=(\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}+\mathsf{T}_{\mathsf{x}})\cdot\breve{\mathnormal{\Gamma}}_{\mathrm{K}}^{\mathrm{J}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})}$,
giving us the ${\mathsf{x}}$-biased duration distribution (equation 36)

${\uppsi\left[\mathnormal{\tau}_{\mathrm{J}}\right]_{\mathsf{x},\mathrm{K}}(\mathnormal{\tau})}=\frac{{\mathnormal{\tau}}+\mathsf{T}_{\mathsf{x}}}{{\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}}+\mathsf{T}_{\mathsf{x}}}{\uppsi\left[\mathnormal{\tau}_{\mathrm{J}}\right]_{\mathrm{K}}(\mathnormal{\tau})}.$
(60)

Finally, we have (equation 43),

${\left\langle
f(\mathnormal{\tau}_{\mathrm{J}})\right\rangle_{\mathsf{x},\mathrm{K}}}=\frac{{\left\langle\mathnormal{\tau}_{\mathrm{J}}f(\mathnormal{\tau}_{\mathrm{J}})\right\rangle_{\mathrm{K}}}+\mathsf{T}_{\mathsf{x}}{\left\langle
f(\mathnormal{\tau}_{\mathrm{J}})\right\rangle_{\mathrm{K}}}}{{\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}}+\mathsf{T}_{\mathsf{x}}}.$
(61)

These formulae assume that the rate has been constant infinitely far into the
past, at least in that each object’s lifespan is shorter than the age of
${\mathrm{K}}$. Otherwise, an additional cutoff is needed for the integration
in equation 59, as is well known in the field of stellar populations (e.g.,
Miller & Scalo, 1979). In the limit that all the objects are more short-lived
than the host’s age at the start of ${\mathsf{x}}$,
$\mathsf{\Theta}_{\mathsf{x}}-\mathnormal{\vartheta}_{\mathrm{K}}$, equation
61 still applies. On the other hand, if most all the $\mathrm{J}$ objects live
longer than
$\mathsf{\Theta}_{\mathsf{x}}-\mathnormal{\vartheta}_{\mathrm{K}}$, it can be
shown the bias disappears: almost every object that has ever been born is
still around, so we get a fair sample of them.

Figure 6: Lifespan bias by a window ${\mathsf{x}}$ in a population of objects
with exponentially distributed lifespans. Selected objects are in blue (bright
blue within the window). On the left, the objects are scattered in spacetime,
and their lifespans are intervals in time. Those with longer
$\mathnormal{\tau}_{\mathrm{J}}$ are more likely to reach the window, which
covers a fixed span of real time (blue shading). On the right, the objects are
represented by points in a simplified haystack (spatial position not
included). The window covers a bigger $\mathnormal{\vartheta}_{\mathrm{J}}$
range as $\mathnormal{\tau}_{\mathrm{J}}$ increases.

When the observational window is short compared to typical lifespans, equation
61 effectively weights the biased average by another power of
$\mathnormal{\tau}_{\mathrm{J}}$. Every biased moment depends on the next
higher unbiased moment, save the zeroth where a cancellation occurs. For the
mean lifespan in particular,
${\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{K}}}={\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}}+{\mathbb{V}\left[\mathnormal{\tau}_{\mathrm{J}}\right]_{\mathrm{K}}}/({\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}}+\mathsf{T}_{\mathsf{x}})\geq{\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}}$.
Hence, one can replicate the result in Kipping et al. (2020) that
${\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{K}}}\approx
2{\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}}$ for
exponentially distributed $\mathnormal{\tau}_{\mathrm{J}}$. Or, one can show
that the lifespan’s selection-relative mean is ill-behaved for a power-law
distribution
${\uppsi\left[\mathnormal{\tau}_{\mathrm{J}}\right]_{\mathrm{K}}}\propto\mathnormal{\tau}_{\mathrm{J}}^{-\alpha}$
extending to infinity unless $\alpha>3$.

Lifespan bias can be thought of in two ways (Figure 6). In real spacetime,
“bigger” objects are more likely to cross into a window with fixed duration,
having a larger “cross-section.” In the haystack, the window itself becomes
wider for longer lifespans and thus has a longer reach for these objects. A
similar bias applies to band-limited selection by frequency: wideband
broadcasts are more likely to be intercepted, a significant fact in the chord
model presented later.

Figure 6 helps illustrate the different types of means. Each dot represents a
specific, realized object;
$\left\langle\mathnormal{\tau}_{\mathrm{j}}\right\rangle$ is the simple mean
of the lifespan for that object ${\mathrm{j}}$, which is entirely determined
by the dot’s position in the haystack. Then
$\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle=\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{J}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}$
is a _function_ ranging over the entire space that returns the average
lifespan for an object given its parameters – it returns the
$\mathnormal{\tau}_{\mathrm{J}}$ coordinate in the right-hand panel. A
realized host ${\mathrm{k}}$ has an accompanying distribution of objects over
the haystack:
$\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{k}}$
averages over the entire haystack, while
$\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{k}}$
averages only over the blue-shaded region. One such realization of the
distribution is shown in the figure. Likewise, for a random host
${\mathrm{K}}$, the distribution itself is a function of
$\boldsymbol{\mathnormal{w}}_{\mathrm{K}}$, as are
$\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$
and
$\left\langle\mathnormal{\tau}_{\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{K}}(\boldsymbol{\mathnormal{w}}_{\mathrm{K}})$.

##  4 Assumptions about randomness

Technosignatures are the result of complex biological, social, and
technological phenomena, with a panoply of factors shaping their populations.
The full interplay responsible for broadcast characteristics may be
analytically intractable. Nonetheless, treating broadcasts as the result of a
hierarchy of conditionally independent point processes illuminates our
understanding of the ETI populations.

###  4.1 “Random” ETIs

Random variables are a very general concept (Section 2.1) and can describe