Research Paper

McCullen Sandora

# Information Gain as a Tool for Assessing Biosignature Missions

Benjamin Fields1,2† Sohom Gupta1,3† McCullen Sandora1∗ †Equal contribution
1Blue Marble Space Institute of Science 2Wheaton College and 3Indian Institute
of Science Education and Research Kolkata<EMAIL_ADDRESS>
(xx xxxx xxxx; xx xxxx xxxx; xx xxxx xxxx)

###### Abstract

We propose the mathematical notion of _information gain_ as a way of
quantitatively assessing the value of biosignature missions. This makes it
simple to determine how mission value depends on design parameters, prior
knowledge, and input assumptions. We demonstrate the utility of this framework
by applying it to a plethora of case examples: the minimal number of samples
needed to determine a trend in the occurrence rate of a signal as a function
of an environmental variable, and how much cost should be allocated to each
class of object; the relative impact of false positives and false negatives,
with applications to Enceladus data and how best to combine two signals; the
optimum tradeoff between resolution and coverage in the search for lurkers or
other spatially restricted signals, with application to our current state of
knowledge for solar system bodies; the best way to deduce a habitability
boundary; the optimal amount of money to spend on different mission aspects;
when to include an additional instrument on a mission; the optimal mission
lifetime; and when to follow/challenge the predictions of a habitability
model. In each case, we generate concrete, quantitative recommendations for
optimising mission design, mission selection, and/or target selection.

###### keywords:

biosignatures, statistics, astrobiology

††journal: International Journal of Astrobiology

##  1 Introduction

The overarching goal of the field of astrobiology is to search for signs of
life elsewhere in the universe. As it is a highly interdisciplinary endeavour,
the strategies proposed to achieve this are multifaceted, spanning a range of
solar system and exoplanet targets and detection techniques. As such, it is
sometimes challenging even for seasoned practitioners to assess the value of
upcoming missions. Recently, there have been efforts to address this. The
Ladder of Life Detection [37], Confidence of Life Detection score [20], and
Standards of Evidence framework [34] aim to provide a language for
communicating the evidential strength of any observed biosignature, both to
the public and amongst scientists. The nine axes of merit has been proposed as
a framework for describing the utility of any technosignature detection
mission in [49]. [32] provides a risk versus payoff framework for life
detection missions, which even factors in the mission success probability.

While many of these frameworks are fantastic at isolating the relevant mission
aspects necessary for judging the merits of a mission, they are fundamentally
descriptive processes, and leave up to the judgement of the practitioner how
to prioritise each mission dimension. As such, they are not suited for any
final decision making process, whether it be selecting between missions or
optimising mission design.

In this note we propose a simple method of assessing mission value in terms of
_information gain_ , and demonstrate the utility of this formalism through
considering several idealised mission designs. This framework allows to distil
the specifics of a mission down to a single number, which is useful for many
purposes. At the outset, we should acknowledge the shortcomings of this
procedure: clearly, there is no unique way of ranking missions; our approach
is capable of providing one such quantification, but other factors are surely
important. While our approach is encompasses many mission aspects, it
specifically disregards, for instance, any ancillary benefits a mission would
provide apart from biosignature detection. Secondly, we wholeheartedly disavow
mistaking performance metrics for the indefinite concept of mission value. Any
practitioner interested in using a method like this should be well aware of
the dangers inherent in using performance metrics for decision making- this is
exemplified in Goodhart’s law, whereby when a quantitative proxy is used to
drive decision making, there is a tendency for people to optimise for the
proxy itself, sometimes to the detriment of the original intended goal [10].
As such, we do not endorse a plan to ‘maximise information gain’, nor do we
believe it would be particularly useful to very precisely calculate the
information of any particular mission. The intent of our framework is to
provide a rough heuristic that is easily calculable in a manner that makes the
dependence on mission parameters highly apparent, so that these can be quickly
and straightforwardly varied to determine their effect on the overall mission
value.

In the remainder of this section we go into more detail on the process of
calculating information gain for a given mission setup. In the following
sections, we demonstrate the utility and flexibility of this approach by
applying it to several disparate scenarios, and deriving concrete,
quantitative recommendations for optimising mission design. In section 2 we
apply this formalism to the challenge of determining whether the occurrence
rate of a biosignature depends on some system variable. In section 3 we
consider the effects false positives and false negatives have on information.
In section 4 we consider a tradeoff between resolution and coverage in
searching for a biosignature. Section 5 contains more applications, including:
how to best determine the limiting value of some system parameter for life,
how to apportion budget to various aspects of a mission, when to include an
instrument on a mission, how to determine the optimal mission lifetime, and
when to follow a habitability model, and when to challenge it.

###  1.1 Bayes Estimation and Information gain

In this work we propose to operationalize the dictum of many proposed
missions, to “maximise science return”, in the following way- by equating
science return with _information gain_ , in the information theoretic sense,
we may write the science return as

$S=\int df\,p(f)\,\log p(f)$ (1)

This explicitly incorporates a value for how to reduce raw mission data, in
the form of a bit stream, down to the distilled representation relevant for
our immediate purposes. As such, this framework is more general, and can
suitably be applied to a variety of scientific goals across disciplines.

As a simple application of the above framework, let us imagine the simplest
setup, with a single biosignature, in the absence of false positives and
negatives, and drawing from a uniform population. We wish to estimate the
fraction of locations possessing this biosignature $f$, given a total number
of systems surveyed by a mission $N_{\text{tot}}$, which returns
$N_{\text{det}}$ detections. If the occurrence rate were known, the number of
detections would follow a Bernoulli distribution, and so when the number of
detections is known instead, the signal occurrence rate $f$ follows the
conjugate distribution $f\sim\beta(N_{\text{tot}},N_{\text{det}},f)$, where
$\beta$ refers to the Beta distribution111This is dependent on the choice of
the prior distribution for $f$, which we take here to be uniform. However,
using other priors does not change the behaviour of the population mean and
variance by much[47]. [47].

$\displaystyle\beta(N_{\text{tot}},N_{\text{det}},f)$ $\displaystyle=$
$\displaystyle
B(N_{\text{tot}},N_{\text{det}})\,f^{N_{\text{det}}}(1-f)^{N_{\text{tot}}-N_{\text{det}}}$
$\displaystyle B(N_{\text{tot}},N_{\text{det}})$ $\displaystyle=$
$\displaystyle\frac{(N_{\text{tot}}+1)!}{N_{\text{det}}!(N_{\text{tot}}-N_{\text{det}})!}$
(2)

This distribution has the following mean and variance:

$\mu=\frac{N_{\text{det}}+1}{N_{\text{tot}}+2},\hskip
7.22743pt\sigma^{2}=\frac{(N_{\text{det}}+1)(N_{\text{tot}}-N_{\text{det}}+1)}{(N_{\text{tot}}+2)^{2}(N_{\text{tot}}+3)}$
(3)

If the mean is well separated from 0, this has $\mu\rightarrow
f_{o}=N_{\text{det}}/N_{\text{tot}}$ and $\sigma^{2}\rightarrow
f_{o}(1-f_{o})/N_{\text{tot}}$. If the mean is consistent with 0 to observed
uncertainty, $\mu\rightarrow 1/N_{\text{tot}}$ and $\sigma^{2}\rightarrow
1/N_{\text{tot}}^{2}$. While simple, already this setup indicates how
precision depends on survey size, for multiple different regimes. This can be
related to science yield (information gain) through eqn. (1).

For a broad class of distributions, including the setup above, information
reduces to $S\approx\log\sigma$, so maximising information gain becomes
equivalent to minimising variance, which sets the measurement uncertainty. In
this limit, 1 bit of information is gained if the error bars for the measured
quantity are decreased by a factor of 2.

The above expression assumes initial ignorance. For the case where we are
updating knowledge obtained from a previous mission, we can instead use the
Kullback-Leibler (KL) divergence as the generalisation:

$\Delta S=KL(p||q)=\int df\,p(f)\,\log\left(\frac{p(f)}{q(f)}\right)$ (4)

This is a measure of information gain as one moves from an initial
distribution $p$ to a final one $q$. This has the property that $KL(p||q)\geq
0$, with equality only if $p=q$, guaranteeing that information is always
gained with new missions. This reduces to the prior case if we take the
initial distribution $q$ to be uniform.

In the case where we are updating our knowledge of biosignature occurrence
rate from a previous mission, both subject to the conditions outlined above,
we have $p\sim\beta(N_{\text{tot}},N_{\text{det}},f)$ and
$q\sim\beta(N_{\text{tot}}^{(0)},N_{\text{det}}^{(0)},f)$ then we can find the
exact formula for information gain from [44]:

$\displaystyle\Delta S$ $\displaystyle=$
$\displaystyle\log\frac{B\left(N_{\text{tot}},N_{\text{det}}\right)}{B\left(N_{\text{tot}}^{(0)},N_{\text{det}}^{(0)}\right)}$
(5)
$\displaystyle+\Delta(N_{\text{tot}}-N_{\text{det}})\psi(N_{\text{tot}}-N_{\text{det}}+1)$
$\displaystyle+\Delta N_{\text{det}}\psi(N_{\text{det}}+1)-\Delta
N_{\text{tot}}\psi(N_{\text{tot}}+2)$

where $\psi(x)=\frac{d\Gamma(x)}{dx}/\Gamma(x)$ is the digamma function. Using
the approximation $\psi(x)\sim\log x-\frac{1}{2x}$, we can see that this is a
generalisation of the formula $\Delta S\approx\log\sigma$ to situations where
we start with some information. Additionally, in the limit where the sample
size is incremented by 1 in an updated experiment, we find $\Delta S\approx
1/N_{\text{tot}}$.

This framework can be compared to other, more rigorous statistical methods of
estimating survey efficacy. If one had data from a set of observations, the
usual procedure would be to perform a battery of statistical tests (eg, Neyman
Pearson, binomial, kernel tests) aimed at determining the precise degree to
which the tested hypothesis is (dis)favoured compared to some null hypothesis.
While rejecting the null hypotheses to some specified confidence level (0.05
is a standard choice) is suitable for large data sets, this procedure must be
treated with care when the data samples are small and difficult to obtain
[30]. A well-known way for estimation of parameters is to use the method of
moments [7], which yields potentially biased but consistent estimators.
Another method is bootstrapping from existing data to estimate the parameters
of the parent distribution via the plug-in principle [36]. However, these are
often of little help for determining the amount of data needed to gain a
specified level of significance, nor which targets we should favor when
collecting this data. Modelling the output of future missions can help to
proactively determine the amount needed, and the authors regard this as a
necessary step of the mission planning process, but this is laborious, must be
redone from scratch for each new mission, and, in our opinions, does not
always lead to an enlightening understanding of why a certain amount of data
is required. Our framework aims to augment these existing procedures and guide
intuition by showing how back of the envelope calculations can lead us to
concrete, actionable recommendations for mission design.

We now demonstrate the wider applicability of our framework by applying it to
extensions of these simple setups.

##  2 Ratio of 2 Signals

As a first application of our framework, we turn our attention to the case
where we want to determine how the occurrence rate of a biosignature depends
on some system parameter. This is a generalisation of the analysis done in
[47], which made the idealisation that all systems being measured were equally
probable of hosting life. Here, we are interested in determining the minimum
number of systems required to be observed to detect a population trend, and
how best to allocate mission time toward systems of varying characteristics to
most effectively distinguish whether a trend exists.

This analysis has many applications. Indeed, we may expect the probability of
biosignatures to depend on planet mass, stellar mass, incident irradiation,
planet temperature, age, metallicity, etc. For illustrative purposes, we focus
our attention on a particular case, the system’s position within our galaxy,
though the analysis will be general enough to easily adapt to any of the other
quantities mentioned.

There are numerous schools of thought on how habitability may depend on
galactic radius (for a review, see [24]). Metallicity (the abundance of
elements higher than hydrogen) is observed to decrease with galactic radius
[12], which may either be a boon or bane for life. On the one hand, a minimum
amount of material is required for planet formation [25], and planets are
expected to be more abundant around more metal-rich stars [4]. On the other
hand, the prevalence of hot Jupiters also has been shown to increase with
metallicity and, depending on the formation mechanism, could preclude the
presence of terrestrial planets in the habitable zone [14]. Additionally,
galactic centres (usually) host an active galactic nucleus (AGN), which spews
high energy radiation that can strip planetary atmospheres [24]. These two
competing factors have lead to the hypothesis that there is a galactic
habitable zone [31], analogous to the circumstellar habitable zone, comprised
of a possibly thin annular region, only within which habitable planets can
exist. This hypothesis has drawn some criticism, not least because stars are
known to migrate radially throughout the course of their evolution, defying
simple attempts at placing their orbit [43].

Here, we treat the GHZ as a hypothesis, and ask what sort of data we would
need to either verify or falsify it. We may divide the alternatives into four
scenarios: (i) null hypothesis- the biosignature occurrence rate does not
depend on galactic radius (ii) Z- the occurrence rate is greater toward the
center (iii) HJ- the occurrence rate is greater toward the edge, and (iv) AGN-
the occurrence rate is 0 inside some critical radius.

A fully statistically rigorous approach would do something along the lines of
fitting a curve for occurrence rate versus galactic radius, then determining
to what degree of certainty the slope is away from 0. Here, we opt for a much
simpler method of bucketing our population into two bins, then comparing the
occurrence rates of each bin and determining if they are different from each
other to the statistical power afforded by the mission. While this simplified
analysis is not a replacement for the full approach, it allows us to easily
track the dependence on mission parameters, and so can be useful for planning
purposes.

We therefore assume we have two populations, either from the same or different
missions. If mission 1 surveys $M_{\text{tot}}$ planets and detects
$M_{\text{det}}$ signals, and mission 2 surveys $N_{\text{tot}}$ planets and
detects $N_{\text{det}}$ signals, the signal occurrence rate in each
population will be given by Beta distributions
$X_{M}\sim\beta(M_{\text{tot}},M_{\text{det}},f)$ and
$X_{N}\sim\beta(N_{\text{tot}},N_{\text{det}},f)$ respectively. Then the
inferred ratio of these two occurrence rates $X=X_{M}/X_{N}$, is given by the
following distribution [41]:

$X\sim\begin{cases}\frac{B(\alpha,\beta_{N})\,{}_{2}F_{1}(\alpha,1-\beta_{M};\alpha+\beta_{N};x)}{B(\alpha_{M},\beta_{M})B(\alpha_{N},\beta_{N})}x^{\alpha_{M}-1},&x\leq
1\\\ \\\
\frac{B(\alpha,\beta_{M})\,{}_{2}F_{1}\left(\alpha,1-\beta_{N};\alpha+\beta_{M};\frac{1}{x}\right)}{B(\alpha_{M},\beta_{M})B(\alpha_{N},\beta_{N})}x^{-\alpha_{N}-1},&x>1\end{cases}$
(6)

where $\alpha_{M}=M_{\text{det}}+1$,
$\beta_{M}=M_{\text{tot}}-M_{\text{det}}+1$, $\alpha_{N}=N_{\text{det}}+1$,
$\beta_{N}=N_{\text{tot}}-N_{\text{det}}+1$, $\alpha=\alpha_{M}+\alpha_{N}$.
The hypergeometric function is
${}_{2}F_{1}(a,b;c;x)=\sum_{n=0}^{\infty}(a)_{n}(b)_{n}/(c)_{n}x^{n}/n!$, with
$(a)_{k}=\Gamma(a+k)/\Gamma(a)$ the k-th order rising Pochhammer symbol.

The moments of this distribution can be written as

$E\left[X^{k}\right]=\frac{(M_{\text{det}}+1)_{k}}{(M_{\text{tot}}+2)_{k}}\frac{(N_{\text{tot}}+2-k)_{k}}{(N_{\text{det}}+1-k)_{k}}$
(7)

From this we can derive the mean

$\mu=\frac{M_{\text{det}}+1}{M_{\text{tot}}+2}\frac{N_{\text{tot}}+1}{N_{\text{det}}}=\frac{f_{M}}{f_{N}}+\mathcal{O}(1/N)$
(8)

where $f_{M}=M_{\textnormal{det}}/M_{\textnormal{tot}}$ and
$f_{N}=N_{\textnormal{det}}/N_{\textnormal{tot}}$

Similarly, the variance can be written

$\displaystyle\sigma^{2}$ $\displaystyle=$
$\displaystyle\frac{(M_{\text{det}}+1)(M_{\text{det}}+2)N_{\text{tot}}(N_{\text{tot}}+1)}{(M_{\text{tot}}+2)(M_{\text{tot}}+3)(N_{\text{det}}-1)N_{\text{det}}}-\mu^{2}$
(9) $\displaystyle=$
$\displaystyle\frac{f_{M}^{2}}{f_{N}^{2}}\left(\frac{1-f_{N}}{f_{N}N_{\text{tot}}}+\frac{1-f_{M}}{f_{M}M_{\text{tot}}}\right)+\mathcal{O}\left(1/N^{2}\right)$

The variance is not symmetric about the two ratios $f_{M}$ and $f_{N}$ since
the associated distribution $X_{M}/X_{N}$ is not symmetric. However, the
quantity $\sigma^{2}/\mu^{2}$ is symmetric, and has a characteristic 1/N
behaviour, but in this case dependent on the number of detected signals
$N_{\text{det}}=f_{N}N_{\text{tot}}$ and $M_{\text{det}}=f_{M}M_{\text{tot}}$,
rather than the total number of systems observed. Note also that this variance
contains two additive contributions from the uncertainties in both surveys.

If one occurrence rate is smaller than the other, a correspondingly larger
number of total systems need to be surveyed to adequately measure any trend
with system parameters, rather than the overall occurrence rate. For rare
signals, $f_{M,N}\ll 1$, the number of detections in each survey should be
roughly equal. This implies that surveys should collect more signals from
regions where the signal is expected to be rarer.

To use this to distinguish between two competing hypotheses predicting
different values of the ratio $f_{M}/f_{N}$, we would want the measurement
error $e=\sqrt{\sigma^{2}}$ to be smaller than the difference in predictions.
So if we are trying to distinguish between the different galactic habitability
scenarios outlined above, we would want $\sigma\lesssim\mu$.

###  2.1 Cost analysis

We now use this analysis to determine the optimal way of apportioning funds to
the two populations. We denote the fraction of funding given to mission 2 as
$c$, and as an example assume the cost functions for these surveys are given
by the same power law relation $M=M_{0}(1-c)^{q}$ and $N=N_{0}c^{q}$, where
$M_{0}$ and $N_{0}$ are the yields that would be obtained if all resources
were put into mission $M$ and $N$, respectively. As we discuss in section 5.2,
we expect $q>1$ if the two populations come from two different missions, as
increasing budget allows multiple mission aspects to be improved
simultaneously. For apportioning observing time within the same mission, we
instead expect $q<1$, as increasing observing time yields diminishing returns,
discussed further in [47]. We wish to minimise the variance in equation (9),
corresponding to maximum information gain. The derivative of eqn (9) yields

$\displaystyle\frac{\partial}{\partial c}\sigma^{2}$ $\displaystyle=$
$\displaystyle\frac{f_{M}^{2}}{f_{N}^{2}}\bigg{[}\frac{-q}{N_{0}}\frac{1-f_{N}}{f_{N}}c^{-(q+1)}$
(10)
$\displaystyle+\frac{q}{M_{0}}\frac{1-f_{M}}{f_{M}}(1-c)^{-(q+1)}\bigg{]}$

This makes use of the approximation that both $M_{\text{det}}$ and
$N_{\text{det}}$ are substantially removed from 0. Setting this to zero and
solving for $c$ gives us the optimal cost

$c=\frac{1}{1+s^{\frac{1}{q+1}}},\quad
s=\frac{N_{0}}{M_{0}}\frac{f_{N}}{f_{M}}\frac{1-f_{M}}{1-f_{N}}$ (11)

Yielding survey sizes

$N=N_{0}\frac{1}{\left(1+s^{\frac{1}{q+1}}\right)^{q}},\quad
M=M_{0}\frac{s^{\frac{q}{q+1}}}{\left(1+s^{\frac{1}{q+1}}\right)^{q}}$ (12)

From here, it can be seen that if $f_{M}=f_{N}$ and
$M_{\text{tot}}=N_{\text{tot}}$, then $c=1/2$, i.e. both surveys should get
equal priority. The relative cost is plotted for generic values of observed
fraction in Fig. 1.

Figure 1: Optimum cost fraction $c$ to minimise variance. Unless specified,
these plots take $q=1$ and $M_{0}=N_{0}$. Here, a low value of $c$ corresponds
to allocating most funds to mission $M$.

This allows us to explore how the optimal cost depends on various parameters.
The recommendations are sensible - the better the observations in population
M, the more cost should be invested on N and vice versa to get maximum
information. If $q$ is large, the optimal cost will deviate more strongly from
1/2 compared to the $q=1$ case, holding other parameters fixed. Similarly if
$q$ is small, we have $c\approx 1/2$ over a broad range of parameter values.
In the lower right panel we see that more money should be spent on missions
that have lower yields.

Lastly, note that from the above expression, we see that the optimal survey
sizes depend on biosignature occurrence rates, which are not necessarily known
beforehand. If these are unknown, a well designed survey would be able to
adapt to information as it arrives, to alter upcoming target selection as
current estimates of the occurrence rates dictate.

##  3 The Tradeoff Between False Positives and False Negatives

In this section, we apply our procedure to solar system planetary biosignature
missions. Here, we address ambiguity in the interpretation of mission results.
This ambiguity can arise from two significant contributing factors. Firstly,
there could be biosignature experiments where a positive result may be a false
positive, e.g. some geochemical trickery masquerading as life. Alternatively,
there may be situations where a negative result could be a false negative,
e.g. there may be entire communities of microbes present which the instrument
is not sensitive enough to detect. Any realistic mission will have to deal
with both. In such a scenario, and assuming finite resources/design
constraints, one key decision which will have to be made is which of the two
sources of ambiguity to prioritise minimising to arrive at a higher overall
certainty of whether or not life is present. As discussed in [15], the need to
mitigate these leads to the drive toward the most definitive biosignatures
possible, and as exhaustive a determination of abiogenesis rate as possible.

Unlike large scale surveys of many exoplanet systems, solar system based
biosignature search strategies are characterised by information depth rather
than information breadth. In other words, in the solar system life detection
mission, we are going to be successively surveying the same planet over and
over again, gaining more and more information and constraining our statistical
parameters. For this reason, the ideal framework from which to approach the
problem is a Bayesian one, which can successively adjust prior probabilities
based on new information [8].

We define the false positive rate as the probability of a detection given
nonlife, $\text{FP}=P(D|NL)$ and the false negative rate as the probability of
a nondetection given life, $\text{FN}=P(ND|L)$. The quantities needed for the
information can be derived via Bayes’ theorem:

$P(L\mid D,FP,FN,f)=\frac{P(D\mid L,FP,FN,f)P(L\mid FP,FN,f)}{P(D\mid
FP,FN,f)}$ (13)

Where

$P(L\mid FP,FN,f)=P(L)=f$ (14)

is the probability of life existing in the location where the probe searched.
The probability that the instrument registers a signal is

$P(D\mid FP,FN,f)=(1-\text{FN})f+\text{FP}(1-f)$ (15)

Therefore, the posterior probability of life given a detection is given by

$P(L\mid D,FP,FN,f)=\frac{(1-\text{FN})f}{(1-\text{FN})f+\text{FP}(1-f)}$ (16)

and

$P(L|ND,FP,FN,f)=\frac{\text{FN}f}{\text{FN}f+(1-\text{FP})(1-f)}$ (17)

To arrive at a decision making framework, we approach the problem in terms of
information gain. Here, the information gain measures the uncertainty present
in data–the larger the Shannon Entropy, the lower the certainty of the data
and the higher the likelihood that it may be random. In this case the
probabilities are conditional, so we use the Shannon Entropy for conditional
probabilities, given in terms of the event of detection, nondetection, and the
presence of life below. This is the information of the posterior probability
distributions, weighted by their probability of occurrence.

$\displaystyle S(\text{FP},\text{FN})=-\int
dfp_{f}(f)\bigg{[}P(D)\,s\big{(}P(L|D)\big{)}$
$\displaystyle+P(ND)\,s\big{(}P(L|ND)\big{)}\bigg{]}$ (18)

where $s(p)=p\log p+(1-p)\log(1-p)$.

Lastly, as the probability of life $f$ is not known ahead of time, our
expression requires that we integrate over all possible values. This requires
a prior distribution $p_{f}(f)$, which incorporates our current estimates on
the probability of life existing in a particular locale, given the current
uncertainties in the probability of life emerging, as well as any geological
history which may have impacted the site’s habitability [51, 32]. This is an
inescapable part of the calculation, and so to investigate the dependence of
our results on any assumptions we make about this quantity we explore four
different options: (i) the value of $f$ is arbitrarily taken to be .001 (ii)
the value of $f$ is taken to be .5 (iii) the probability of any value of $f$
is uniformly distributed, and (iv) the probability is log-uniform,
$p_{f}(f)\sim 1/f$. This last option requires a minimum value for $f$, which
we take to be $10^{-10}$ in our calculations. The choice of this value is
arbitrary, but mainly influences the overall scale of the expected information
gain through $S\sim 1/\log(p_{\text{min}})$, rather than the dependence on the
variables FN and FP. The expected information gain for these three choices is
displayed in Fig. 2.

Figure 2: Heat map of the conditional entropy in terms of FP and FN. The top
two plots take particular values for the probability of life existing at the
sampling location $f$. The bottom two integrate over a uniform and log-uniform
distribution for $f$, respectively. The white dashed line in each separates
regions where it is more beneficial to decrease FP (above the line) and FN
(below the line).

In these plots, the blue regions throughout the midsection have highest
entropy, corresponding to maximum ambiguity in result interpretation. The red
regions in the upper right and lower left corners are both places where the
information entropy approaches zero, corresponding to certain result
interpretation. The rightmost red region is a pathological point in parameter
space where both FP or FN approach 1. In our parameterization, this region
indicates that the signal (or lack thereof) is almost sure to convey the
opposite of what the instrument was designed to do, and the experiment is
certain to be wrong. This is what every biosignature mission should
deliberately avoid.

In the bottom left corner, we approach our second region where $S\rightarrow
0$, this time where FP and FN both approach zero, representing the ideal
scenario. Such unambiguous results will be unobtainable for any realistic
mission, but these plots do provide a metric that can be used to evaluate
missions with different values of FP and FN. For a preexisting mission design
that can be improved upon incrementally, the recommendation given by this
analysis is to go in the direction of maximum gradient, which depends on the
mission’s position in the plot. Through these plots, we have placed a white
dashed line demarcating the region where it is more important to prioritise
false positives (above the curve) from the region where it is more important
to prioritise false negatives (below). This line is exactly equal to the
diagonal FP$=$FN for the uniform and $f=.5$ cases, but is substantially
different in the others. Notably, in the region where both FP and FN are
small, it remains more important to prioritise false positives. This is a
consequence of our assumption that $f\ll 1$; the opposite conclusion would
hold if we consider $f\approx 1$.

This procedure, is, of course, reliant on a method of estimating false
positive and negative rates for a mission, both of which may be difficult in
practice. FP can be estimated by characterising the instrumental sensitivity,
and accurately accounting for all abiotic signal sources with careful
geochemical modelling. This is often fraught with controversy (for a recent
review see [22]), as historical examples can attest (see e.g. [5, 52, 3]). FN,
on the other hand, may be just as difficult to ascertain, as life may be
present in far lower abundances than Earth based analogue environments
indicate, yielding uncertainty in overall signal strength [11]. Additionally,
extraterrestrial life may have entirely different metabolisms, and so may not
produce a targeted biosignature [26].

As further illustration of the application of this formalism, we next turn to
two more examples: the combination of multiple instruments, and the recent
detection of methane on Enceladus.

###  3.1 Combining Signals from Multiple Instruments

Here we investigate the effect of combining signals to mitigate false
positives and negatives, which can provide redundancy and ameliorate the
difficulties in interpretation present for a single signal in isolation.
Generally, this can be done in a variety of different ways, to either increase
the coverage or certainty of a detection. We illustrate the various approaches
in a simple setup, and show that our framework yields recommendations for
which choice is preferable.

Let’s restrict our attention to the combination of two signals. The first is
fixed, say the detection of a certain biosignature (say, methane), which
carries with it a probability of false positive and false negative. We then
have two options for including a second instrument. In option A, we’re
concerned with false positives, so we include a second instrument to detect a
secondary signal which we expect to be present in the case of life (say,
homochirality). We then claim detection of life only if both these signals
register. While this increases our confidence in a detection, it comes at an
increased risk of false negatives, as if either instrument fails to register,
we would reject the signal of the other.

Alternatively, we may be more concerned with false negatives, and so may
include an additional instrument to measure a third biosignature (say, a
spectral red edge), which would be complimentary to the first. This improves
our original coverage, and so alleviates concern for false negatives. However,
this also increases the chances of false positives, for if either instrument
registers a signal, we would claim detection.

The question we wish to address is which of these two options is preferable,
given the parameters of the setup. For simplicity, we assume that the false
positive and false negative rates of all three instruments are identical, to
avoid a proliferation of variables. We also assume that the signals which they
measure are uncorrelated with each other (as a high degree of correlation
would render additional instruments not worthwhile). In this case, we have

$\displaystyle P_{A}(\text{FP})=\text{FP}^{2},\quad
P_{A}(\text{FN})=1-(1-\text{FN})^{2}$ $\displaystyle
P_{B}(\text{FP})=1-(1-\text{FP})^{2},\quad P_{B}(\text{FN})=\text{FN}^{2}$
(19)

With this, we display the difference in information gained from these two
setups in Fig. 3, using the value $f=.001$. We notice that if
$\text{FP}>\text{FN}$, then scenario $A$ is preferred, where both signals must
register to count as a detection. Also of note is that the difference in the
two setups is greatest when either one of the two probabilities is either very
small or very large.

Figure 3: Difference in two signal entropy for two different signal
combination strategies. If this difference is positive, it is more beneficial
to use the “and” strategy, so that both signals must register to count as a
detection. If the difference is negative, the “or” strategy, where either
signal registering would count as a detection, is preferable.

###  3.2 Direct Application: The Enceladus Data

The Cassini spacecraft, as part of its mission to study Saturn and its moons,
passed through a plume erupting from the south pole of Enceladus to collect a
sample. Instrumental analysis of the sample revealed a relatively high
abundance of methane (CH4), molecular hydrogen (H2) and carbon dioxide (CO2)
[50]. Methane is considered a potential biosignature because on Earth,
microbial communities around hydrothermal vents obtain energy through chemical
disequilibrium present in H2 emanating from the vents, resulting in the
release of methane as a waste product, a process known as methanogenesis [33].
However, this is not an unambiguous signal, as methane can also be produced
abiotically in hydrothermal vent systems via serpentization [35]. One method
of distinguishing between these two scenarios is then to compare the ratio of
fluxes, as biogenic production would yield a higher relative methane rate than
would occur abiotically.

This situation represents an ideal test case for our decision making tool–an
ambiguous biosignature result, with potential for both false positives and
false negatives in the data.

An analysis of the Cassini findings by [2] compared outgassing rates of
simulated populations of archaea and abiotic serpentization, and claimed
moderate support for the biotic hypothesis on Bayesian grounds. The model also
indicated that at the population sizes and rates of methanogenesis demanded by
the data, the archaea would have a negligible effect on the abundance of
H2–therefore, the high concentration of H2 present in the data did not count
against the biotic methanogenesis hypothesis.

It should be noted that this model still carries considerable ambiguity, and a
substantial portion of their model runs produce the observed signal
abiotically. Regardless, they present quantitative estimates for the
probabilities of false positives and false negatives in actual data.
Therefore, it is well suited for an application of our decision making tool.

According to the biogeochemical models in [2], the Cassini plume flybys
resulted in FP=.76 and FN=.24. Here FP and FN are determined by the limits of
instrumental sensitivity; from [50] the sensitivity threshold is estimated to
be about 10x smaller than the observed values. This is illustrated in Fig. 4,
where we place a white star to indicate where in the diagram out current
knowledge of these biosignatures lies.

Figure 4: White star corresponds to the estimated FP and FN of the Enceladus
Cassini flythrough mission. The underlying plot is the same as shown in Fig.
2, with uniform prior for $f$.

##  4 Lurkers

As a further application, we apply our approach to the spatial problem, where
we are concerned with the tradeoff between coverage and resolution when
looking for a biosignature. Much of the language we use in this section is
couched in terms of technosignature “lurkers”, but our formalism is general
and can equally be applied to microbial colonies, molecular biomarkers, etc.

###  4.1 Background and Context

An ongoing area of technosignature research considers physical artefacts and
probes which may be hiding in our solar system, referred to as “lurkers” [6].
The idea is that if extraterrestrial civilisations are present in our galaxy,
they may have seeded the galaxy with probes, which, either defunct, dormant or
active, could be present in our solar system [21]. The plausibility of this
was argued by Hart and Tippler, for the time it takes self replicating von
Neumann devices to cover the galaxy is modelled to be much shorter than the
age of the galaxy itself [19]. Therefore, in spite of the considerable energy
requirements and engineering challenges inherent in interstellar travel, a
reasonable case can be made for past extraterrestrial visitation of our solar
system which may have left behind artefacts [17]. Hart and Tippler’s argument
frames the absence of such lurkers as justification for the Fermi Paradox and
pessimism about the prevalence of intelligent life in general [19]. However,
as deduced by [21], only a small fraction of the solar system’s total volume
has been imaged at sufficient resolution to rule out the presence of lurkers.
Therefore it remains an open question warranting further investigation and
dedicated search programs.

This motivates us to apply our formalism to this setup to determine how to
effectively allocate resources in order to maximise our chances of finding
signs of lurkers. While a survey with unlimited resources may be able to scour
every square inch of a planet (or moon, asteroid, etc.), any realistic mission
will have budgetary and lifetime constraints. The two extreme strategies would
then be to perform a cursory inspection across the entire planet area, but
potentially miss signals with insufficient strength, or to perform a high
resolution search in a small region of the planet. Here we show that the ideal
mission forms a compromise between these two extremes, and deduce the optimal
design to maximise chances of success (which in our language corresponds to
the mission yielding most information).

###  4.2 Mathematical Framework

The setup we consider consists of a survey of resolution $R$, which is the two
dimensional spatial resolution of a survey. Our survey covers a fraction of
the planet area $f_{R}$. We assume that our survey results in no detection (as
otherwise, our concern with interpretive statistics becomes lessened
considerably). We would like to compute the probability that at least one
object is present, given a null result, $p(N>0|\text{null})$, and want to
optimise mission parameters so that this quantity is as small as possible
given some constraints. Of course, this is equivalent to the formulation where
we try to maximise the quantity $p(N=0|\text{null})$.

To compute this, first note that, in the scenario where there are $N$ objects
uniformly spaced on the planet surface, all of size $D$, we have

$p(\text{null}|N,D)=(1-f_{R})^{N}\theta(D-R)+1\,\theta(R-D)$ (20)

Here $\theta(x)$ is the Heaviside function, which is 1 for positive values and
0 for negative values. From here, we can see that a null result is guaranteed
if our mission resolution is above the object size. If the mission does have
sufficient resolution, we see that the probability of a null signal diminishes
with increasing survey coverage or number of objects.

Inverting this expression requires priors for both the number of objects and
their size. If the prior distribution of object sizes is $p_{D}(D)$, this
expression can be integrated to find

$\displaystyle p(\text{null}|N)$ $\displaystyle=$ $\displaystyle\int
dD\,p_{D}(D)\,p(\text{null}|N,D)$ (21) $\displaystyle=$ $\displaystyle
1-\left(1-(1-f_{R})^{N}\right)P_{D}(D>R)$

And if the prior distribution of number of objects is $p_{N}(N)$, the total
probability of nondetection is

$p(\text{null})=\sum p(\text{null}|N)\,p_{N}(N).$ (22)

Then by Bayes’ theorem,

$p(N|\text{null})=\frac{p(\text{null}|N)p(N)}{p(\text{null})}.$ (23)

Note that here, in the limit $R\rightarrow\infty$, the survey is completely
uninformative, and we have $p(N|\text{null})\rightarrow p(N)$.

Then, the probability of at least one object being present given a null result
is

$p(N>0|\text{null})=1-\frac{p_{N}(N=0)}{p(\text{null})}.$ (24)

In the limit of complete survey $f_{R}\rightarrow 1$, we find

$p(N>0|\text{null})\rightarrow\frac{P_{D}(D<R)}{\Theta+P_{D}(D<R)},\quad\Theta=\frac{P_{N}(N=0)}{P_{N}(N>0)}$
(25)

reproducing the results of [21].

To simplify this general analysis, we now specialise to the case where the
number of objects is constrained to be either 0 or $N^{*}$,
$p_{N}(N)=(1-p^{*})\delta_{N,0}+p^{*}\delta_{N,N^{*}}$, where $\delta_{i,j}$
is the Kronecker delta. Then we have

$p(N>0|\text{null})=\frac{p^{*}\,\left(P_{D}(D<R)+(1-f_{R})^{N^{*}}\,P_{D}(D>R)\right)}{1-p^{*}\,\left(1-(1-f_{R})^{N^{*}}\right)\,P_{D}(D>R)}$
(26)

Figure 5: The probability that lurkers exist on a surface as a function of
mean diameter over resolution $D/R$ and fraction of surface covered $f_{R}$.
The vertical dashed line corresponds to mean lurker diameter, and the dashed
curve corresponds to a constraint on mission design. The white star is the
point of lowest $p(N>0|\text{null})$ restricting to the constraint. The top
two and bottom left plots take a normal distribution for lurker diameter, and
the lower right plot takes a power law.

We display this expression for various parameter values in Fig. 5, which plots
the ratio of mean lurker size to resolution $D/R$ versus the coverage fraction
$f_{R}$. In the first three panes, we assume the distribution of lurker
diameters follows a normal distribution (with mean 1 and standard deviation
.5). The aim of a mission should be to get as close to the ‘red region’ of
these plots as possible given mission constraints. We overlay a constraint
curve $A\sim\tan^{-1}(1/R)$ to encapsulate a typical area/resolution tradeoff,
but this form specifies a family of curves with differing base values. For a
given constraint curve, this will single out a unique mission design that
minimises this quantity (thus maximising science return). This point is
denoted by the white star in each of these plots.

Comparing the top two plots, we can see that the optimum point does not depend
on the lurker probability $p^{*}$ at all, a fact that can be derived from the
form of eqn. (26). Comparing to the lower left plot, we can see that the
optimum point is shifted toward lower $f_{R}$ and higher $D/R$ for larger
$N^{*}$, which is reasonable, as there would be a greater probability of
spotting a lurker in a given area if more are present on the planet surface.
The bottom right plot assumes a power law distribution (with $p(D)\propto
D^{-2}$) of lurker sizes to highlight the effect this has on our
recommendation; again, this shifts the optimum point toward lower $f_{R}$ and
higher $D/R$.

###  4.3 Direct Application to Solar System Bodies

Here, we will demonstrate a direct application of this statistical framework,
by examining the recommendations we find for a series of different bodies in
the solar system. We use the spatial resolutions of planetary missions and
surveys to establish upper limits on the probability of lurkers which may have
gone undetected up until now, allowing us to make a series of recommendations
for each solar system body/region of space. For this, we display the
resolution and fraction of area covered for multiple missions to bodies around
our solar system in Table 1.

Table 1: Upper limit diameters for lurkers on the surfaces of solar system bodies and regions of surveyed space. Original concept for this table and all numbers adapted from [29]. Location | R | $f_{R}$ | Mission | Source
---|---|---|---|---
Mercury | 250 m | 1 | MESSENGER | [23]
Venus | 10 m | $10^{-9}$ | Venera | [18]
| 250 m | 1 | Magellan | [48]
Earth | 1 cm | 1/3 | various | [18]
Moon | 0.5 m | 1 | LRO | [46]
Mars | 150 m | .86 | MRO | [53]
| 1 m | .01 | HiRISE | [28]
Ceres | 35 m | .97 | Dawn | [39]
Europa (Jupiter) | 230 m | .01 | Galileo | [13]
Enceladus (Saturn) | 110 m | .96 | Cassini | [45]
Ganymede (Uranus) | 3.5 km | .94 | Galileo | [40]
Triton (Neptune) | 3 km | 1/3 | Voyager 2 | [1]
Pluto | 660 m | .45 | New Horizons | [38]
| 77 m | .003 | New Horizons | [27]

The estimates quoted are for illustrative purposes only, and there are some
nuances which the table does not account for. Our model makes a simplifying
assumption by assuming that lurkers will be unambiguously distinguishable from
background geological features. This may not necessarily be true, especially
if deliberately camouflaged. For the same reason, even excluding the
possibility of deliberate camouflage or stealth, using diameter as our only
proxy for detectability has its limitations because the lurkers’ albedo
relative to that of its surroundings is also a factor. A low albedo lurker on
a dark planetary surface may blend in, appearing to be a shadow of another
feature or a dark rock. Conversely, a lurker with an albedo that contrasts
more with surrounding geology may be easier to detect. Additionally, these are
remote sensing missions, and so any subsurface/sub-ocean area would be
inaccessible and are not accounted for in this application.

Figure 6: Probability that lurkers exist on the various solar system bodies
listed in Table 1. This assumes an a priori probability of 1/2, and considers
the scenario where 10 lurkers would be present, with a mean diameter of 10
meters.

The values in Table 1 are displayed alongside our estimate for
$p(N>0|\text{null})$ in Fig. 6. As fiducial values, we use $p^{*}=.5$,
$N^{*}=10$, and $\langle D\rangle=10$ m, which is a reasonable upper estimate
for the size of an interstellar probe [17]. From this, we can draw a number of
conclusions:

  * •

Of the bodies considered, we are most certain that no lurkers exist on the
moon, followed by Earth. This is because although the Earth has been mapped to
higher resolution on the continents, the oceans prevent coverage for the
majority of the surface.

  * •

The gradient at each point dictates which aspect of a particular mission
should be improved. In a nutshell, whichever is worse between the covered
fraction and resolution is what should be improved, but this formalism allows
for that comparison to be made quantitatively.

  * •

Several bodies have near-complete low resolution maps, with small patches of
high resolution. This allows us to compare which are more valuable for
confirming the absence of lurkers. For Mars, for example, the high resolution
images give us most confidence that lurkers are absent.

  * •

Of the bodies considered, we are least certain about the absence of lurkers on
the outer solar system moons. Areas even further out, such as the Kuiper belt
and Oort cloud, are even less certain.

  * •

Despite mapping efforts, many of these bodies still have
$p(N>0|\text{null})\approx.5$, which is our reference probability that lurkers
are present. Therefore we are nowhere close to being able to claim that no
lurkers are present in the solar system.

We may concern ourselves with the overall probability that lurkers are present
in the total solar system, which is given by the compound probability of each
body separately,
$p(N>0|\text{null})_{\text{tot}}=1-\prod_{b\in\text{bodies}}(1-p(N>0|\text{null})_{b})$.
This total probability is most sensitive to the body with lowest probability,
suggesting that if our goal is to minimise this quantity, we ought to
prioritise the bodies we are least certain about first.

##  5 Further Applications

###  5.1 What is the best way to determine a habitability boundary?

Supposing we do begin to detect biosignatures, we will become interested in
the problem of determining the range of conditions for which life can persist.
Though there are a number of variables that certainly play a role
(temperature, planet size, star size, atmospheric mass, water content,
ellipticity, obliquity, etc.), in this section we focus on a single abstract
quantity $t$. We wish to outline a procedure for determining the range of $t$
which can sustain life. For simplicity we focus on determining the lower
bound, denote the current range as $(0,t_{1})$, and treat the variable as
uniformly distributed222if a quantity is not uniformly distributed, its
cumulative distribution function can be used as $t$ everywhere in this
section..

In the scenario where all systems with habitable characteristics are
guaranteed to possess biosignatures, this problem reduces to the standard
method of bisecting the remaining interval until the desired precision is
reached. The subtlety here is that if a planet does not exhibit the
biosignature, we would not be able to infer that life cannot exist within that
range, because presumably life does not inhabit every potentially habitable
environment. Let us discuss the certain case first as a warm up, to outline
how information gain can be used to arrive at this classic result.

As stated, before our measurement we know that the lower bound for $t$ lies
somewhere in the range $(0,t_{1})$. We will perform a measurement of a system
with $t_{0}$. A priori, the probability of a detection (the probability that
the lower bound is less than $t_{0}$) is $P(\text{det})=t_{0}/t_{1}\equiv
c_{0}$, and the probability of no detection is $P(\text{miss})=1-c_{0}$. If a
signal is detected, then the updated distribution of possible lower bounds is
$P_{\text{det}}=\mathcal{U}(0,t_{0})$, and if no signal is detected, the
distribution becomes $P(t_{\text{lower}})=\mathcal{U}(t_{0},t_{1})$. The
expected information gain from this measurement is

$\displaystyle\langle S\rangle$ $\displaystyle=$ $\displaystyle
S(P_{\text{det}})P_{\text{det}}+S(P_{\text{miss}})P_{\text{miss}}$ (27)
$\displaystyle=$ $\displaystyle-c_{0}\log
c_{0}-\left(1-c_{0}\right)\log\left(1-c_{0}\right)$

This quantity is maximised at $t_{0}=t_{1}/2$, recovering the classic
bisection method.

Now, let’s generalise this to the case where life is only present on a
fraction of all habitable planets, denoted $f$ (independent of $t$ in this
section). In this case, the probability of detection is given by
$P_{\text{det}}=fc_{0}$, which is the product of the probability of the lower
bound being smaller than the observed system, and that life is present. The
absence of a detection could be either because the lower bound is greater than
the system’s value, or because no life happens to be present. So, the
probability of no detection is given by the sum of two terms,
$P_{\text{miss}}=P(t_{0}<t_{l})+P(t_{l}<t_{0})(1-f)=1-fc_{0}$. The expected
information gain is given by

$\displaystyle\langle S\rangle=fc_{0}\log c_{0}+(1-c_{0})\log(1-fc_{0})$
$\displaystyle+(1-f)c_{0}\log\left(\frac{1-fc_{0}}{1-f}\right)$ (28)

After some manipulation, it is found that this quantity is maximized at

$c_{0}=\frac{(1-f)^{1/f-1}}{1+f(1-f)^{1/f-1}}$ (29)

When $f\rightarrow 1$, this expression goes to $1/2$, reproducing the certain
case. When $f\rightarrow 0$ it goes to $1/e$, reminiscent of the classic
secretary problem, which aims to determine an optimal value in complete
absence of information.

The expected information gained by an optimal measurement is

$\langle S\rangle|_{\text{opt}}=\log\left(1+f(1-f)^{1/f-1}\right)$ (30)

This goes to $1/e$ for the certain case, representing 1 nat of information
gain, and $f/e$ for small $f$, which is 1 nat multiplied by the detection
probability.

We can also determine the convergence rate toward the true lower bound by
looking at the expected maximum possible value for a given time step. Recall
that in the certain case, this is given by $1/2t_{0}+1/2t_{1}=3/4t_{1}$, so
that this converges as $(3/4)^{N}$ after $N$ time steps. In the uncertain
case, the formula instead yields $t_{0}fc_{0}+t1(1-fc_{0})$. We do not display
the full expression after substituting the optimal value of $c_{0}$ here, but
instead report its limit for small $f$, which becomes $(1-(e-1)/e^{2}f)t_{1}$.
This series converges as approximately $\text{exp}(-.232fN)$, which is
considerably slower than the certain case.

###  5.2 What is the optimal amount of money to spend on different mission
aspects?

Another application is in determining how much money to spend on each aspect
of a mission. In this section we will abstractly define a number of mission
aspects $\\{a_{i}\\}$, with the total number of observed planets scaling as
$N_{\text{tot}}=N_{0}\prod_{i}a_{i}^{q_{i}}$. Here, we assume that the total
number scales with each aspect to some power. For example, the aspects could
correspond to telescope diameter $D_{t}$, mission lifetime $T$, spectral
resolution $\Delta\lambda$, etc., as found in [47] where
$N_{\text{tot}}\propto D_{t}^{12/7}T^{3/7}\Delta\lambda^{3/7}$. In the next
subsection we will treat a step-wise increase as well. Now, we assume that a
mission has a fixed total budget $c_{\text{tot}}$ apportioned amongst each
aspect as $c_{\text{tot}}=\sum_{i}c_{i}$. We would like to determine the
optimal way of assigning each $c_{i}$ so as to maximise the number of observed
planets, as well as the resultant scaling of mission yield with cost.

This is a straightforward convex optimisation problem, that can be solved
inductively by finding the location where the gradient of the total yield
vanishes for every component budget. The solution is

$c_{i}=\frac{q_{i}}{q_{\text{tot}}}c_{\text{tot}},\quad
N_{\text{tot}}=N_{0}\frac{\prod_{i}q_{i}^{q_{i}}}{q_{\text{tot}}^{q_{\text{tot}}}}\,c_{\text{tot}}^{q_{\text{tot}}}$
(31)

Here, we have defined $q_{\text{tot}}=\sum_{i}q_{i}$, and assume nothing other
than that each exponent is positive.

This yields a concrete optimal fraction of budget to yield to each mission
aspect. Particularly noteworthy is the fact that the total yield scales
superlinearly with budget for all practical applications of this formalism.
This implies, among other things, that the cost per planet decreases
dramatically with increased scale. The cost per information gain, however,
does not, but is instead minimised at the value

$c_{\text{b}}=\frac{e}{\left(\frac{\prod_{i}q_{i}^{q_{i}}}{q_{\text{tot}}^{q_{\text{tot}}}}\,N_{0}\right)^{1/q_{\text{tot}}}}$
(32)

This minimum occurs because the information return depends only
logarithmically on total number, and so these diminishing returns must be
balanced against initial startup costs. For the reference scenario above,
taking spectral resolution, telescope diameter and mission length (to each
scale linearly with cost) gives $q_{\text{tot}}=18/7$, and that the optimal
cost is $\$10^{9}$ when $N_{0}=121.8$.

Several things to note about this expression: optimal pricing decreases slowly
for missions with higher yield potential, and is smaller for missions that
scale faster. If followed, this gives the optimal number of planets for a
mission to observe as $N_{opt}=e^{q_{\text{tot}}}$, which results in
$q_{\text{tot}}$ nats of information. However, minimising cost per information
gain in this manner neglects the cost associated with mission development,
which can be significantly higher than the costs for the duration of the
mission. This is included in section 5.4, where we find that the recommended
number of systems to observe is substantially larger.

###  5.3 When should an additional instrument be included?

In the previous subsection, we focused on mission aspects that affected the
mission yield continuously and monotonically. Some aspects do not scale this
way; a particular case is which an aspect is either included or not, and its
effectiveness cannot be improved other than it simply being present. An
example would be whether to include a starshade for a space mission, or
whether to include an instrument capable of making a specific measurement that
would lead to higher yields.

First, we will consider the case that the type of biosignatures being measured
are not affected by the inclusion of a binary aspect, but that only the total
yield is. In this case, we can say that the total yield is
$N_{\text{tot}}(b,c)=N_{0}(b)c^{q_{\text{tot}}}$, where $b$ is the binary
aspect, and $N_{0}(b)=N_{0}$ if the aspect is absent and $N_{0}(b)=N_{1}$ if
it is present. If the cost of the binary aspect is $c_{b}$, then this
instrument should be included when

$N_{1}\left(c_{\text{tot}}-c_{b}\right)^{q_{\text{tot}}}>N_{0}\,c_{\text{tot}}^{q_{\text{tot}}}$
(33)

It may also happen that a binary aspect does not affect the overall number of
planets observed, but allows the measurement of a different class of
biosignatures. This is the case, for example, when deciding whether to include
an instrument that can detect oxygen. In this scenario, the yield without this
instrument is given by $N_{\text{tot}}$, whereas with the instrument it is
given by $f_{\text{bio}}N_{\text{tot}}^{2}$. In this case, the instrument
should be included when

$f_{\text{bio}}\,N_{0}\,\left(c_{\text{tot}}-c_{b}\right)^{2q_{\text{tot}}}>c_{\text{tot}}^{q_{\text{tot}}}$
(34)

From this, we can see that the number of planets observed should be at least
as large as $1/f_{\text{bio}}$, plus some to counteract the fact that fewer
systems will be observed with the instrument. If the occurrence rate is too
low, the mission will not be expected to observe a signal anyway, and so the
measurement of the second class of biosignatures would be a waste.

###  5.4 The marginal value theorem and mission lifetimes

In this subsection we address the question of how long a mission should be
run. The yield scales sublinearly with mission lifetime, as the brightest,
most promising stars are targeted first, leaving the stars which require
longer integration times for later in the mission. This gives diminishing
returns as the mission goes on.

The marginal value theorem from optimal foraging theory [9] can be applied to
determine the point at which a mission should be abandoned in favour of a new
mission with greater yield potential. Indeed, collecting exoplanet data
closely resembles the process of sparse patch foraging, wherein samples are
collected from a finite resource in order of desirability until the yield of a
patch is poor enough to warrant the time investment to travel to another. The
analogy is completed when we consider that it takes considerable time to
design and construct missions. Though different missions are always being run
in parallel, they consume a fixed portion of the budget which could otherwise
have gone to some other mission.

We will consider a mission at fixed cost that takes a development period
$t_{\text{dev}}$ before it begins collecting data. Then the yield as a
function of time is $N(t)=N_{0}(t-t_{\text{dev}})^{q_{t}}$, where it should be
understood that when this quantity is not positive, it is 0. It is convenient
to nondimensionalize the variables in this expression by measuring time in
years, so that $N_{0}$ represents the number of planets returned in the first
year of operation. To determine the optimal stopping time we use the marginal
value theorem, which maximises the average information gain per time, this
sets $\dot{S}(t_{0})=S(t_{0})/t_{0}$. This can then be solved for $t_{0}$,
yielding

$t_{operation}=\frac{1}{W_{0}\left(N_{0}^{1/q_{t}}t_{\text{dev}}/e\right)}t_{\text{dev}}$
(35)

Here, $W_{0}(x)$ is the Lambert W function, which is the solution to the
equation $W_{0}e^{W_{0}}=x$, and asymptotes to a logarithm for large
arguments. So, it can be seen that the operation time scales not quite
linearly with development time, and decreases with increasing first year
yield. The total science return over the mission lifetime is given by

$S(t_{0})=q_{t}\left[W_{0}\left(N_{0}^{1/q_{t}}t_{\text{dev}}/e\right)+1\right]=q_{t}\left[\frac{t_{dev}}{t_{op}}+1\right]$
(36)

###  5.5 When is it best to follow a prediction, and when is it best to
challenge it?

We now turn our attention to the scenario where, according to some leading
model of habitability, a particular locale should be devoid of life. According
to this model, any attempt at measuring a biosignature around such a location
would be fruitless, and would have been better spent collecting data from a
safer bet. However, we are unlikely to ever be $100\%$ confident in any of our
habitability models, and so it is possible that testing the model will result
in it being overturned. Here, we show that our framework allows us to
determine a criterion for when this habitability model should be obeyed, and
when it should be challenged, based off of the confidence one assigns to its
truthfulness. A concrete example would be whether to check hot Jupiters for
biosignatures, as by most reasonable accounts they do not fulfil the
requirements for life [16].

As a toy example, let us consider the case where there are two possibilities:
the leading theory, which dictates that life is impossible on some type of
system, and a second theory, which assigns a nonzero chance that life may
exist. The confidence we assign to the standard theory is denoted $p$. The
prior distribution for the fraction of systems harbouring a signal is then

$p_{b}(f)=p\,\delta(f)+(1-p)\,\delta(f-f_{2})$ (37)

Here $\delta$ is the delta function, which equals 0 when its argument is
nonzero and integrates to 1. More thoroughly, we could consider a range of
values for $f_{2}$, but this significantly complicates the analysis, and leads
to little new insight.

Now we determine the expected amount of information gained from a measurement
of a system. According to the above prior, the probability of measuring no
signal is $P_{0}=p+(1-p)(1-f_{2})$, and the probability of measuring a signal
is $P_{1}=(1-p)f_{2}$. Then the expected information gain $\langle\Delta
S\rangle=P_{0}S(p(f|\text{no
detection}))+P_{1}S(p(f|\text{detection}))-S(p_{b}(f))$ is:

$\displaystyle\langle\Delta
S\rangle=(1-p)(1-f_{2})\log\left(\frac{(1-p)(1-f_{2})}{p+(1-p)(1-f_{2})}\right)$
$\displaystyle-p\log(p+(1-p)(1-f_{2}))-(1-p)\log(1-p)$ (38)

If we further take the limits $p\rightarrow 1$, $f_{2}\rightarrow 0$, this
expression simplifies somewhat:

$\langle\Delta
S\rangle\rightarrow(1-p)\Big{[}f_{2}(1-\log(1-p))+(1-f_{2})\log(1-f_{2})\Big{]}$
(39)

From this, it can be seen that the expected amount of information gain is
proportional to $(1-p)$, the probability that the theory’s recommendation is
incorrect.

Figure 7: Information gain for measurement of an unlikely signal as a function
of possible signal occurrence rate $f_{2}$ and confidence in the theory
proclaiming its impossibility $p$.

Eqn. 38 is displayed in Fig. 7, from where it can be seen that expected
information is maximised when $f_{2}$ is large and $p$ is small. As example
values, take $\langle\Delta S(p=.9,f_{2}=.1)\rangle=.02$, $\langle\Delta
S(p=.99,f_{2}=.1)\rangle=.005$.

To determine whether this measurement should be made, we need to compare it to
an alternative measurement yielding a different amount of expected information
gain, and see which is larger. In the standard case of measuring a
biosignature of frequency $f$ well localised from 0, the information is
$S=1/2\log(f(1-f)/N)$ after $N$ measurements, and so the information gain is
$\Delta S=1/(2N)$. So, for the case $p=.99$, $f_{2}=.1$, the expected value
for the risky measurement exceeds the standard after the standard measurement
has been made 107 times. This suggests a strategy of collecting measurements
around likely places first, until the gains become small enough that measuring
the unlikely signal becomes more interesting, on the off chance that it could
invalidate the current theory of habitability.

The above assumed that no measurement of the unlikely signal had been
attempted previously. We may wish to determine how many times to challenge a
theory before accepting it. For this scenario, if $N$ previous measurements
have been made, the prior will be

$p_{b}(f)\propto p\,\delta(f)+(1-p)(1-f_{2})^{N}\delta(f-f_{2})$ (40)

With constant of proportionality enforcing that this is normalised to 1. The
expected information gain is, in the limit of $N\rightarrow\infty$,

$\langle\Delta
S\rangle\rightarrow\frac{1-p}{p}\,N\,f_{2}\,(1-f_{2})^{N}\,\log\left(\frac{1}{1-f_{2}}\right)$
(41)

Figure 8: Information gain of an unlikely signal after a series of $N$
unsuccessful measurements. Above $N\gtrsim 1/f_{2}$ the likelihood of
measuring a signal becomes exponentially suppressed.

The full expression is plotted in Fig. 8. From here, we can clearly see a
steep dropoff at $N>1/f_{2}$, beyond which empirical evidence strongly
indicates the absence of a signal.

##  6 Discussion

In this work, we have utilised the concept of _information gain_ as a method
to assess the value of biosignature missions, and demonstrated that this is
capable of generating nontrivial recommendations for mission design and target
prioritisation, across various domains. Throughout, this has resembled less of
a first-principles approach, whereby any mission setup can automatically be
loaded into a master equation to straightforwardly generate recommendations,
but rather a style of thinking about missions in terms of the quality of the
knowledge that they generate as a function of mission parameters. The concept
of information explicitly depends on the quantities that are abstracted out of
the raw data any mission would return, and so at the outset any analysis of
this sort relies on value judgements on the part of the practitioner. We have
illustrated that this process does not always yield a unique encapsulation of
the data, as for instance when we chose to define how we detect population
trends through the distribution of the occurrence rate ratio, rather than some
other measure. While there is no guarantee that the recommendations will be
independent of the way we set up the problem, the advantage this framework has
is in making the mission goals and assumptions explicit when deriving
recommendations.

At this point we’d like to reiterate that we are ardently against using
information gain as a quantity to maximise for designing missions, since
attempts to substitute quantitative proxies for vague overarching goals often
results in pathological maximisation procedures, often to the detriment of the
original intent [10]. Rather, we advocate the use of this framework for
generating rules of thumb and an intuitive understanding for when certain
mission aspects ought to be improved upon. Additionally, we note that this
framework explicitly depends upon an honest and accurate assessment of mission
capabilities. Even with rigorous design practices, often there ends up being
uncertainties in instrumental sensitivities, background noise, and
plausibility of modelling scenarios that prevent perfect assessment of the
relevant parameters. On top of this, human biases preclude even perfect
experts from being able to make impartial assessments for any new venture
[42]. Nevertheless, our formalism is useful for determining the sensitivity of
our recommendations to these unknowns.

We hope that by demonstrating the practicality of this framework to a variety
of cases, the reader will find it productive to apply to other, new problems.
To this end, our approach has often been the following: first, we seek out a
continuous quantity that can be measured with a particular mission setup.
Next, we compute the distribution of values compatible with observations,
given the model parameters and any other additional assumptions that must be
used. Then, lastly, we vary these mission parameters, subject to any relevant
mission constraints such as finite budget/lifetime/sample size, and determine
the combination of mission parameters that equivalently maximizes information,
minimizes uncertainty or maximizes probability of the desired outcome. Our
hope is that this may be useful as a tool to aid in determining the design
parameters of future missions, providing transparency and impartiality in
selecting among upcoming missions, and ultimately maximizing our chances of
successfully detecting biosignatures.

Data Availability. Code and data used in this paper are located here:
github.com/SennDegrare/Bio-signatures.

Declaration of Interests. The authors report no conflict of interest.
[Acknowledgements]We would like to thank Jacob Haqq Misra, Joe Lazio and Joe
Silk for fruitful discussions.

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