Epistemology and anomaly detection in astrobiology

Abstract

We examine the epistemological foundations of a leading technique in the search for evidence of life on exosolar planets. Specifically, we consider the “transit method” for spectroscopic analysis of exoplanet atmospheres, and the practice of treating anomalous chemical compositions of the atmospheres of exosolar planets as indicators of the potential presence of life. We propose a methodology for ranking the anomalousness of atmospheres that uses the mathematical apparatus of support vector machines, and which aims to be agnostic with respect to the particular chemical biosignatures of life. We argue that our approach is justified by an appeal to the “hinge” model of epistemic justification first proposed by as reported by Wittgenstein (On certainty, Blackwell, Oxford, 1969). We then compare our approach to previous work due to Walker et al. (Astrobiology 18(6):779–824, 2018) and Cleland (Astrobiology 19(6):722–729, 2019a; Does ‘life’ have a definition?, Cambridge University Press, New York, 2019b).

Appendices

Appendix A

Our measure of anomalousness is defined precisely as follows. Let $\vec {w}_{i}$ be a vector with n entries, where each entry $w_{ij}$ is the abundance of gas j from some subset of Seager et al.’s list in planet i’s atmosphere (namely, some subset of the set of gases that can be measured in a given context). Let the set of observed composition vectors be $\mathbf {W}$. Our proposed anomalousness measure draws on support vector machine (SVM) based approaches to anomaly detection in high-dimensional space, as proposed by Banerjee et al. (2006). Specifically, we propose to measure the anomalousness $\mathcal {A}(\vec {w}_{i},\mathbf {W})$ of the atmospheric composition of a given planet $W_{i}$, in the context of a set of planets $\mathbf {W}$ with cardinality m, using the following equation:

$$\begin{aligned} \mathcal {A}(\vec {w}_{i},\mathbf {W})=1-2\sum _{j|j\ne i}^{m}\alpha _{j}\exp \bigg (\frac{-||\vec {w}_{i}-\vec {w}_{j}||^{2}}{\sigma ^{2}}\bigg ) + \sum _{j|j\ne i}^{m}\sum _{k|k\ne i}^{m}\alpha _{j}\alpha _{k}\exp \bigg (\frac{-||\vec {w}_{j}-\vec {w}_{k}||^{2}}{\sigma ^{2}}\bigg ) \end{aligned}$$

(6)

See Banerjee et al. (2006) for a detailed derivation of this particular measure of anomalousness. Each $\alpha _{j}$ and $\alpha _{k}$ is a positive, real-number entry in a vector $\vec {\alpha }$ in which each entry is a specific weighting parameter for each planet. The norm $||\vec {w}_{i}-\vec {w}_{j}||$ denotes the Euclidean distance between the vectors $\vec {w}_{i}$ and $\vec {w}_{j}$. The scale parameter $\sigma$ sets the shape of the distribution of the anomalousness of the planets in the dataset. Thus, if $\vec {w}_{i}$ is an abnormally large distance from all of the other data vectors in the set, then the first term of the equation will be larger and the second term will be smaller, and so $\mathcal {A}(\vec {w}_{i},\mathbf {W})$ will be larger, all else being equal. This justifies the claim that higher the value of $\mathcal {A}(\vec {w}_{i},\mathbf {W})$, the greater the anomalousness of the atmospheric composition of the atmosphere of the planet $W_{i}$.

The scale parameter $\sigma$ and the weight vector $\vec {\alpha }$ can be estimated from the available data. To estimate $\sigma$, we use a technique from Ghafoori et al. (2018). Let $\delta _{min}$ be the minimal Euclidean distance of any vector $\vec {w_{i}}$ from its nearest neighbor in the data set $\mathbf {W}$. Let $\delta _{avg}$ be the average Euclidean distance of all vectors in $\mathbf {W}$ from their nearest neighbor, excluding the vector that is $\delta _{min}$ from its nearest neighbor. The scale parameter $\sigma$ is estimated as follows:

$$\begin{aligned} \sigma = \frac{-\ln (\delta _{min}/\delta _{avg})}{\delta _{avg}^{2}-\delta _{min}^{2}} \end{aligned}$$

(7)

This method for estimating $\sigma$ has proven successful at detecting anomalies in a series of empirical tests, using standard machine-learning data sets (Ghafoori et al., 2018, pp. 5064-5069). Following Schölkopf et al. (2001), we can set the weight vector $\vec {\alpha }$ by solving the following optimization problem:

$$\begin{aligned} \min _{\vec {\alpha }}\sum _{j=1}^{m}\sum _{k=1}^{m}\alpha _{j}\alpha _{k}\exp \bigg (\frac{-||\vec {w}_{j}-\vec {w}_{k}||^{2}}{\sigma ^{2}}\bigg )\text {, s.t.} \ \sum _{i=1}^{m}\alpha _{i}=1 \end{aligned}$$

(8)

This ensures that $\mathcal {A}(\vec {w}_{i},\mathbf {W})$ assigns as high anomalousness as possible to each planet $W_{i}$.

Appendix B

To demonstrate the potential applicability of our proposed anomaly-detection method, we tested our method on simulated data. The simulation proceeded in the following steps:

• 1. Generate 90 gas abundance vectors by repeatedly sampling from a Dirichlet distribution $\text {Dir}(1000,\vec {u})$, where $\vec {u}=[u_{1},\dots ,u_{1000}]$ is a concentration parameter such that $u_{i}=10$ for $i\le 100$ and $u_{i}=1$ for $i>100$. This ensures that abundance vectors sampled from this distribution are expected to have abundances concentrated near the first 100 entries. This is meant to simulate observations from a typical, lifeless planet, on which certain gases are much more likely to be common than others. Call this sample of abundance vectors $\mathbf {W_{A}}$.

• 2. Generate 10 gas abundance vectors by repeatedly sampling from a Dirichlet distribution $\text {Dir}(1000,\vec {v})$, where $\vec {v}=[v_{1},\dots ,v_{1000}]$ is a concentration parameter such that $v_{i}=x$ for $i\le 100$ and $v_{i}=1$ for $i>100$. For higher values of x, abundance vectors sampled from this distribution are expected to have abundances concentrated near the first 100 entries. This is meant to simulate observations from a typical, lifeless planet, on which certain gases are much more likely to be common than others. For lower values of x, all possible abundance vectors become increasingly likely. This is meant to simulate observations from planets that are possible hosts of life, where many more combinations of gas abundances are assumed to be possible. Call this sample of abundance vectors $\mathbf {W_{B}}$.

• 3. Form the full data set $\mathbf {W}=\mathbf {W_{A}}\cup \mathbf {W_{B}}$.

• 4. Compute the parameters $\sigma$ and $\vec {\alpha }$ of the anomalousness measure for the data set $\mathbf {W}$.

• 5. Calculate the anomalousness measure $\mathcal {A}(\vec {w},\mathbf {W})$ for each element of $\mathbf {W}$.

• 6. Identify the ten most anomalous abundance vectors in $\mathbf {W}$.

• 7. Record as y the proportion of the ten most anomalous vectors that are elements of $\mathbf {W_{B}}$.

• 8. Repeat steps 1-7 for all possible values of x in the interval (1, 10), in increments of .2.

• 9. Repeat steps 1-8 fifty times, for a total of 460 data points.

Recall that lower values of the parameter x, which determines the concentration parameter for the distribution from which elements of $\mathbf {W_{B}}$ are sampled, are meant to simulate cases in which the elements of $\mathbf {W_{B}}$ represent planets that are possible hosts of life. Thus, we expect that as x increases, the proportion of abundance vectors with top-ten anomalousness in $\mathbf {W}$ that are also elements of $\mathbf {W_{B}}$ should decrease.

Fig. 2

Plot showing the relationship between the value of x, which determines the concentration parameter for the distribution from which elements of $\mathbf {W_{B}}$ are sampled, and the proportion of abundance vectors with top-ten anomalousness in $\mathbf {W}$ that are also elements of $\mathbf {W_{B}}$

Indeed, this is what we observe in the results of our simulation. Figure 2 shows the proportion of top-ten anomalies from the set $\mathbf {W}$ that are also elements of the set $\mathbf {W_{B}}$, for each value of x in each simulation. For values of x lower than or equal to 6, the proportion of top-ten anomalous vectors that are elements of $\mathbf {W_{B}}$ is tightly clustered around 1. For values of x greater than 6, this same proportion is declining linearly in x. We take this to be positive evidence for the in-principle utility of our measure for detecting the kinds of anomalies that we are interested in. It should be noted that while there is a fairly sharp transition in correctly identifying $W_B$, knowing the value of x (or the analogous parameter(s) in a given setting) at which this transition occurs in an applied setting will require careful thought. As pointed out above, correct identification depends on the degree of anomalousness and thus, such methods are expected to fail when life is not very anomalous. Code for this simulation is available at https://github.com/davidbkinney/astrobiologyappendixB.