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Probabilistic Graphical Models in Astronomy

Abigail Sheerin, Giuseppe Vinci

TL;DR

This paper advocates using probabilistic graphical models to analyze the growing complexity of astronomical data. It surveys undirected (MRF/GGM) and directed (DAG) frameworks, detailing estimation approaches such as hypothesis testing, Graphical LASSO with EBIC tuning, and nonparametric normal-score transformations, as well as the PC algorithm for structure learning and CPDAG representations. Applying these methods to exoplanet and host-star data from the NASA Exoplanet Archive, the authors demonstrate that conditional dependence graphs largely recover astrophysically meaningful relationships, often clarifying what marginal correlations obscure. The work highlights the potential for graphical models to serve as a principled, exploratory tool in astrostatistics and outlines avenues for extensions to non-Gaussian data, Bayesian methods, and incomplete data.

Abstract

The field of astronomy is experiencing a data explosion driven by significant advances in observational instrumentation, and classical methods often fall short of addressing the complexity of modern astronomical datasets. Probabilistic graphical models offer powerful tools for uncovering the dependence structures and data-generating processes underlying a wide array of cosmic variables. By representing variables as nodes in a network, these models allow for the visualization and analysis of the intricate relationships that underpin theories of hierarchical structure formation within the universe. We highlight the value that graphical models bring to astronomical research by demonstrating their practical application to the study of exoplanets and host stars.

Probabilistic Graphical Models in Astronomy

TL;DR

This paper advocates using probabilistic graphical models to analyze the growing complexity of astronomical data. It surveys undirected (MRF/GGM) and directed (DAG) frameworks, detailing estimation approaches such as hypothesis testing, Graphical LASSO with EBIC tuning, and nonparametric normal-score transformations, as well as the PC algorithm for structure learning and CPDAG representations. Applying these methods to exoplanet and host-star data from the NASA Exoplanet Archive, the authors demonstrate that conditional dependence graphs largely recover astrophysically meaningful relationships, often clarifying what marginal correlations obscure. The work highlights the potential for graphical models to serve as a principled, exploratory tool in astrostatistics and outlines avenues for extensions to non-Gaussian data, Bayesian methods, and incomplete data.

Abstract

The field of astronomy is experiencing a data explosion driven by significant advances in observational instrumentation, and classical methods often fall short of addressing the complexity of modern astronomical datasets. Probabilistic graphical models offer powerful tools for uncovering the dependence structures and data-generating processes underlying a wide array of cosmic variables. By representing variables as nodes in a network, these models allow for the visualization and analysis of the intricate relationships that underpin theories of hierarchical structure formation within the universe. We highlight the value that graphical models bring to astronomical research by demonstrating their practical application to the study of exoplanets and host stars.
Paper Structure (12 sections, 27 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 27 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Abstract.
  2. Introduction
  3. Probabilistic Graphical Models
  4. Undirected Graphs
  5. Estimating a GGM via multiple hypothesis testing
  6. Estimating a GGM via penalized MLE
  7. Estimating a GGM from non-Gaussian data
  8. Directed Graphs
  9. DAGs
  10. Estimating DAGs
  11. Analysis of exoplanets and host stars data
  12. Discussion

Figures (3)

  • Figure 1: (a): An undirected graph where any two variables are not connected by an edge if and only if they are independent conditionally on all other variables. For example, $X_1$ and $X_5$ are independent conditionally on all other variables. (b): A directed acyclic graph (DAG) where a variable $X_i$ is independent of a variable $X_j$ conditionally on the parents of $X_i$ if and only if $X_j$ is neither a parent nor a descendant of $X_i$. For example, $X_6$ is the only parent of $X_2$ so, conditionally on $X_6$, $X_2$ is independent of $(X_1,X_3,X_4)$, but it is dependent on its descendant $X_5$.
  • Figure 2: (a): Histograms of the ten variables listed in Table \ref{['table:exoplanets']} (diagonal plots) standardized via Equation \ref{['eq:normalscore']}; scatterplots of each variable versus every other (upper off-diagonals); and contour plots of estimated bivariate Gaussian distributions (lower diagonals). (b): Heatmap of the sample marginal correlations $\hat{r}_{ij}$ (upper diagonals; Equation \ref{['eq:corrmle']}) and sample partial correlations $\hat{\rho}_{ij}$ (lower diagonals; Equation \ref{['eq:parcorhat']}) of the ten variables, where blue colors denote negative correlations, red colors denote positive correlations, and grey color denotes correlations not significantly different from zero (FWER$\le$1%, Bonferroni correction). Values of the correlations and partial correlations are shown as rounded signed percentages.
  • Figure 3: (a): Marginal correlation graph based on 1% FWER control via Bonferroni correction (Table \ref{['table:exoplanetsempirical']}, upper diagonals). (b): Marginal correlation graph based on 1% FDR control. (c): Partial correlation (conditional dependence) graphs based on 1% FWER control via Bonferroni correction (Equation \ref{['eq:bonferroniedgeset']}; Table \ref{['table:exoplanetsempirical']}, lower diagonals). (d): Partial correlation graph based on 1% FDR control (Equation \ref{['eq:fdredgeset']}). (e): Partial correlation graph based on GLASSO (Equation \ref{['eq:glassoE']}) with EBIC tuning parameter selection (Equation \ref{['eq:ebic']}). In all graphs in (a)--(e), blue edges denote negative correlations, red edges denote positive correlations, the thickness of each edge is proportional to the magnitude of the correlation, and the radius of each node is proportional to the sum of the magnitudes of the correlations relative to the node. (f): CPDAG obtained via PC algorithm (Algorithm \ref{['algo:pc']}, $\alpha=0.01$).