# Hierarchical probabilistic inference of the color-magnitude diagram and shrinkage of stellar distance uncertainties

## TL;DR

This paper introduces a hierarchical probabilistic model that enhances stellar distance estimates by integrating noisy parallax and photometric data through a data-driven color-magnitude diagram, improving accuracy especially for uncertain measurements.

## Contribution

It presents a novel hierarchical model that infers a noise-deconvolved color-magnitude diagram without relying on stellar models, significantly improving distance estimates for Gaia stars.

## Key findings

- Distance SNR improved by 19% on average
- Uncertainty reduced by 36% on average
- 8% of objects have SNR doubled

## Abstract

We present a hierarchical probabilistic model for improving geometric stellar distance estimates using color--magnitude information. This is achieved with a data driven model of the color--magnitude diagram, not relying on stellar models but instead on the relative abundances of stars in color--magnitude cells, which are inferred from very noisy magnitudes and parallaxes. While the resulting noise-deconvolved color--magnitude diagram can be useful for a range of applications, we focus on deriving improved stellar distance estimates relying on both parallax and photometric information. We demonstrate the efficiency of this approach on the 1.4 million stars of the Gaia TGAS sample that also have APASS magnitudes. Our hierarchical model has 4~million parameters in total, most of which are marginalized out numerically or analytically. We find that distance estimates are significantly improved for the noisiest parallaxes and densest regions of the color--magnitude diagram. In particular, the average distance signal-to-noise ratio and uncertainty improve by 19~percent and 36~percent, respectively, with 8~percent of the objects improving in SNR by a factor greater than 2. This computationally efficient approach fully accounts for both parallax and photometric noise, and is a first step towards a full hierarchical probabilistic model of the Gaia data.