Distortion Estimates.pdf

Distortion estimates for approximate Bayesian inference are crucial for assessing the quality of posterior approximations. Researchers propose a generic diagnostic tool that checks the quality of a posterior approximation specifically at the observed data. They introduce and estimate a family of "distortion maps" that act on univariate marginals of the multivariate approximate posterior. The distortion map transports the approximate marginal posterior CDF onto the corresponding exact posterior CDF without requiring the exact posterior.

The distortion map is estimated using a normalizing flow, which is an optimal transport constructed from a sequence of transformations. The estimated distortion map contains diagnostic information about the approximation error in the approximate marginal CDF. If the distortion map differs substantially from the identity map, the magnitude and location of any distortion are of interest.

The researchers also extend their diagnostics to bivariate marginal distributions and provide examples of estimated distortion surfaces. The approach has benefits, including being a generic and invertible mapping between functions, and having a simple simulation-based fitting scheme. However, it restricts itself to diagnostics for low-dimensional marginal distributions.

Key aspects of the approach include:

Estimating a distortion map that acts on univariate marginals of the approximate posterior
Using a normalizing flow to estimate the distortion map
Extending diagnostics to bivariate marginal distributions
Providing a simple simulation-based fitting scheme
* Restricting itself to diagnostics for low-dimensional marginal distributions

The proposed approach provides a useful tool for assessing the quality of posterior approximations in Bayesian inference, particularly when the approximation scheme is computationally efficient.