Understanding the difficulties of posterior predictive estimation

Abhinav Agrawal, Justin Domke
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:662-702, 2025.

Abstract

Predictive posterior densities (PPDs) are essential in approximate inference for quantifying predictive uncertainty and comparing inference methods. Typically, PPDs are estimated by simple Monte Carlo (MC) averages. In this paper, we expose a critical under-recognized issue: the signal-to-noise ratio (SNR) of the simple MC estimator can sometimes be extremely low, leading to unreliable estimates. Our main contribution is a theoretical analysis demonstrating that even with exact inference, SNR can decay rapidly with an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of test data relative to training data. Through several examples, we empirically verify these claims and show that these factors indeed lead to poor SNR and unreliable PPD estimates (sometimes, estimates are off by hundreds of nats even with a million samples). While not the primary focus, we also explore an adaptive importance sampling approach as an illustrative way to mitigate the problem, where we learn the proposal distribution by maximizing a variational proxy to the SNR. Taken together, our findings highlight an important challenge and provide essential insights for reliable estimation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-agrawal25a, title = {Understanding the difficulties of posterior predictive estimation}, author = {Agrawal, Abhinav and Domke, Justin}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {662--702}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/agrawal25a/agrawal25a.pdf}, url = {https://proceedings.mlr.press/v267/agrawal25a.html}, abstract = {Predictive posterior densities (PPDs) are essential in approximate inference for quantifying predictive uncertainty and comparing inference methods. Typically, PPDs are estimated by simple Monte Carlo (MC) averages. In this paper, we expose a critical under-recognized issue: the signal-to-noise ratio (SNR) of the simple MC estimator can sometimes be extremely low, leading to unreliable estimates. Our main contribution is a theoretical analysis demonstrating that even with exact inference, SNR can decay rapidly with an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of test data relative to training data. Through several examples, we empirically verify these claims and show that these factors indeed lead to poor SNR and unreliable PPD estimates (sometimes, estimates are off by hundreds of nats even with a million samples). While not the primary focus, we also explore an adaptive importance sampling approach as an illustrative way to mitigate the problem, where we learn the proposal distribution by maximizing a variational proxy to the SNR. Taken together, our findings highlight an important challenge and provide essential insights for reliable estimation.} }
Endnote
%0 Conference Paper %T Understanding the difficulties of posterior predictive estimation %A Abhinav Agrawal %A Justin Domke %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-agrawal25a %I PMLR %P 662--702 %U https://proceedings.mlr.press/v267/agrawal25a.html %V 267 %X Predictive posterior densities (PPDs) are essential in approximate inference for quantifying predictive uncertainty and comparing inference methods. Typically, PPDs are estimated by simple Monte Carlo (MC) averages. In this paper, we expose a critical under-recognized issue: the signal-to-noise ratio (SNR) of the simple MC estimator can sometimes be extremely low, leading to unreliable estimates. Our main contribution is a theoretical analysis demonstrating that even with exact inference, SNR can decay rapidly with an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of test data relative to training data. Through several examples, we empirically verify these claims and show that these factors indeed lead to poor SNR and unreliable PPD estimates (sometimes, estimates are off by hundreds of nats even with a million samples). While not the primary focus, we also explore an adaptive importance sampling approach as an illustrative way to mitigate the problem, where we learn the proposal distribution by maximizing a variational proxy to the SNR. Taken together, our findings highlight an important challenge and provide essential insights for reliable estimation.
APA
Agrawal, A. & Domke, J.. (2025). Understanding the difficulties of posterior predictive estimation. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:662-702 Available from https://proceedings.mlr.press/v267/agrawal25a.html.

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