# Bayesian model comparison with the Hyv"arinen score: computation and consistency

arXiv: https://arxiv.org/abs/1711.00136 · HTML: https://mlsanity.com/paper/1711.00136/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2017.

## Built on by (most cited first, 10 of 10)

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- [Sample Complexity Bounds for Score-Matching: Causal Discovery and Generative Modeling](https://mlsanity.com/paper/2310.18123/) (2023)
- [Generalised Bayesian Inference for Discrete Intractable Likelihood](https://mlsanity.com/paper/2206.08420/) (2022)
- [Robust Score-Based Quickest Change Detection](https://mlsanity.com/paper/2407.11094/) (2024)
- [Locking and Quacking: Stacking Bayesian model predictions by log-pooling and superposition](https://mlsanity.com/paper/2305.07334/) (2023)
- [Direct Bayesian Additive Regression Trees for Conditional Average Treatment Effects in Regression Discontinuity Designs](https://mlsanity.com/paper/2603.03819/) (2026)
- [Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data](https://mlsanity.com/paper/2609.01783/) (2026)

## Abstract (arXiv)

The Bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out-of-sample predictive scores under the logarithmic scoring rule. However, when some of the candidate models involve vague priors on their parameters, the log-Bayes factor features an arbitrary additive constant that hinders its interpretation. As an alternative, we consider model comparison using the Hyv"arinen score. We propose a method to consistently estimate this score for parametric models, using sequential Monte Carlo methods. We show that this score can be estimated for models with tractable likelihoods as well as nonlinear non-Gaussian state-space models with intractable likelihoods. We prove the asymptotic consistency of this new model selection criterion under strong regularity assumptions in the case of non-nested models, and we provide qualitative insights for the nested case. We also use existing characterizations of proper scoring rules on discrete spaces to extend the Hyv"arinen score to discrete observations. Our numerical illustrations include L'evy-driven stochastic volatility models and diffusion models for population dynamics.