On micromodes in Bayesian posterior distributions and their implications for MCMC
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910014374936576 |
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| author | Agrawal, Sanket Grazzi, Sebastiano Roberts, Gareth O. |
| author_facet | Agrawal, Sanket Grazzi, Sebastiano Roberts, Gareth O. |
| contents | We investigate the existence and severity of local modes in posterior distributions from Bayesian analyses. These are known to occur in posterior tails resulting from heavy-tailed error models such as those used in robust regression. To understand this phenomenon clearly, we consider in detail location models with Student-$t$ errors in dimension $d$ with sample size $n$. For sufficiently heavy-tailed data-generating distributions, extreme observations become increasingly isolated as $n \to \infty$. We show that each such observation induces a unique local posterior mode with probability tending to $1$. We refer to such a local mode as a micromode. These micromodes are typically small in height but their domains of attraction are large and grow polynomially with $n$. We then connect this posterior geometry to computation. We establish an Arrhenius law for the time taken by one-dimensional piecewise deterministic Monte Carlo algorithms to exit these micromodes. Our analysis identifies a phase transition where a misspecified and overly underdispersed model causes exit times to increase sharply, leading to a pronounced deterioration in sampling performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06931 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On micromodes in Bayesian posterior distributions and their implications for MCMC Agrawal, Sanket Grazzi, Sebastiano Roberts, Gareth O. Statistics Theory Computation 62E20, 62-08, 62F15, 65C05 We investigate the existence and severity of local modes in posterior distributions from Bayesian analyses. These are known to occur in posterior tails resulting from heavy-tailed error models such as those used in robust regression. To understand this phenomenon clearly, we consider in detail location models with Student-$t$ errors in dimension $d$ with sample size $n$. For sufficiently heavy-tailed data-generating distributions, extreme observations become increasingly isolated as $n \to \infty$. We show that each such observation induces a unique local posterior mode with probability tending to $1$. We refer to such a local mode as a micromode. These micromodes are typically small in height but their domains of attraction are large and grow polynomially with $n$. We then connect this posterior geometry to computation. We establish an Arrhenius law for the time taken by one-dimensional piecewise deterministic Monte Carlo algorithms to exit these micromodes. Our analysis identifies a phase transition where a misspecified and overly underdispersed model causes exit times to increase sharply, leading to a pronounced deterioration in sampling performance. |
| title | On micromodes in Bayesian posterior distributions and their implications for MCMC |
| topic | Statistics Theory Computation 62E20, 62-08, 62F15, 65C05 |
| url | https://arxiv.org/abs/2602.06931 |