Improved dimension dependence in the Bernstein von Mises Theorem via a new Laplace approximation bound
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929574896467968 |
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| author | Katsevich, Anya |
| author_facet | Katsevich, Anya |
| contents | The Bernstein-von Mises theorem (BvM) gives conditions under which the posterior distribution of a parameter $θ\inΘ\subseteq\mathbb R^d$ based on $n$ independent samples is asymptotically normal. In the high-dimensional regime, a key question is to determine the growth rate of $d$ with $n$ required for the BvM to hold. We show that up to a model-dependent coefficient, $n\gg d^2$ suffices for the BvM to hold in two settings: arbitrary generalized linear models, which include exponential families as a special case, and multinomial data, in which the parameter of interest is an unknown probability mass functions on $d+1$ states. Our results improve on the tightest previously known condition for posterior asymptotic normality, $n\gg d^3$. Our statements of the BvM are nonasymptotic, taking the form of explicit high-probability bounds. To prove the BvM, we derive a new simple and explicit bound on the total variation distance between a measure $π\propto e^{-nf}$ on $Θ\subseteq\mathbb R^d$ and its Laplace approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06899 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Improved dimension dependence in the Bernstein von Mises Theorem via a new Laplace approximation bound Katsevich, Anya Statistics Theory The Bernstein-von Mises theorem (BvM) gives conditions under which the posterior distribution of a parameter $θ\inΘ\subseteq\mathbb R^d$ based on $n$ independent samples is asymptotically normal. In the high-dimensional regime, a key question is to determine the growth rate of $d$ with $n$ required for the BvM to hold. We show that up to a model-dependent coefficient, $n\gg d^2$ suffices for the BvM to hold in two settings: arbitrary generalized linear models, which include exponential families as a special case, and multinomial data, in which the parameter of interest is an unknown probability mass functions on $d+1$ states. Our results improve on the tightest previously known condition for posterior asymptotic normality, $n\gg d^3$. Our statements of the BvM are nonasymptotic, taking the form of explicit high-probability bounds. To prove the BvM, we derive a new simple and explicit bound on the total variation distance between a measure $π\propto e^{-nf}$ on $Θ\subseteq\mathbb R^d$ and its Laplace approximation. |
| title | Improved dimension dependence in the Bernstein von Mises Theorem via a new Laplace approximation bound |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2308.06899 |