Differentiability and overlap concentration in optimal Bayesian inference
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929676852658176 |
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| author | Chen, Hong-Bin Issa, Victor |
| author_facet | Chen, Hong-Bin Issa, Victor |
| contents | In this short note, we consider models of optimal Bayesian inference of finite-rank tensor products. We add to the model a linear channel parametrized by $h$. We show that at every interior differentiable point $h$ of the free energy (associated with the model), the overlap concentrates at the gradient of the free energy and the minimum mean-square error converges to a related limit. In other words, the model is replica-symmetric at every differentiable point. At any signal-to-noise ratio, such points $h$ form a full-measure set (hence $h=0$ belongs to the closure of these points). For a sufficiently low signal-to-noise ratio, we show that every interior point is a differentiable point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08786 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Differentiability and overlap concentration in optimal Bayesian inference Chen, Hong-Bin Issa, Victor Probability Information Theory Statistics Theory In this short note, we consider models of optimal Bayesian inference of finite-rank tensor products. We add to the model a linear channel parametrized by $h$. We show that at every interior differentiable point $h$ of the free energy (associated with the model), the overlap concentrates at the gradient of the free energy and the minimum mean-square error converges to a related limit. In other words, the model is replica-symmetric at every differentiable point. At any signal-to-noise ratio, such points $h$ form a full-measure set (hence $h=0$ belongs to the closure of these points). For a sufficiently low signal-to-noise ratio, we show that every interior point is a differentiable point. |
| title | Differentiability and overlap concentration in optimal Bayesian inference |
| topic | Probability Information Theory Statistics Theory |
| url | https://arxiv.org/abs/2501.08786 |