Connections between sequential Bayesian inference and evolutionary dynamics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913733396135936 |
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| author | Pathiraja, Sahani Wacker, Philipp |
| author_facet | Pathiraja, Sahani Wacker, Philipp |
| contents | It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research in which this precise connection is rigorously established in the continuous time setting. Here we focus on a partial differential equation known as the Kushner-Stratonovich equation describing the evolution of the posterior density in time. Of particular importance is a piecewise smooth approximation of the observation path from which the discrete time filtering equations, which are shown to converge to a Stratonovich interpretation of the Kushner-Stratonovich equation. This smooth formulation will then be used to draw precise connections between nonlinear stochastic filtering and replicator-mutator dynamics. Additionally, gradient flow formulations will be investigated as well as a form of replicator-mutator dynamics which is shown to be beneficial for the misspecified model filtering problem. It is hoped this work will spur further research into exchanges between sequential learning and evolutionary biology and to inspire new algorithms in filtering and sampling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connections between sequential Bayesian inference and evolutionary dynamics Pathiraja, Sahani Wacker, Philipp Probability Populations and Evolution Methodology Machine Learning 35R60, 92-10, 60G35, 62F15, It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research in which this precise connection is rigorously established in the continuous time setting. Here we focus on a partial differential equation known as the Kushner-Stratonovich equation describing the evolution of the posterior density in time. Of particular importance is a piecewise smooth approximation of the observation path from which the discrete time filtering equations, which are shown to converge to a Stratonovich interpretation of the Kushner-Stratonovich equation. This smooth formulation will then be used to draw precise connections between nonlinear stochastic filtering and replicator-mutator dynamics. Additionally, gradient flow formulations will be investigated as well as a form of replicator-mutator dynamics which is shown to be beneficial for the misspecified model filtering problem. It is hoped this work will spur further research into exchanges between sequential learning and evolutionary biology and to inspire new algorithms in filtering and sampling. |
| title | Connections between sequential Bayesian inference and evolutionary dynamics |
| topic | Probability Populations and Evolution Methodology Machine Learning 35R60, 92-10, 60G35, 62F15, |
| url | https://arxiv.org/abs/2411.16366 |