Sampling, variational Bayesian inference, and conditioned stochastic differential equations
Sampling, variational Bayesian inference, and conditioned stochastic differential equations
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DOI：
10.1109/cdc45484.2021.9683159
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发表时间：
2021-12
期刊：
影响因子：
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通讯作者：
T. Coleman;M. Raginsky
中科院分区：
文献类型：
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作者：
T. Coleman;M. Raginsky
我们考虑概率生成模型中的抽样和统计推理问题，其中潜在目标是一个有限维的扩散过程。一般来说，很难得到对数似然的精确表达式，因此人们不得不求助于所谓的变分推理，其中使用度量变化来提出一个可处理的上界。我们首先使用W. Fleming的对数变换表明，构造对数似然的变分近似的问题可以解释为最优控制问题，其中变分近似的选择相当于在原始扩散上添加漂移。然后，我们使用F. Baudoin的条件随机微分方程的形式来分析这类控制问题。讨论了该问题与熵最优输运和随机极大值原理的关系。
We consider the problem of sampling and statistical inference in probabilistic generative models, where the latent object is a finite-dimensional diffusion process. In general, it is difficult to obtain exact expressions for the log-likelihood, so one has to resort to so-called variational inference, where a change of measure is used to come up with a tractable upper bound. We first show, using W. Fleming’s logarithmic transformation, that the problem of constructing a variational approximation to the log-likelihood can be interpreted as an optimal control problem, where the choice of a variational approximation amounts to adding a drift to the original diffusion. We then analyze this class of control problems using the formalism of conditioned stochastic differential equations due to F. Baudoin. We discuss the relation of this problem to entropic optimal transport and to the stochastic maximum principle.