We propose the stochastic optimal path which solves the classical optimal path problem by a probability-softening solution. This unified approach transforms a wide range of DP problems into directed acyclic graphs in which all paths follow a Gibbs distribution. We show the equivalence of the Gibbs distribution to a message-passing algorithm by the properties of the Gumbel distribution and give all the ingredients required for variational Bayesian inference of a latent path, namely Bayesian dynamic programming (BDP). We demonstrate the usage of BDP in the latent space of variational autoencoders (VAEs) and propose the BDP-VAE which captures structured sparse optimal paths as latent variables. This enables end-to-end training for generative tasks in which models rely on unobserved structural information. At last, we validate the behaviour of our approach and showcase its applicability in two real-world applications: text-to-speech and singing voice synthesis.

翻译：我们提出随机最优路径，通过概率软化解决方案求解经典最优路径问题。这一统一方法将大量动态规划问题转化为所有路径服从吉布斯分布的有向无环图。我们利用Gumbel分布的性质论证了吉布斯分布与消息传递算法的等价性，并给出了潜变量路径变分贝叶斯推断（即贝叶斯动态规划）所需的所有要素。我们展示了BDP在变分自编码器潜空间中的应用，并提出BDP-VAE模型，将结构化稀疏最优路径作为潜变量进行捕捉。这使得依赖未观测结构信息的生成式任务能够实现端到端训练。最后，我们验证了该方法的行为特性，并在文本转语音和歌声合成两个实际应用中展示了其适用性。