In this paper, we study the problem of personalized product search under streaming scenarios. We address the problem by proposing a Dynamic Bayesian Metric Learning model, abbreviated as DBML, which can collaboratively track the evolutions of latent semantic representations of different categories of entities (i.e., users, products and words) over time in a joint metric space. In particular, unlike previous work using inner-product metric to model the affinities between entities, our DBML is a novel probabilistic metric learning approach that is able to avoid the contradicts, keep the triangle inequality in the latent space, and correctly utilize implicit feedbacks. For inferring dynamic embeddings of the entities, we propose a scalable online inference algorithm, which can jointly learn the latent representations of entities and smooth their changes across time, based on amortized inference. The inferred dynamic semantic representations of entities collaboratively inferred in a unified form by our DBML can benefit not only for improving personalized product search, but also for capturing the affinities between users, products and words. Experimental results on large datasets over a number of applications demonstrate that our DBML outperforms the state-of-the-art algorithms, and can effectively capture the evolutions of semantic representations of different categories of entities over time.