Bayesian deep neural networks (DNN) provide a mathematically grounded framework to quantify uncertainty in their predictions. We propose a Bayesian variant of policy-gradient based reinforcement learning training technique for image captioning models to directly optimize non-differentiable image captioning quality metrics such as CIDEr-D. We extend the well-known Self-Critical Sequence Training (SCST) approach for image captioning models by incorporating Bayesian inference, and refer to it as B-SCST. The "baseline" reward for the policy-gradients in B-SCST is generated by averaging predictive quality metrics (CIDEr-D) of the captions drawn from the distribution obtained using a Bayesian DNN model. This predictive distribution is inferred using Monte Carlo (MC) dropout, which is one of the standard ways to approximate variational inference. We observe that B-SCST improves all the standard captioning quality scores on both Flickr30k and MS COCO datasets, compared to the SCST approach. We also provide a detailed study of uncertainty quantification for the predicted captions, and demonstrate that it correlates well with the CIDEr-D scores. To our knowledge, this is the first such analysis, and it can pave way to more practical image captioning solutions with interpretable models.