Bayesian deep neural networks (DNNs) can provide a mathematically grounded framework to quantify uncertainty in predictions from image captioning models. 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" 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. We infer this predictive distribution using Monte Carlo (MC) dropout approximate variational inference. We show that B-SCST improves CIDEr-D scores on Flickr30k, MS COCO and VizWiz image captioning datasets, compared to the SCST approach. We also provide a 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 improve the interpretability of image captioning model outputs, which is critical for practical applications.