The paper introduces BDLOB, a Bayesian deep convolutional neural network for predicting short-term price movements from limit order book (LOB) data. By applying dropout variational inference to the DeepLOB architecture, the model not only improves predictive performance through stochastic regularization but also provides uncertainty estimates via Monte Carlo dropout. The authors demonstrate that these uncertainty measures, specifically predictive entropy, can be effectively used for position sizing in trading strategies, leading to higher profits and reduced risk compared to strategies relying solely on softmax probabilities. The model is evaluated on five stocks from the London Stock Exchange over a full year, with millions of observations, showing consistent improvements in accuracy and trading returns. The work is the first to apply Bayesian neural networks to LOB data, highlighting the potential of uncertainty-aware deep learning in high-frequency trading.