Current material genomics approaches face challenges in efficiently exploring the high dimensional design space of polymer-bonded explosives (PBX), which play a critical role in understanding their sensitivity under shock loading conditions. These challenges lead to high computational costs and limited insights. Our innovative Bayesian learning framework addresses this by effectively navigating the complex design space of PBX, thereby reducing the computational expenses typically associated with physics-based simulations. This is particularly relevant for assessing the shock-to-detonation transition (SDT), a key factor in evaluating PBX sensitivity. For this purpose, our study focuses on how microstructural variations in PBX, such as void properties, shape, orientation, volume fraction, and spatial arrangements, impact detonation, assessing these factors against safety and ignition metrics such as distance to detonation and shock pressure. We first generate diverse synthetic microstructures to enhance the design space. Next, we use 2-point statistics for microstructural descriptors and apply Principal Component Analysis for dimensionality reduction. Finally, a Batch Active Learning algorithm on a Multi-Output Gaussian Process Regression model is employed to explore structure-property relationships with fewer simulations. The results highlighted that our framework can rapidly learn the primary microstructure attributes dominating the SDT behavior while reducing the uncertainty of predictions on the candidate microstructures by effectively exploring complex variations and guiding simulations. This maximizes information gain and minimizes computational costs, enhancing our understanding of PBX microstructures and setting new standards for material design in engineering.