The recent trend in adapting ultra-energy-efficient (but error-prone) nanomagnetic devices to non-Boolean computing and information processing (e.g. stochastic/probabilistic computing, neuromorphic, belief networks, etc) has resulted in rapid strides in new computing modalities. Of particular interest are Bayesian networks (BN) which may see revolutionary advances when adapted to a specific type of nanomagnetic devices. Here, we develop a novel nanomagnet-based computing substrate for BN that allows high-speed sampling from an arbitrary Bayesian graph. We show that magneto-tunneling junctions (MTJs) can be used for electrically programmable 'sub-nanosecond' probability sample generation by co-optimizing voltage-controlled magnetic anisotropy and spin transfer torque. We also discuss that just by engineering local magnetostriction in the soft layers of MTJs, one can stochastically couple them for programmable conditional sample generation as well. This obviates the need for extensive energy-inefficient hardware like OP-AMPS, gates, shift-registers, etc to generate the correlations. Based on the above findings, we present an architectural design and computation flow of the MTJ network to map an arbitrary Bayesian graph where we develop circuits to program and induce switching and interactions among MTJs. Our discussed framework can lead to a new generation of stochastic computing hardware for various other computing models, such as stochastic programming and Bayesian deep learning. This can spawn a novel genre of ultra-energy-efficient, extremely powerful computing paradigms, which is a transformational advance.