Bayesian neural networks (BNNs) have been proposed to address the problems of overfitting and overconfident decision making, common in conventional neural networks (NNs), due to their ability to model and express uncertainty in their predictions. However, BNNs require multiple inference passes to produce the necessary posterior distributions used to generate these highly desirable uncertainty estimates. As such, BNNs require not only an efficient, high-performance multiply-accumulation (MAC) operation but also an efficient Gaussian random number generator (GRNG) with high-quality statistics. In this article, an NN accelerator chip, leveraging a multi-bit analog compute-in-memory (CiM) static random-access memory (SRAM) macro, with a tightly coupled and highly efficient GRNG scheme, is presented in the Intel 22FFL process. The CiM macro achieves a peak energy efficiency of 32.2 TOP/sW, with 8-bit precision, while ensuring accurate on-chip matrix-vector multiplications (MVMs) with a computation error less than 0.5%. The variable precision GRNG achieves a peak throughput of 7.31 GSamp/s for an energy efficiency of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim$</tex-math> </inline-formula> 1 TSamp/J. Overall, our proposed system achieves a peak energy efficiency of 1170 GOP/s/W, a 35–133 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times$</tex-math> </inline-formula> improvement over the state-of-the-art BNN accelerators, with 98.14% accuracy for the MNIST dataset.