While modern machine learning has transformed numerous application domains, its growing computational demands increasingly constrain scalability and efficiency, particularly on embedded and resource-limited platforms. In practice, neural networks must not only operate efficiently but also provide reliable predictions under distributional shifts or unseen data. Bayesian neural networks offer a principled framework for quantifying uncertainty, yet their computational overhead further compounds these challenges. This work advances resource-efficient and robust inference for both conventional and Bayesian neural networks through the joint pursuit of algorithmic and hardware efficiency. The former reduces computation through model compression and approximate Bayesian inference, while the latter optimizes deployment on digital accelerators and explores analog hardware, bridging algorithmic design and physical realization. The first contribution, Galen, performs automatic layer-specific compression guided by sensitivity analysis and hardware-in-the-loop feedback. Analog accelerators offer efficiency gains at the cost of noise; this work models device imperfections and extends noisy training to nonstationary conditions, improving robustness and stability. A second line of work advances probabilistic inference, developing analytic and ensemble approximations that replace costly sampling, integrate into a compiler stack, and optimize embedded inference. Finally, probabilistic photonic computing introduces a paradigm where controlled analog noise acts as an intrinsic entropy source, enabling fast, energy-efficient probabilistic inference directly in hardware. Together, these studies demonstrate how efficiency and reliability can be advanced jointly through algorithm-hardware co-design, laying the foundation for the next generation of trustworthy, energy-efficient machine-learning systems.