Battery health monitoring and management is of extreme importance for the performance and cost of electric vehicles. This paper is concerned with machine-learning-enabled battery state-of-health (SOH) indication and prognosis. The sample entropy of short voltage sequence is used as an effective signature of capacity loss. Advanced sparse Bayesian predictive modeling (SBPM) methodology is employed to capture the underlying correspondence between the capacity loss and sample entropy. The SBPM-based SOH monitor is compared with a polynomial model developed in our prior work. The proposed approach allows for an analytical integration of temperature effects such that an explicitly temperature-perspective SOH estimator is established, whose performance and complexity is contrasted to the support vector machine (SVM) scheme. The forecast of remaining useful life is also performed via a combination of SBPM and bootstrap sampling concepts. Large amounts of experimental data from multiple lithium-ion battery cells at three different temperatures are deployed for model construction, verification, and comparison. Such a multi-cell setting is more useful and valuable than only considering a single cell (a common scenario). This is the first known application of combined sample entropy and SBPM to battery health prognosis.