This paper discusses the application of information-theoretic concepts to the backward filtering of time series using moving averages. We identify moving averages as time-dependent expectation values derived from maximum entropy probability kernels that are subject to relevant constraints. Constraining the width of the kernel results in the simple moving average, while constraining the typical timescale yields the exponential moving average. With a martingale constraint, we derive a moving average corresponding to a risk-neutral valuation scheme for financial time series. By expanding this framework to generalized forms of entropy, we introduce a broad family of maximum-entropy-based moving averages.