For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes’ law. Duality of risk sensitivity to the multiplier version of min–max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I–III, the minimizing players’ worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.