  Suppose that a random variable $X$ of interest is observed perturbed by
independent additive noise $Y$. This paper concerns the "the least favorable
perturbation" $\hat Y_\ep$, which maximizes the prediction error
$E(X-E(X|X+Y))^2$ in the class of $Y$ with $ \var (Y)\leq \ep$. We find a
characterization of the answer to this question, and show by example that it
can be surprisingly complicated. However, in the special case where $X$ is
infinitely divisible, the solution is complete and simple. We also explore the
conjecture that noisier $Y$ makes prediction worse.
