In most examples of inference and prediction, the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model M that formalizes assumptions about how x and y are related. M will typically have two parts: structural assumptions S, such as the form of the link function and the choice of error distribution in a generalized linear model, and parameters ` whose meaning is specific to a given choice of S. It is common in statistical theory and practice to acknowledge parametric uncertainty about ` given a particular assumed structure S; it is less common to acknowledge structural uncertainty about S itself. A widely used approach, in fact, involves enlisting the aid of x to specify a plausible single "best " choice S for S, and then proceeding as if S were known to be correct. In general this approach fails to fully assess and propagate structural uncertainty, and may lead to miscalibrated uncertainty assessments about y given x. When miscalibration occurs it will often be in the direction of understatement of inferential or predictive uncertainty about y, leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty. In this paper I discuss a Bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible, by virtue of recent computational advances, and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the U.S. Space Shuttle.