conditional on θ A , p( z | θ A , A). The observables distribution typically involves both z and θ A : p( Y T | z , θ A , A). Clearly one could also have a hierarchical prior distribution for θ A in this context as well. Latent variables are convenient, but not essential, devices for describing the dis- tribution of observables, just as hyperparameters are convenient but not essential in constructing prior distributions. The convenience stems from the fact that the likeli- hood function is otherwise awkward to express, as the reader can readily verify for the stochastic volatility model. In these situations Bayesian inference then has to con- front the problem that it is impractical, if not impossible, to evaluate the likelihood function or even to provide an adequate numerical approximation. Tanner and Wong (1987) provided a systematic method for avoiding analytical integration in evaluating the likelihood function, through a simulation method they described as data augmenta- tion. Section 5.2.2 provides an example.