Lecture 18: Variable Augmentation & MCMC Foundations

Recall that Gibbs sampling is frequently used in the following two (related) settings: • It is difficult or impossible to sample x = (x1,..., xp) directly, but is possible to conditionally sample xi | x1,..., xi−1, xi+1,..., xp for all i = 1,..., p. • It is difficult or impossible to sample x directly, but there exists a “latent variable ” y such that it is possible to conditionally sample x | y and y | x. The latter setting is referred to as variable augmentation. Observe that, in this case, the Gibbs sampler returns samples of the form (x, y); marginalizing to obtain samples xmay be accomplished by simply ignoring the y component of each (x, y) pair. We consider several concrete examples of variable augmentation. (a) (b) (c) Figure 1: Probit regression model augmented with variable Z = (Z1,..., ZN): (a) with the observed data shaded (b) ‘observed ’ data associated with the Gibbs conditional β | X,Y, Z shaded (c) ‘observed ’ data associated with the Gibbs conditional Z | β,X, Y shaded. Example 1 (Probit Regression). A probit model assigns to eachXi ∈ Rp a random variable Yi ∈ {0, 1} using the law Pr(Yi = 1) = Φ(βTXi), where Φ is the cumulative distribution function for the standard normal. Assuming (for convenience) a flat prior on β, and X = (X1,..., XN) with outcomes Y = (Y1,..., YN): p(β | X,Y) ∝ p(Y | X,β)p(β) = N∏ i=