For the hierarchical model outlined m Section 19 .1, the conditional independence structure suggests a Gibbs sampling (Gelfand and Smith, 1990; see also Spiegelhalter et al., 1995: this volume). Let 0 = (01 , .. ,,BM) and y = (Yl,··•,YM)- Let [0; I.] denote the full conditional distribution of 0;, that is, the distribution of 0; given all other quantities in the model (see Gilks, 1995: this volume). Unfortunately, the nonlinear model f ( 0;, t;j) leads to a form for [0; I . ] that is non-standard and is known

up to a normalizing constant. VVe first give, for completeness, the form of the other conditional distributions. For further details, see Gelfand a!. (1990) and Wakefield et al. (1994)

Extension to a second-stage multivariate t-distribution with fixed degrees of freedom v and variance-covariance matrix is straightforward if we exploit the fact that such a distribution can expressed as a scale mixture of normals (Racine-Poon, 1992; Bernardo and Smith, 1994). For this, M additional parameters >. = ( A1 , ... , AM) are introduced with second-stage distribution 0i ~ N(µ, >.; 1LJ), where the univariate Ai parameters are assigned a Ga( f, ~) distribution at the third stage. Defining

v; = (0; - µ?E-1 (0; - µ) + v. Sampled values for A; substantially less than LO indicate that individual i may be outlying.