. (1989) consider these data under normal linear regression and Student regression and show support for the latter..

Apply Student t regression (Section 5.7) to the stack loss data in Example 4.4, with degrees of freedom ν an unknown. Lange et al. (1989) consider these data under normal linear regression and Student regression and show support for the latter. In fact they report an estimate ν = 1.1.

data, also much analysed, illustrate both predictor redundancy and observation outliers. They relate to percent of unconverted ammonia escaping from a plant during 21 days of operation in a stage in the production of nitric acid. The three predictors are as follows: x_{2}, airflow, a measure of the rate of operation of the plant; x_{3}, the inlet temperature of cooling water circulating through coils in a countercurrent absorption tower; and x_{4}, which is proportional to the concentration of acid in the tower. Small values of y correspond to efficient absorption of the nitric oxides. Previous analysis suggests x_{4} as most likely to be redundant and observations {3, 4, 21} as most likely to be outliers.

Here two methods for variable selection are considered and combined with outlier detection as in (4.7), with ω = 0.1 and η = 7. The assumed priors for β_{j} are N(0, 1000), while β_{1} ∼ N(20, 1000) and 1/σ^{2} ∼ Ga(1, 0.001). The product of the selection indicator and the sampled value of the coefficient is denoted by κ_{j} = δ_{j}β_{j}.

In the first model, variable selection is based on binary indicators δ_{j} ∼ Bern(0.5), j = 2, . . . , 4. A two-chain run of 10 000 iterations (1000 burn-in) shows highest posterior probabilities of outlier status for observations 4 and 21, namely 0.74 and 0.94, as compared to prior probabilities of 0.10. The posterior probabilty that δ_{2} = 1 is 1 (relating to the first predictor x_{2}), while those for the second and third predictors are 0.47 and 0.04. While the posterior density of κ_{2} is clearly confined to positive values, those for κ_{3} and κ_{4} straddle zero. One may obtain Bayes factors on various models by considering the K = 2^{3} models corresponding to combinations of δ^{(t)}_{j1} = 1 and δ^{(t)}_{j2} = 0 and accumulating over the iterations.