Robustness of Bayesian factor analysis estimates

The focus of this dissertation is the study of the robustness of the posterior distribution in the Bayesian factor analysis problem to variations in the assumptions about the prior distribution.

We will adopt the Bayesian approach to factor analysis developed in 1989 (Press and Shigemasu). For Bayesian analysis, the prior distribution permits the analyst to identify the model by bringing prior information about model to bear. At the same time, the analyst would generally prefer that conclusions about the model not be too sensitive to the prior assumptions. Therefore we would like to study the robustness of estimates in the Bayesian factor analysis model with respect to variation in the priors. Berger (1986) mentioned that an attractive method of modeling uncertainty in the prior distribution is through the use of $\epsilon$-contamination classes, i.e, classes of distributions which have the form: $\pi$ = (1 - $\epsilon)\pi\sb0 + \epsilon q, \pi\sb0$ being the base elicited prior, q being a "contamination prior," and $\epsilon$ reflecting the amount of error in $\pi\sb0$ that is possible.

We will use normal theory for the sampling distribution, and adopt a model with a full disturbance covariance matrix. Using an $\epsilon$-contamination class, with vague and natural conjugate priors for the parameters, we find that the marginal posterior distribution of the factor score is a linear combination of matrix T-distributions, in large samples. Factor loadings are estimated conditional on the estimated factor scores. Also, disturbance variance and covariance terms are estimated conditional on estimated factor scores and factor loading matrices.

The idea of robust Bayesian analysis is that it must be shown that the inference or decision to be made is essentially the same for any prior in $\Gamma$(a certain prior class). To quantify robustness we used several methods: (1) Absolute value of the means of estimates in each column of the factor loading matrix, (2) Mahalanobis Distance, $D\sp2,$ (3) Norm Distances, between two Bayes estimates, along with the changing of parameters in the prior distributions. (4) Directional derivatives that show the approximate ways to calculate the differences along with the directions on parameters.