We are concerned with the problem of image segmentation, in which each pixel is assigned to one of a predefined finite number of labels. In Bayesian image analysis, this requires fusing together local predictions for the class labels with a prior model of label images. Following the work of Bouman and Shapiro (1994), we consider the use of tree-structured belief networks (TSBNs) as prior models. The parameters in the TSBN are trained using a maximum-likelihood objective function with the EM algorithm and the resulting model is evaluated by calculating how efficiently it codes label images. A number of authors have used Gaussian mixture models to connect the label field to the image data. We compare this approach to the scaled-likelihood method of Smyth (1994) and Morgan and Bourlard (1995), where local predictions of pixel classification from neural networks are fused with the TSBN prior. Our results show a higher performance is obtained with the neural networks. We evaluate the classification results obtained and emphasize not only the maximum a posteriori segmentation, but also the uncertainty, as evidenced e.g., by the pixelwise posterior marginal entropies. We also investigate the use of conditional maximum-likelihood training for the TSBN and find that this gives rise to improved classification performance over the ML-trained TSBN.

Recently Hinton (1999) has introduced the Products of Experts (PoE) model in which several individual probabilistic models for data are combined to provide an overall model of the data. Be(cid:173) low we consider PoE models in which each expert is a Gaussian. Although the product of Gaussians is also a Gaussian, if each Gaus(cid:173) sian has a simple structure the product can have a richer structure. We examine (1) Products of Gaussian pancakes which give rise to probabilistic Minor Components Analysis, (2) products of I-factor PPCA models and (3) a products of experts construction for an AR(l) process. Recently Hinton (1999) has introduced the Products of Experts (PoE) model in which several individual probabilistic models for data are combined to provide an overall model of the data. In this paper we consider PoE models in which each expert is a Gaussian. It is easy to see that in this case the product model will also be Gaussian. However, if each Gaussian has a simple structure, the product can have a richer structure. Using Gaussian experts is attractive as it permits a thorough analysis of the product architecture, which can be difficult with other models, e.g. models defined over discrete random variables. Below we examine three cases of the products of Gaussians construction: (1) Prod(cid:173) ucts of Gaussian pancakes (PoGP) which give rise to probabilistic Minor Compo(cid:173) nents Analysis (MCA), providing a complementary result to probabilistic Principal Components Analysis (PPCA) obtained by Tipping and Bishop (1999); (2) Prod(cid:173) ucts of I-factor PPCA models; (3) A products of experts construction for an AR(l) process. Products of Gaussians If each expert is a Gaussian pi(xI8i ) '" N(J1i' ( i), the resulting distribution of the product of m Gaussians may be expressed as By completing the square in the exponent it may be easily shown that p(xI8) N(/1;E, (2:), where (E l = 2::1 (i l . To simplify the following derivations we will assume that pi(xI8i ) '" N(O, (i) and thus that p(xI8) '" N(O, (2:). J12: i ° can be obtained by translation of the coordinate system. 1 Products of Gaussian Pancakes A Gaussian "pancake" (GP) is a d-dimensional Gaussian, contracted in one dimen(cid:173) sion and elongated in the other d - 1 dimensions. In this section we show that the maximum likelihood solution for a product of Gaussian pancakes (PoGP) yields a probabilistic formulation of Minor Components Analysis (MCA). 1.1 Covariance Structure of a GP Expert Consider a d-dimensional Gaussian whose probability contours are contracted in the direction w and equally elongated in mutually orthogonal directions VI , ... , vd-l.We call this a Gaussian pancake or GP. Its inverse covariance may be written as