Continuous Sigmoidal Belief Networks Trained using Slice Sampling

These include Boltzmann machines (Hinton and Sejnowski 1986), binary sigmoidal belief networks (Neal 1992) and Helmholtz machines (Hinton et al. 1995; Dayan et al. 1995). However, some hidden variables, such as translation or scaling in images of shapes, are best represented using continuous values. Continuous-valued Boltzmann machines have been developed (Movellan and McClelland 1993), but these suffer from long simulation settling times and the requirement of a "negative phase" during learning. Tibshirani (1992) and Bishop et al. (1996) consider learning mappings from a continuous latent variable space to a higher-dimensional input space. MacKay (1995) has developed "density networks" that can model both continuous and categorical latent spaces using stochasticity at the topmost network layer. In this paper I consider a new hierarchical top-down connectionist model that has stochastic hidden variables at all layers; moreover, these variables can adapt to be continuous or categorical. The proposed top-down model can be viewed as a continuous-valued belief network, which can be simulated by performing a quick top-down pass (Pearl 1988).