Graphical models, such as Bayesian networks and Markov random fields, represent statistical dependencies of variables by a graph. Local “belief propagation” rules of the sort proposed by Pearl (1988) are guaranteed to converge to the correct posterior probabilities in singly connected graphs. Recently, good performance has been obtained by using these same rules on graphs with loops, a method we refer to as loopy belief propagation . Perhaps the most dramatic instance is the near Shannon-limit performance of “Turbo codes,” whose decoding algorithm is equivalent to loopy propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give sufficient conditions for convergence and show that when belief propagation converges, it gives the correct posterior means for all graph topologies, not just networks with a single loop. These results motivate using the powerful belief propagation algorithm in a broader class of networks and help clarify the empirical performance results.

Graphical models, such as Bayesian networks and Markov networks, represent joint distributions over a set of variables by means of a graph. When the graph is singly connected, local propagation rules of the sort proposed by Pearl (1988) are guaranteed to converge to the correct posterior probabilities. Recently a number of researchers have empirically demonstrated good performance of these same local propagation schemes on graphs with loops, but a theoretical understanding of this performance has yet to be achieved. For graphical models with a single loop, we derive an analytical relationship between the probabilities computed using local propagation and the correct marginals. Using this relationship we show a category of graphical models with loops for which local propagation gives rise to provably optimal maximum a posteriori assignments (although the computed marginals will be incorrect). We also show how nodes can use local information in the messages they receive in order to correct their computed marginals. We discuss how these results can be extended to graphical models with multiple loops and show simulation results suggesting that some properties of propagation on single-loop graphs may hold for a larger class of graphs. Specifically we discuss the implication of our results for understanding a class of recently proposed error-correcting codes known as turbo codes.

Performance of human subjects in a wide variety of early visual processing tasks improves with practice. HyperBF networks (Poggio and Girosi 1990) constitute a mathematically well-founded framework for understanding such improvement in performance, or perceptual learning, in the class of tasks known as visual hyperacuity. The present article concentrates on two issues raised by the recent psychophysical and computational findings reported in Poggio et al. (1992b) and Fahle and Edelman (1992). First, we develop a biologically plausible extension of the HyperBF model that takes into account basic features of the functional architecture of early vision. Second, we explore various learning modes that can coexist within the HyperBF framework and focus on two unsupervised learning rules that may be involved in hyperacuity learning. Finally, we report results of psychophysical experiments that are consistent with the hypothesis that activity-dependent presynaptic amplification may be involved in perceptual learning in hyperacuity.