This paper introduces Expectation Propagation (EP), a new deterministic approximation technique for Bayesian networks. EP unifies and extends two previous methods: assumed-density filtering (ADF) and loopy belief propagation (LBP). ADF is a one-pass, sequential method for computing approximate posterior distributions, while LBP is useful for discrete belief networks but is limited by its exact belief state propagation. EP approximates belief states by retaining only expectations (mean and variance) and iterates until these expectations are consistent throughout the network, making it applicable to hybrid networks with both discrete and continuous nodes. The paper demonstrates EP's effectiveness through experiments with Gaussian mixture models, showing it to be more accurate than methods like Laplace's method, variational Bayes, and Monte Carlo. EP is also efficient for training Bayes Point Machine classifiers, a Bayesian approach to linear classification. The algorithm processes each data point in \(O(d^2)\) time, where \(d\) is the dimensionality, and can be extended to use arbitrary inner product functions. EP's iterations always have a fixed point, which can be found by minimizing an energy function. The paper concludes by highlighting EP's potential for other networks and its superior speed and accuracy compared to existing methods.This paper introduces Expectation Propagation (EP), a new deterministic approximation technique for Bayesian networks. EP unifies and extends two previous methods: assumed-density filtering (ADF) and loopy belief propagation (LBP). ADF is a one-pass, sequential method for computing approximate posterior distributions, while LBP is useful for discrete belief networks but is limited by its exact belief state propagation. EP approximates belief states by retaining only expectations (mean and variance) and iterates until these expectations are consistent throughout the network, making it applicable to hybrid networks with both discrete and continuous nodes. The paper demonstrates EP's effectiveness through experiments with Gaussian mixture models, showing it to be more accurate than methods like Laplace's method, variational Bayes, and Monte Carlo. EP is also efficient for training Bayes Point Machine classifiers, a Bayesian approach to linear classification. The algorithm processes each data point in \(O(d^2)\) time, where \(d\) is the dimensionality, and can be extended to use arbitrary inner product functions. EP's iterations always have a fixed point, which can be found by minimizing an energy function. The paper concludes by highlighting EP's potential for other networks and its superior speed and accuracy compared to existing methods.