1 Introduction -- 2 Probabilistic Inference in Signal Processing -- 2.1 Introduction -- 2.2 The likelihood function -- 2.3 Bayesian data analysis -- 2.4 Prior probabilities -- 2.5 The removal of nuisance parameters -- 2.6 Model selection using Bayesian evidence -- 2.7 The general linear model -- 2.8 Interpretations of the general linear model -- 2.9 Example of marginalization -- 2.10 Example of model selection -- 2.11 Concluding remarks -- 3 Numerical Bayesian Inference -- 3.1 The normal approximation -- 3.2 Optimization -- 3.3 Integration -- 3.4 Numerical quadrature -- 3.5 Asymptotic approximations -- 3.6 The Monte Carlo method -- 3.7 The generation of random variates -- 3.8 Evidence using importance sampling -- 3.9 Marginal densities -- 3.10 Opportunities for variance reduction -- 3.11 Summary -- 4 Markov Chain Monte Carlo Methods -- 4.1 Introduction -- 4.2 Background on Markov chains -- 4.3 The canonical distribution -- 4.4 The Gibbs sampler -- 4.5 The Metropolis-Hastings algorithm -- 4.6 Dynamical sampling methods -- 4.7 Implementation of simulated annealing -- 4.8 Other issues -- 4.9 Free energy estimation -- 4.10 Summary -- 5 Retrospective Changepoint Detection -- 5.1 Introduction -- 5.2 The simple Bayesian step detector -- 5.3 The detection of changepoints using the general linear model -- 5.4 Recursive Bayesian estimation -- 5.5 Detection of multiple changepoints -- 5.6 Implementation details -- 5.7 Multiple changepoint results -- 5.8 Concluding Remarks -- 6 Restoration of Missing Samples in Digital Audio Signals -- 6.1 Introduction -- 6.2 Model formulation -- 6.3 The EM algorithm -- 6.4 Gibbs sampling -- 6.5 Implementation issues -- 6.6 Relationship between the three restoration methods -- 6.7 Simulations -- 6.8 Discussion -- 6.9 Concluding remarks -- 7 Integration in Bayesian Data Analysis -- 7.1 Polynomial data -- 7.2 Decay problem -- 7.3 General model selection -- 7.4 Summary -- 8 Conclusion -- 8.1 A review of the work -- 8.2 Further work -- A The General Linear Model -- A.1 Integrating out model amplitudes -- A.1.1 Least squares -- A.1.2 Orthogonalization -- A.2 Integrating out the standard deviation -- A.3 Marginal density for a linear coefficient -- A.4 Marginal density for standard deviation -- A.5 Conditional density for a linear coefficient -- A.6 Conditional density for standard deviation -- B Sampling from a Multivariate Gaussian Density -- C Hybrid Monte Carlo Derivations -- C.1 Full Gaussian likelihood -- C.2 Student-t distribution -- C.3 Remark -- D EM Algorithm Derivations -- D.l Expectation -- D.2 Maximization -- E Issues in Sampling Based Approaches to Integration -- E.1 Marginalizing using the conditional density -- E.2 Approximating the conditional density -- E.3 Gibbs sampling from the joint density -- E.4 Reverse importance sampling -- F Detailed Balance -- F.1 Detailed balance in the Gibbs sampler -- F.2 Detailed balance in the Metropolis Hastings algorithm. -- F.3 Detailed balance in the Hybrid Monte Carlo algorithm. -- F.4 Remarks -- References.