Probability theory: scientific inference, Bayes’ theorem, probability and frequency, marginalization, fundamentals of logic, logic functions. Bayesian inference: parameter estimation, nuisance parameters, Occam’s razor, ignorance priors, systematic errors, spectral analysis, Bayesian inference with Posisson sampling. Frequentist statistical inference: sampling theory, probability distributions, discrete and continuous distributions, central limit theorem, chi-squared distribution, sample variance, Student’s t distribution, confidence intervals. Hypothesis testing: testing with the chi-squared statistic, goodness-of-fit test, problems with frequentist hypothesis and Bayesian resolution. Model fitting: Bayesian inference with Gaussian errors, linear model fitting, regression analysis, model parameter errors, correlated data errors; nonlinear model fitting: iterative linearization, Levenberg-Marquardt method. Markov chain Monte Carlo: Metropolis-Hastings algorithm, simulated and parallel tampering, automated MCMC. Neural networks: feed-forward network functions, network training, error backpropagation, Hessian matrix, regularization, Bayesian neural networks. Kernel methods: dual representations, building kernels, radial basis function networks, Gaussian processes. Sparse kernel machines: maximum margin classifiers, support vector machines (SVM) and relevance vector machine (RVM), RVM for regression and classification. Graphical models: Bayesian networks, conditional independence, Markov random fields, inference in graphical models. Mixture models and EM: K-means clustering, mixtures of Gaussians, maximum likelihood, expectation-maximization (EM) for Bayesian linear regression. Approximate inference: variational inference, variational mixture of Gaussians, variational linear regression, variational logistic regression, expectation propagation.