This content of this roadmap follows Prof. Jordan's lectures/textbook. I'll update it as I/we work through the material. Send me an email if you'd like to contribute (colorado _AT_ berkeley _DOT_ edu) ## Conditional Independence and Factorization * Much of our early discussion focused on [[conditional independence]] in the context of [directed graphical models (Bayes nets)](Bayesian networks) and [undirected graphical models (Markov random fields - MRFs)](Markov random fields) * We can use the [[Bayes Ball]] algorithm to determine conditional independencies in Bayes nets. * We can use simple [reachability algorithms](http://en.wikipedia.org/wiki/Reachability) to determine conditional independencies in MRFs * We briefly discussed [[factor graphs]], which provide a more fine-grained representation of the independencies in a MRF ## Exact Inference * The [[variable elimination]] algorithm is based on interchanging sums and products in the definitions of marginals or partition functions. * The [sum product algorithm](sum_product_on_trees) is a belief propagation algorithm based on dynamic programming. It has the advantage over naive variable elimination in that it reuses computations to compute marginals for all nodes in the graph * [[junction trees]] generalize the sum product algorithm to arbitrary graphs by grouping variables together into cliques such that the cliques form a tree. ## Sampling-based inference * [[rejection sampling]] is a Monte Carlo method for sampling from a potentially complex distribution p(x) given a simpler distribution q(x) * [[importance sampling]] is a way of estimating expectations under an intractable distribution p by sampling from a tractable distribution q and reweighting the samples according to the ratio of the probabilities * We discussed some standard [[Markov chain Monte Carlo]] methods: * [[Metropolis-Hastings algorithm]] is a very general method for approximately sampling from a distribution p by defining a Markov chain which has p as a stationary distribution * [[Gibbs sampling]] is an MCMC algorithm where each random variable is iteratively resampled from its conditional distribution given the remaining variables -- it can be viewed [as a special case of Metropolis-Hastings](gibbs_as_mh) * we also touched on determining [[MCMC convergence]] * we can use [simulated annealing](simmulated_annealing) to determine a [MAP estimates](map_parameter_estimation) * Some fancier MCMC algorithms we touched on include: * [hybrid Monte Carlo](Hamiltonian Monte Carlo ) * [slice sampling](slice_sampling) * [[reversible jump MCMC]] * [[sequential Monte Carlo]] ## Statistical Concepts We discussed Bayesian vs frequentist inference; some topics we touched on include: * [[maximum likelihood]] and [[asymptotics of maximum likelihood]] and the various prerequisites for these concepts * [bias-variance decomposition](bias_variance_decomposition) * [[Bayesian model averaging]] ## Linear Regression and the Least Mean Squares algorithm * We discussed [[linear regression]] and its [closed form solution](linear_regression_closed_form) and their various prerequisites