This content of roadmap follows Prof. Jordan's lectures/textbook. ## 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 but can perform many redundant calculations. * [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 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~~+ * [[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 a 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)~~+ * [[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 also touched on determining [[MCMC convergence]] + * we can use [simulated annealing](simmulated_annealing) to determine a (MAP estimate)[map_parameter_estimation] + * Some fancier MCMC algorithms we touched oninclude: + * [hybrid Monte Carlo](Hamiltonian Monte Carlo ) + * [slice sampling](slice_sampling) + * [[reversible_jump_mcmc]] + * [[sequential_monte_carlo]]