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• Variational_Bayesian_methods abstract "Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As is typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes:To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.To derive a lower bound for the marginal likelihood (sometimes called the "evidence") of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved variables). This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data. (See also the Bayes factor article.)In the former purpose (that of approximating a posterior probability), variational Bayes is an alternative to Monte Carlo sampling methods — particularly, Markov chain Monte Carlo methods such as Gibbs sampling — for taking a fully Bayesian approach to statistical inference over complex distributions that are difficult to directly evaluate or sample from. In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior.Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be solved analytically.For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed. However, deriving the set of equations used to iteratively update the parameters often requires a large amount of work compared with deriving the comparable Gibbs sampling equations. This is the case even for many models that are conceptually quite simple, as is demonstrated below in the case of a basic non-hierarchical model with only two parameters and no latent variables.".
• Variational_Bayesian_methods wikiPageExternalLink variational.html.
• Variational_Bayesian_methods wikiPageExternalLink index.html.
• Variational_Bayesian_methods wikiPageExternalLink vbayes.
• Variational_Bayesian_methods wikiPageExternalLink itila.
• Variational_Bayesian_methods wikiPageExternalLink fox_vbtut.pdf.
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• Variational_Bayesian_methods wikiPageLength "48410".
• Variational_Bayesian_methods wikiPageOutDegree "135".
• Variational_Bayesian_methods wikiPageRevisionID "670235730".
• Variational_Bayesian_methods wikiPageWikiLink Bayes_factor.
• Variational_Bayesian_methods wikiPageWikiLink Bayesian_inference.
• Variational_Bayesian_methods wikiPageWikiLink Bayesian_network.
• Variational_Bayesian_methods wikiPageWikiLink Bijection.
• Variational_Bayesian_methods wikiPageWikiLink Calculus_of_variations.
• Variational_Bayesian_methods wikiPageWikiLink Categorical_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Category:Bayesian_statistics.
• Variational_Bayesian_methods wikiPageWikiLink Circular_dependency.
• Variational_Bayesian_methods wikiPageWikiLink Completing_the_square.
• Variational_Bayesian_methods wikiPageWikiLink Conditional_independence.
• Variational_Bayesian_methods wikiPageWikiLink Conditional_probability_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Conditionally_independent.
• Variational_Bayesian_methods wikiPageWikiLink Conjugate_prior.
• Variational_Bayesian_methods wikiPageWikiLink Convex_function.
• Variational_Bayesian_methods wikiPageWikiLink Covariance_matrix.
• Variational_Bayesian_methods wikiPageWikiLink Credible_interval.
• Variational_Bayesian_methods wikiPageWikiLink David_J.C._MacKay.
• Variational_Bayesian_methods wikiPageWikiLink David_J._C._MacKay.
• Variational_Bayesian_methods wikiPageWikiLink Dirichlet_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Expectation-maximization.
• Variational_Bayesian_methods wikiPageWikiLink Expectation-maximization_algorithm.
• Variational_Bayesian_methods wikiPageWikiLink Expectation_maximization.
• Variational_Bayesian_methods wikiPageWikiLink Expectation_propagation.
• Variational_Bayesian_methods wikiPageWikiLink Expectation–maximization_algorithm.
• Variational_Bayesian_methods wikiPageWikiLink Expected_value.
• Variational_Bayesian_methods wikiPageWikiLink Exponential_family.
• Variational_Bayesian_methods wikiPageWikiLink Gamma_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Gaussian-Wishart_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Gaussian-gamma_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Gaussian_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Gaussian_mixture_model.
• Variational_Bayesian_methods wikiPageWikiLink Generalized_filtering.
• Variational_Bayesian_methods wikiPageWikiLink Gibbs_sampling.
• Variational_Bayesian_methods wikiPageWikiLink Global_maximum.
• Variational_Bayesian_methods wikiPageWikiLink Graphical_model.
• Variational_Bayesian_methods wikiPageWikiLink Hyperparameter.
• Variational_Bayesian_methods wikiPageWikiLink Hyperparameters.
• Variational_Bayesian_methods wikiPageWikiLink Independence_(probability_theory).
• Variational_Bayesian_methods wikiPageWikiLink Independent_and_identically_distributed_random_variables.
• Variational_Bayesian_methods wikiPageWikiLink Independent_identically_distributed.
• Variational_Bayesian_methods wikiPageWikiLink Integral.
• Variational_Bayesian_methods wikiPageWikiLink Iteration.
• Variational_Bayesian_methods wikiPageWikiLink Iterative.
• Variational_Bayesian_methods wikiPageWikiLink Joint_probability.
• Variational_Bayesian_methods wikiPageWikiLink Joint_probability_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Kullback–Leibler_divergence.
• Variational_Bayesian_methods wikiPageWikiLink Latent_variable.
• Variational_Bayesian_methods wikiPageWikiLink Limit_of_a_sequence.
• Variational_Bayesian_methods wikiPageWikiLink Lower_bound.
• Variational_Bayesian_methods wikiPageWikiLink Machine_learning.
• Variational_Bayesian_methods wikiPageWikiLink Marginal_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Marginal_likelihood.
• Variational_Bayesian_methods wikiPageWikiLink Marginal_probability.
• Variational_Bayesian_methods wikiPageWikiLink Markov_chain_Monte_Carlo.
• Variational_Bayesian_methods wikiPageWikiLink Maxima_and_minima.
• Variational_Bayesian_methods wikiPageWikiLink Maximum_a_posteriori.
• Variational_Bayesian_methods wikiPageWikiLink Maximum_a_posteriori_estimation.
• Variational_Bayesian_methods wikiPageWikiLink Maximum_likelihood.
• Variational_Bayesian_methods wikiPageWikiLink Mean.
• Variational_Bayesian_methods wikiPageWikiLink Mixture_model.
• Variational_Bayesian_methods wikiPageWikiLink Mode_(statistics).
• Variational_Bayesian_methods wikiPageWikiLink Model_evidence.
• Variational_Bayesian_methods wikiPageWikiLink Model_selection.
• Variational_Bayesian_methods wikiPageWikiLink Moment_(mathematics).
• Variational_Bayesian_methods wikiPageWikiLink Monte_Carlo_method.
• Variational_Bayesian_methods wikiPageWikiLink Monte_Carlo_sampling.
• Variational_Bayesian_methods wikiPageWikiLink Multinomial_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Multivariate_Gaussian_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Multivariate_normal_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Nonlinear.
• Variational_Bayesian_methods wikiPageWikiLink Nonlinear_system.
• Variational_Bayesian_methods wikiPageWikiLink Normal-Wishart_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Normal-gamma_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Normal-inverse-gamma_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Normal-scaled_inverse_gamma_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Normal_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Normalization_constant.
• Variational_Bayesian_methods wikiPageWikiLink Normalizing_constant.
• Variational_Bayesian_methods wikiPageWikiLink One-to-one_correspondence.
• Variational_Bayesian_methods wikiPageWikiLink Parameter.
• Variational_Bayesian_methods wikiPageWikiLink Partition_of_a_set.
• Variational_Bayesian_methods wikiPageWikiLink Posterior_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Posterior_probability.
• Variational_Bayesian_methods wikiPageWikiLink Precision_(statistics).
• Variational_Bayesian_methods wikiPageWikiLink Precision_matrix.
• Variational_Bayesian_methods wikiPageWikiLink Prior_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Prior_probability.
• Variational_Bayesian_methods wikiPageWikiLink Probability_density_function.
• Variational_Bayesian_methods wikiPageWikiLink Probability_distribution.
• Variational_Bayesian_methods wikiPageWikiLink Random_variable.

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