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Approximate Bayesian Learning Using Bregman Divergence

Approximate Bayesian Learning Using Bregman Divergence. Kazuho Watanabe Nara Institute of Science and Technology. Outline. General framework of LVA using Bregman divergence Its application to latent variable models. Local Variational Approximation (LVA) for Bayesian Learning

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Approximate Bayesian Learning Using Bregman Divergence

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  1. Approximate Bayesian Learning Using Bregman Divergence Kazuho Watanabe Nara Institute of Science and Technology

  2. Outline • General framework of LVA using Bregman divergence • Its application to latent variable models Local Variational Approximation (LVA) for Bayesian Learning logistic regression (Jaakkola & Jordan, 2000) sparse linear model (Seeger, 2008) Contents:

  3. Bayesian Learning Training data: Parameter: Model: Prior: Posterior: intractable Marginal likelihood: Predictive distribution:

  4. Example: Logistic Regression Likelihood: Prior: intractable

  5. Local Variational Approximation Normalize Variational parameter Approximating Posteriors

  6. Local Variational Approximation (Example) (Jaakkola & Jordan, 2000) Convex w.r.t. quadratic function of w : normal Convex w.r.t. : normal linear function of w

  7. General Framework of LVA based on Convexity ,    :convex functions Bregman divergence Example:

  8. Free Energy Bounds : Free energy Upper bound minimization :posterior :approximation EMalgorithm: Lower bound maximization :posterior :approximation

  9. Latent Variable Models Latent variables: Model: Ex) Gaussian mixture model, Hidden Markov model

  10. LVA for Latent Variable Models Variational free energy Approximating posterior Variational Bayes

  11. Asymptotic Analysis of Variational Free Energy :Entropy i.i.d. Variational Bayes cf.) Bayes (S.Watanabe, 2001) : log-sum inequality

  12. Conclusion General framework for local variational approximation using Bregman divergence Its application to latent variable models

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