Last updated: 2020-10-27

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Rmd 5a6cc2a DongyueXie 2020-10-27 wflow_publish(c(“analysis/index.Rmd”, “analysis/kl.Rmd”, “analysis/polya_gamma.Rmd”,

This paper provides a connection between Jaakkola and Jordan (2000) and Polya-gamma augmentation for logistic regression.

VB

David Blei’s paper, Variational inference: A review for statisticians, has been cited a lot as good VB introduction.(I guess I should re-read it several times, and often). VB tries to find a tractable approximation of the posterir distribution \(p(\theta|y)\) where \(\theta\) is coeffients and \(y\) is observed data.

The approximation is posed as an optimization problem, by minimizing KL divergence, \[ KL(q(\theta)||p(\theta|y)) = \int q(\theta)\log\frac{q(\theta)}{p(\theta|y)}d\theta = \int q(\theta)\log\frac{q(\theta)p(y)}{p(y,\theta)}d\theta. \]

(On why its \(KL(q(\theta)||p(\theta|y))\) instead of \(KL(p(\theta|y)||q(\theta))\))

The evidence lower bound ELBO is \[ELBO(q(\theta)) = \int q(\theta))\log\frac{p(y,\theta)}{q(\theta)} = \log p(y) - KL(q(\theta)||p(\theta|y)) \\= E(\log p(y|\theta)) - KL(q(\theta)||p(\theta))\]

The optimal \(q_j(\theta_j)\) is proportional to the exp expected log of complete likelihood, \[q^*_j(\theta_j)\propto exp\{E_{-j}\log p(\theta_j|\theta_{-j},y)\}.\]

(Why? ELBO\((q_j) = E_j E_{-j}\log p(\theta_j,\theta_{-j},y) - E_j \log q_j(\theta_j)+constant \\= -E_j\log\frac{q_j}{\exp{(E_{-j}\log p(\theta_j,\theta_{-j},y))}}\))

Mean–field VB for conditionally conjugate exponential family

Suppose each complete conditional is in the exponential family \[p(\theta_j|\theta_{-j},y) = h(\theta_j)\exp\{\eta_j(\theta_{-j},y)^T\theta_j - a(\eta_j(\theta_{-j},y))\}\] where \(h\) is a base measure and \(a(\cdot)\) is the log normalizer and \(\eta\) is the canonical parameter.

(The exponential family is \(f(y;\theta,\phi) = \exp\{(y\theta-b(\theta))/a(\phi)+c(y,\phi)\}\). If the dispersion parameter \(\phi\) is known, this is an exponential family model with canonical parameter \(\theta\). often \(a(\phi)=1\), \(c(y,\phi) = c(y)\))

Then the update is \[q_j(\theta_j)\propto \exp\{\log h(\theta_j)+E(\eta_j(\theta_{-j},y))^T\theta_j - E(a(\eta_j(\theta_{-j},y)))\} \\ \propto h(\theta_j)\exp\{E(\eta_j(\theta_{-j},y))^T\theta_j\}\]

Let \(\beta\) be the global coefficients and \(z\) be the local variables, and \(y\) the observed data, then the joint distribution of \(y,\beta,z\) is \[p(y,\beta,z) = p(\beta)\Pi_i p(z_i|\beta)p(y_i|z_i,\beta) = p(\beta)\Pi_i p(y_i,z_i|\beta)\] where \(\Pi_i p(y_i,z_i|\beta)\) is from an exponential family and \(p(\beta)\) is a conjugate prior for the density.

logistic likelihood