Variational Gaussian Copula Inference

We use Gaussian copulas (combined with fixed/free-form margins) as automated inference engines for variational approximation in generic hierarchical Bayesian models (the only two model-specific terms are the log likelihood & prior term and its derivatives). We evaluate the peculiarities reproduced in the univariate margins and the posterior dependence captured broadly across latent variables.

Matlab code for the paper

Shaobo Han, Xuejun Liao, David B. Dunson, and Lawrence Carin, <a href="http://people.ee.duke.edu/~lcarin/VGC_AISTATS2016.pdf"> "Variational Gaussian Copula Inference"</a>, The 19th International Conference on Artificial Intelligence and Statistics (AISTATS 2016), Cadiz, Spain, May, 2016

Examples

Demo 1: Marginal Adaptation (Skew normal, Student's t, Beta, Gamma)

>> demo_SkewNormal
>> demo_StudentT
>> demo_Gamma
>> demo_Beta

The accuracy of marginal approximation for real, positive real, and truncated [0,1] variables is shown as follows,

<a href="url"><img src="https://github.com/shaobohan/VariationalGaussianCopula/blob/master/figure/margins.png" align="center" height="200" width="800"></a>

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Demo 2: Bivariate Log-Normal

>> demo_BivariateLN

We approximate bivariate log-normal distributions using a bivariate Gaussian copula with (1) fixed-form log-normal distributed margins (2) free-form Bernstein polynomial based margins,

<a href="url"><img src="https://github.com/shaobohan/VariationalGaussianCopula/blob/master/figure/lognormal.png" align="center" height="300" width="800"></a>

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Demo 3: Horseshoe Shrinkage

Baseline comparisons include:

• Gibbs sampler
• Mean-field VB
• VGC-LN-full: Gaussian copula with log-normal margins
• VGC-LN-diag: Independence copula with Log-normal margins
• VGC-BP-full: Gaussian copula with Bernstein polynomial margins

>> demo_Horseshoe

<a href="url"><img src="https://github.com/shaobohan/VariationalGaussianCopula/blob/master/figure/horseshoe.png" align="center" height="380" width="660"></a>

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Demo 4: Poisson Log-Linear Regression

MCMC sampler is implemented in <a href="http://mcmc-jags.sourceforge.net/"> JAGS</a>:

>> demo_JAGS_PoissonLogLinear

Variational Gaussian copula (VGC) inference:

>> demo_VGC_PoissonLogLinear

The univaraite margins and pairwise posteriors (JAGS v.s. VGC-BP) are shown below:

<a href="url"><img src="https://github.com/shaobohan/VariationalGaussianCopula/blob/master/figure/VGC-JAGS.png" align="center" height="550" width="800"></a>

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Citations

If you find this code helpful, please cite the work using the following information:

@inproceedings{VGC_2016,
  title={Variational Gaussian Copula Inference},
  author={Shaobo Han and Xuejun Liao and David B. Dunson and Lawrence Carin},
  booktitle={The 19th International Conference on Artificial Intelligence and Statistics (AISTATS)},
  year={2016},
}