for MNIST Data
-------------------------------

![\[fig:mnist\]Representative examples from randomly generated digits that show significant improvement after the refining step using SAGD.](mnist-comparison){width="0.95\columnwidth"}

In the last experiment, we consider the MNIST handwritten digits data set, and fit generative models on it. The dimension of the latent space is set to 20, and the generative network is a combination of convolutional filters and fully-connected layers. We first train a VAE model for 500 epochs with a batch size of 200, and then run SAGD for 100 epochs for fine-tuning. In SAGD, twenty independent chains are used to compute the approximate gradient, each with five burn-in’s.

Since SAGD basically refines the generative network of VAE, we can directly compare their output images. We randomly generate 100 digits from the trained VAE model and the debiased model, respectively, and in Figure \[fig:mnist\] we show some representative pairs of generated digits, with VAE-trained ones on the top, and SAGD-refined ones on the bottom. It is clear that the SAGD refining step improves the quality of the generated images. For example, in the first column of Figure \[fig:mnist\], VAE shows an ambiguous digit between “9” and “7”, but the refined one is a definite “7”.

Conclusion
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In 