The FACES dataset

For the third density estimation experiment, we used a subset of 576 images from a normalized face images dataset [Philips, Moon, Rauss, Rizvi 1997]. These images were downsampled to 48 variables (pixels) and 5 gray levels. We divided the data randomly into $N_{train}=500$ and $N_{test}=76$ examples; of the 500 training examples, 50 were left out as a validation set and used to select $m$ and $\alpha$ for the MT and MF models. The results in table 5 show the mixture of trees as the clear winner. Moreover, the MT achieves this performance with almost 5 times fewer parameters than the second best model, the mixture of 24 factorial distributions. Note that an essential ingredient of the success of the MT both here and in the digits experiments is that the data are ``normalized'', i.e., a pixel/variable corresponds approximately to the same location on the underlying digit or face. We do not expect MTs to perform well on randomly chosen image patches.

Table 5: Density estimation results for the mixtures of trees and other models on a the FACES data set. Average and standard deviation over 10 trials.

Model	Train likelihood	Test likelihood
	[bits/data point]	[bits/data point]
Mixture of trees $m=2, \alpha=10$	52.77 $\pm$0.33	56.29 $\pm$ 1.67
Mixture of factorials $m=24, \alpha=100$	56.34 $\pm$ 0.48	64.41 $\pm$ 2.11
Base rate	75.84	74.27
gzip	-	103.51