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This is the video for NIPS 2017 paper: estimating mutual information for discrete-continuous
mixtures.
The motivation of this work is to compute the mutual information between two random
variables X and Y from samples.
Mutual information can be applied for gene network inference, or explain neural networks
through information bottleneck method.
However, when computing mutual information in real applications, we sometimes observe
a negative number, which contradicts with the fact that mutual information is always
non-negative.
The problem is that some random variable have a mixture of discrete and continuous distributions.
As far as we know, there are no existing mutual information estimators for mixed random variables
with theoretical guarantees.
The goal of this paper is to develop an algorithm to compute mutual information for such discrete-continuous
mixtures with theoretical guarantee.
Note that mutual information is well-defined for any probability measure, so proposed algorithm
should work for any probability measure.
Our proposed algorithm is based on k-nearest-neighbor methods.
Let rho_i be the k-nearest-neighbor distance for sample i. k tilde i is the number of samples
within such distance.
Usually k tilde i equals to k, unless the sample comes from a discrete distribution.
k_x,i is the number of samples whose X-distance is smaller than rho_i and k_y,i is similarly
defined.
Our algorithm is given in the box.
We can prove that, if k is chosen correctly as a function of n, then under certain assumptions
on the probability measure, our mutual information estimator is L-2 consistent.
Those assumptions are mild enough to cover almost every real world problem.
We run several experiments to demonstrate the performance of our algorithm.
In synthetic experiments, we test the following cases -- 1.
X is discrete and Y is continuous, 2.(X,Y) is a mixture of continuous and discrete distributions.
We can see that the mean squared error of our algorithm beat other algorithms if enough
samples are provided.
We also test our algorithm on gene network inference problem.
There are twenty genes influencing each other through a network.
We compute the mutual information between pairs of genes, and claim that they are connected
if mutual information is greater than a threshold.
Gene data will be dropped out to 0 with certain probability p.
Our algorithm has a larger area under ROC for different dropout level p.
For more details of our paper, please attend the spotlight presentation on Tuesday 3:45
at Hall C, and poster on Tuesday evening number 220.
Thank you for watching.
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