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Recently, i have read a paper (Thesis et al., 2015) about dequantization method, which is a technique to transform a discrete variable to continuous variable.

Theis, L., Oord, A. v. d., and Bethge, M. A note on the evaluation of generative models. arXiv preprint arXiv:1511.01844, 2015.

However, i cannot understand a concepts that was introduced in 3.1 paragraph of the paper as followings:

when there are two conditions, (1) D-dimensional x is discrete variable, taking on values in {0,1,2,...,255} (2) The dequantized data x is given by y = x+u, where u is drawn uniformly from [0,1[^D

Authors said that they defined the relation between model density q(y)and probability mass function Q on x is as followings:

enter image description here

But, i cannot understand why this equation or definition is satisfied.

Please any body help me!!!!

sungho park
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  • Actually, In this paper, authors used like this expression: "we have defined" above equation. My question is that is it reasonable of defined equation? – sungho park May 18 '20 at 02:44

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This equation is in fact the convolution of the data/model density q, and the uniform distribution on $[0,1]^D$, thus the resulting density $Q$ is the density corresponding to the sum of the random variables $X\sim q(x)$ and $U\sim Uniform([0,1]^D)$.

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