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If we have two separate probability distributions P(x) and Q(x) over the same random variable x, we can measure how different these two distributions are using the Kullback-Leibler (KL) divergence...

The above statement is from Deep Learning by Ian Goodfellow and Yoshua Bengio and Aaron Courville and I have the following question:

As far as I have understood, a random variable is defined considering a specific probability distribution in mind, it takes the value of a random outcome in that distribution. Perhaps I'm wrong in my understanding. My question is:

How can you have two separate probability distributions on the same random variable?

Kindly help me resolve this confusion. Thanks!

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Long story short: just ignore that part ("over the same random variable $x$").


  1. You are correct: a random variable is characterized by its distribution.

  2. You should think of the KL divergence as measuring a "distance" between two distributions $P$ and $Q$. No need to involve random variables.

  3. "over the same random variable $x$" might be referring to some of the following:

    • the two distributions $P$ and $Q$ should be on the same space. (Say, if $x$ is a real number, then they should be distributions on $\mathbb{R}$. If $x$ is one of $4$ discrete values, then $P$ and $Q$ should be distributions on those $4$ values.) Note that another technical issue is that you need $P(x)=0$ for $x$ such that $Q(x)=0$, to avoid division by zero.

    • maybe $x$ is some random variable on some space (see previous bullet) but with a yet-to-be specified distribution. $P$ and $Q$ are possible distributions for $x$.

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The uniqueness of the distribution of a random variable $\mathbf{x}$ implicitly implies that you consider a given measure $\mathbb{P}$.

Consider a probability space $(\Omega,\mathcal{F}, \mathbb{P})$, and a measurable space $(X,\mathcal{X})$. A random variable $\mathbf{x}$ is defined on $(\Omega,\mathcal{F}, \mathbb{P})$ as a measurable function $\mathbf{x}:~\Omega \to X$. Then $\mathbb{P}_{\mathbf{x}}=\mathbb{P}\circ \mathbf{x}^{-1}$ is a measure, and classically it is defined as the distribution of $\mathbf{x}$. Consider now the probability space $(\Omega,\mathcal{F}, \mathbb{Q})$, and the same function $\mathbf{x}:~\Omega \to X$. Then $\mathbb{Q}_{\mathbf{x}}=\mathbb{Q}\circ \mathbf{x}^{-1}$ is a also a measure. Therefore, if you define the random variable as a function, without a specific measure but only considering the measurable space $(\Omega,\mathcal{F})$, two different measures will give two different distributions. The KL divergence compares two measures, for a single measurable function $\mathbf{x}$.

However classically random variables are defined for a given probability measure $\mathbb{P}$, therefore have a given distribution $\mathbb{P}_{\mathbf{x}}$.

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Ideally there is only one "correct" probability measure for any real-world process.   However, there may be many competing theoretical models for that process.   So we need tools to compare the probability distributions of those models ; to find the best fit to experimental results and such.

That is roughly what you are dealing with.

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