# reparameterization trick on joint random variables (speculative…)

10 August 2017

goal: estimate the gradient $\frac{\partial}{\partial \theta} \E[f_\theta((X, Y)_\theta)]$ where the joint random variable $(X, Y)_\theta$ (which is parameterized by $\theta$) can be factorized (need to wait until i learn more about product measures :-( …) as

• independent random variable $X_\theta$ and
• conditional random variable $(Y \given x)_\theta$.

hence (?), $(X, Y)_\theta = (X_\theta, (Y \given X_\theta)_\theta)$.

let’s assume that we can find a random variables $A, B$ and a mappings $g_\theta, h_\theta$ such that \begin{align} g_\theta(A) &= X_\theta \\ h_\theta(B, x) &= (Y \given x)_\theta. \end{align}

hence $(X, Y)_\theta = (g_\theta(A), h_\theta(B, g_\theta(A)))$ and we can write

which can be approximated using a monte carlo estimator \begin{align} \hat I_N &= \frac{1}{N} \sum_{n = 1}^N \frac{\partial}{\partial \theta} \left[f_\theta(g_\theta(a_n), h_\theta(b_n, g_\theta(a_n)))\right] \end{align} where $a_n$ and $b_n$ are independent samples of random variables $A$ and $B$ respectively.

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