# 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

$\frac{\partial}{\partial \theta} \E[f_\theta((X, Y)_\theta)] = \frac{\partial}{\partial \theta} \E\left[f_{\theta}(g_\theta(A), h_\theta(B, g_\theta(A)))\right] = \E\left[\frac{\partial}{\partial \theta} f_{\theta}(g_\theta(A), h_\theta(B, g_\theta(A)))\right]$

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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