Transforms of random variates

Taylor expansion

Not complicated but subtle (Gustafsson and Hendeby 2012).

Consider a general nonlinear differentiable transformation \(g\) and its second order Taylor expansion. Consider the mapping \(g:\mathbb{R}^{n_{x}}\to\mathbb{R}^{n_{z}}\) applied to a variable \(x,\) defining \(z:=g(x).\) Let \(\mathrm{E}(x)=\mu_{x}\) and \(\operatorname{Var}(x)=P_{x}.\) The Hessian of the \(i^{\text {th }}\) component of \(g\) is denoted \(g_{i}^{\prime \prime}.\) \([x_i]_i\) is a vector where the \(i\)th element is \(x_i\). We will approximate \(z\) using the Taylor expansion, \[z=g\left(\mu_{x}\right)+g^{\prime}\left(\mu_{x}\right)\left(x-\mu_{x}\right)+\left[\frac{1}{2}\left(x-\mu_{x}\right)^{T} g_{i}^{\prime \prime}\left(\mu_{x}\right)\left(x-\mu_{x}\right)\right]_{i}.\] Leaving aside questions of when this is convergent for now. Then the first moment of \(z\) is given by \[ \mu_{z}=g\left(\mu_{x}\right)+\frac{1}{2}\left[\operatorname{tr}\left(g_{i}^{\prime \prime}\left(\mu_{x}\right) P_{x}\right)\right]_{i} \] Further, let \(x \sim \mathcal{N}\left(\mu_{x}, P_{x}\right)\), then the second moment of \(z\) is given by \[ P_{z}=g^{\prime}\left(\mu_{x}\right) P_{x}\left(g^{\prime}\left(\mu_{x}\right)\right)^{T}+\frac{1}{2}\left[\operatorname{tr}\left(g_{i}^{\prime \prime}\left(\mu_{x}\right) P_{x} g_{j}^{\prime \prime}\left(\mu_{x}\right) P_{x}\right)\right]_{i j} \] with \(i, j=1, \ldots, n_{z}.\)

This is commonly seen in the context of transforms of Gaussians.

Unscented transforms

Typically seen for Gaussian RVs. Ebeigbe et al. (2021) claims to have devised a method for more general RVs.

Stein’s lemma

As seen in Stein’s method. Gives us the special case of certain exponential RVS (typically Gaussian) under certain matched transforms. Long story.

Stochastic Itô-Taylor expansion

Taylor expansions for stochastic processes. See stochastic taylor expansion. tl;dr: Usually more trouble than it is worth.

References