Transforms of random variates



I have a nonlinear transformation of a random process. What is its distribution?

Related: What is the gradient of the transform? That is the topic of the reparameterization trick and other MC grad estimators. This is very commonly seen in the context of transforms of Gaussians when it produces results like the delta method and extended Kalman filtering.

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

Aït-Sahalia, Yacine, Lars Peter Hansen, and José A. Scheinkman. 2010. Operator Methods for Continuous-Time Markov Processes.” In Handbook of Financial Econometrics: Tools and Techniques, 1–66. Elsevier.
Ariffin, Noor Amalina Nisa, and Norhayati Rosli. 2017. Stochastic Taylor Expansion of Derivative-Free Method for Stochastic Differential Equations.” Malaysian Journal of Fundamental and Applied Sciences 13 (3).
Collard, Fabrice, and Michel Juillard. 2001. Accuracy of Stochastic Perturbation Methods: The Case of Asset Pricing Models.” Journal of Economic Dynamics and Control 25 (6-7): 979–99.
Easley, Deanna, and Tyrus Berry. 2020. A Higher Order Unscented Transform.” arXiv:2006.13429 [Cs, Math], June.
Ebeigbe, Donald, Tyrus Berry, Michael M. Norton, Andrew J. Whalen, Dan Simon, Timothy Sauer, and Steven J. Schiff. 2021. A Generalized Unscented Transformation for Probability Distributions.” ArXiv, April, arXiv:2104.01958v1.
Gustafsson, Fredrik, and Gustaf Hendeby. 2008. On Nonlinear Transformations of Stochastic Variables and Its Application to Nonlinear Filtering.” In 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, 3617–20.
———. 2012. Some Relations Between Extended and Unscented Kalman Filters.” IEEE Transactions on Signal Processing 60 (2): 545–55.
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———. 2010. Numerical Solution of Stochastic Differential Equations. Berlin, Heidelberg: Springer Berlin Heidelberg.
Lin, Wu, Mohammad Emtiyaz Khan, and Mark Schmidt. 2019. Stein’s Lemma for the Reparameterization Trick with Exponential Family Mixtures.” arXiv:1910.13398 [Cs, Stat], October.
Majumdar, Rajeshwari, and Suman Majumdar. 2019. On the Conditional Distribution of a Multivariate Normal Given a Transformation – the Linear Case.” Heliyon 5 (2): e01136.
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Rößler, Andreas. 2004. Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes.” Stochastic Analysis and Applications 22 (6): 1553–76.
Roth, Michael, Gustaf Hendeby, and Fredrik Gustafsson. 2016. “Nonlinear Kalman Filters Explained: A Tutorial on Moment Computations and Sigma Point Methods” 11 (1): 24.
Schmitt-Grohe, Stephanie, and Martın Uribe. n.d. “Perturbation Methods for the Numerical Analysis of DSGE Models: Lecture Notes,” 38.
Schmitt-Grohé, Stephanie, and Martı́n Uribe. 2004. Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function.” Journal of Economic Dynamics and Control 28 (4): 755–75.
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Wolter, Kirk M. 2007. Introduction to Variance Estimation. 2nd ed. Statistics for Social and Behavioral Sciences. New York: Springer.

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