Thomas P. Ogden

Approximate Expectations of Nonlinear Multivariate Functions

Let $X$ be a random variable and $f: \mathbb{R} \to \mathbb{R}$ a deterministic function of that random variable. If $f$ is a linear function such that we may write $f(X) = a + b X$ for $a, b \in \mathbb{R}$, then we have $\mathbb{E}(f(X)) = a + b~\mathbb{E}(X)$. If $f$ is nonlinear, we may not be able to write an exact closed form expression for $\mathbb{E}(f(X))$. In this post I’ll show that if $f$ is continuous and sufficiently differentiable, we can approximate the expected value using a Taylor series.

We expand $f(x)$ in a Taylor series around $\mu_X$, the mean of $X$, for example to second order \begin{equation} f(X) = f(\mu_x) + f’(\mu_x) (X - \mu_x) + \tfrac{1}{2} f’’(\mu_x) (X - \mu_x)^2 + \mathcal{O }((X - \mu_x)^3). \end{equation} Then if the moments of $X$ are finite1, we may write the expected value as \begin{equation} \mathbb{E}(f(X)) = f(\mu_X) + f’(\mu_X) \mathbb{E}( (X - \mu_X)) + \tfrac{1}{2} f’’(\mu_X) \mathbb{E}((X - \mu_X)^2) + \mathcal{O}(\mathbb{E}((X - \mu_X)^3)). \end{equation} Now $\mathbb{E}( (X - \mu_x)) = 0$ and $\mathbb{E}((X - \mu_x)^2) = \sigma^2_X$ is the variance so we can write the second-order approximation \begin{equation} \mathbb{E}(f(X)) \approx f(\mu_X) + \tfrac{1}{2} f’’(\mu_X) \sigma^2_X. \end{equation}

Multivariate Functions

Let $\mathbf{X}$ be a vector of $n$ random variables and $g: \mathbb{R}^n \to \mathbb{R}$ be a function of such a random variable vector. If $g(\mathbf{X})$ is continuous and sufficiently differentiable we can in the same way expand a Taylor series around $\mu_\mathbf{X}$, the mean of $\mathbf{X}$, and write \begin{equation} g(X) = g(\mu_\mathbf{X}) + \nabla g(\mu_\mathbf{X}) (\mathbf{X} - \mu_\mathbf{X}) + \tfrac{1}{2} (\mathbf{X} - \mu_\mathbf{X})^T H_g (\mu_\mathbf{X}) (\mathbf{X} - \mu_\mathbf{X}) + \mathcal{O}((\mathbf{X} - \mu_\mathbf{X})^3) \end{equation} where $H_g$ is the Hessian of $g$.

If the moments of each element of $\mathbf{X}$ are finite, we may approximate the expectation of the function as2 \begin{equation} \mathbb{E}(g(\mathbf{X})) \approx g(\mu_\mathbf{X}) + \tfrac{1}{2} \mathrm{Tr} \left[ H_f (\mu_\mathbf{X}) \Sigma_\mathbf{X} \right] \end{equation} where $\Sigma_\mathbf{X}$ is the covariance matrix of $\mathbf{X}$.

For example, if $n = 2$ such that \(\mathbf{X} = \left[ \begin{smallmatrix} X_1 \\ X_2 \end{smallmatrix} \right]\), we have a Hessian

\[H_g (\mu_\mathbf{X}) = \begin{bmatrix} \frac{\partial^2 g}{\partial X_1^2} & \frac{\partial^2 g}{\partial X_1 \partial X_2} \\ \frac{\partial^2 g}{\partial X_2 \partial X_1} & \frac{\partial^2 g}{\partial X_2^2} \end{bmatrix} (\mu_\mathbf{X})\]

and a covariance matrix

\[\Sigma_\mathbf{X} = \begin{bmatrix} \sigma_{X_1}^2 & \mathrm{Cov}(X_1, X_2) \\ \mathrm{Cov}(X_2, X_1) & \sigma_{X_2}^2 \end{bmatrix}.\]

Then we can expand the trace, \(\begin{split} \mathrm{Tr} \left[ H_g (\mu_\mathbf{X}) \Sigma_\mathbf{X} \right] = &\frac{\partial^2 g}{\partial X_1^2} (\mu_\mathbf{X}) \sigma_{X_1}^2 + \frac{\partial^2 g}{\partial X_1 \partial X_2} (\mu_\mathbf{X}) \mathrm{Cov}(X_2, X_1) + \\ &\frac{\partial^2 g}{\partial X_2 \partial X_1} (\mu_\mathbf{X}) \mathrm{Cov}(X_1, X_2) + \frac{\partial^2 g}{\partial X_2^2} (\mu_\mathbf{X}) \sigma_{X_2}^2. \end{split}\)

If $X_1, X_2$ are independent such that $\mathrm{Cov}(X_1, X_2) = \mathrm{Cov}(X_2, X_1) = 0$, we are left with a second order approximation for the expected value of the multivariate function, \begin{equation} \mathbb{E}(g(\mathbf{X})) \approx g(\mu_\mathbf{X}) + \frac{\partial^2 g}{\partial X_1^2} (\mu_\mathbf{X}) \sigma_{X_1}^2 + \frac{\partial^2 g}{\partial X_2^2} (\mu_\mathbf{X}) \sigma_{X_2}^2. \end{equation}

  1. See this answer on stats.stackexchange.com for an explanation as to why that requirement is important and why this method does not work for heavy-tailed distributions. 

  2. For a random vector $\mathbf{X}$ and symmetric matrix $\Lambda$ the expectation of the quadratic form, $\mathbb{E}(\mathbf{X}^T \Lambda \mathbf{X}) = \mathbb{E}(\mathbf{X})^T \Lambda \mathbb{E}(\mathbf{X}) + \mathrm{Tr} \left[ \Lambda \Sigma_\mathbf{X} \right]$.