# Approximate Expectations of Nonlinear Multivariate Functions

2020-11-19

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}

## Footnotes

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].↩︎