# What is Isserlis’ theorem?

I learnt about Isserlis’ theorem (also known as Wick’s probability theorem) at a talk today. The theorem comes from a paper from 1918, which is listed as Reference 1 below. In the words of Reference 2, the theorem

… allows [us] express to the expectation of a monomial in an arbitrary number of components of a zero mean Gaussian vector $X \in \mathbb{R}^d$ in terms of the entries of its covariance matrix only.

We introduce some notation (as in Reference 2) to state the theorem succinctly. Let $A = \{ \alpha_1, \dots, \alpha_N \}$ be a set of integers such that $1 \leq \alpha_i \leq d$ for all $i$. The $\alpha_i$ need not be distinct. For any vector $X \in \mathbb{R}^d$, denote

\begin{aligned} X_A = \prod_{\alpha_i \in A} X_{\alpha_i}, \end{aligned}

with the convention that $X_\emptyset = 1$. Let $\Pi (A)$ denote the set of all pairings of $A$, i.e. partitions of $A$ into disjoint pairs. For a pairing $\sigma \in \Pi (A)$, let $A / \sigma$ denote the set of indices $i$ such that the pairs in $\sigma$ are $\{ (\alpha_i, \alpha_{\sigma(i)}) : i \in A / \sigma \}$.

(As an example, if $A = \{ 1, 2, 3, 4 \}$, one possible pairing $\sigma$ is $\{\{ 1, 3\}, \{ 2, 4\} \}$. For this pairing, a possible choice of $A / \sigma$ is $A / \sigma = \{ 1, 2\}$, with $\sigma(1) = 3$ and $\sigma(2) = 4$.)

We are now ready to state the theorem:

Theorem (Isserlis’ theorem): Let $A = \{ \alpha_1, \dots, \alpha_N \}$ be a set of integers such that $1 \leq \alpha_i \leq d$ for all $i$, and let $X \in \mathbb{R}^d$ be a Gaussian vector with zero mean. If $N$ is even, then

\begin{aligned} \mathbb{E} [X_A] = \sum_{\sigma \in \Pi (A)} \prod_{i \in A / \sigma} \mathbb{E} [X_{\alpha_i} X_{\alpha_{\sigma(i)}}]. \end{aligned}

If $N$ is odd, then $\mathbb{E}[X_A] = 0$.

Here are some special cases of Isserlis’ theorem to demonstrate how to interpret the equation above. If $\alpha_i = i$ for $1 \leq i \leq 4$, there are 3 possible pairings, giving us

\begin{aligned} \mathbb{E}[X_1 X_2 X_3 X_4] = \mathbb{E}[X_1 X_2] \mathbb{E}[X_3 X_4] + \mathbb{E}[X_1 X_3] \mathbb{E}[X_2 X_4] + \mathbb{E}[X_1 X_4] \mathbb{E}[X_2 X_3]. \end{aligned}

If we take $\alpha_i = 1$ for $1 \leq i \leq 4$, there are still 3 possible pairings, and we get

\begin{aligned} \mathbb{E}[X_1 X_1 X_1 X_1] &= \mathbb{E}[X_1 X_1] \mathbb{E}[X_1 X_1] + \mathbb{E}[X_1 X_1] \mathbb{E}[X_1 X_1] + \mathbb{E}[X_1 X_1] \mathbb{E}[X_1 X_1], \\ \mathbb{E}[X_1^4] &= 3 \left(\mathbb{E}[X_1^2] \right)^2. \end{aligned}

This tells us that the 4th moment of a mean-zero 1-dimensional gaussian random variable is 3 times the square of its 2nd moment.

As a final example, if we take $\alpha_1 = \alpha_2 = 1$ and $\alpha_3 = \alpha_4 = 2$, we still have 3 possible pairings, and we get

\begin{aligned} \mathbb{E}[X_1 X_1 X_2 X_2] &= \mathbb{E}[X_1 X_1] \mathbb{E}[X_2 X_2] + \mathbb{E}[X_1 X_2] \mathbb{E}[X_1 X_2] + \mathbb{E}[X_1 X_2] \mathbb{E}[X_1 X_2], \\ \mathbb{E} [X_1^2 X_2^2] &= \mathbb{E}[X_1^2] \mathbb{E}[X_2^2] + 2 \left( \mathbb{E}[X_1 X_2] \right)^2. \end{aligned}

References: