# Expectation of a variable inside the cumulative distribution function of standard normal

Let $\Phi$ and $\phi$ respectively be the cumulative distribution function and probability density function of a standard normal distribution. $\beta$ is a $d \times 1$ vector which follows a multivariate normal distribution $\mathcal{N}\left(\mu, \Sigma\right)$. How can I compute the following expectation: $$\mathbb{E}_{\beta}\left[\log \Phi \left(x^{T}\beta\right)\right]$$

I wonder if this is the same as $\log \Phi\left(x^{T}\mu\right)$.

• Do you know how to do it when it is $\mu$ instead of $\beta$? – RustyStatistician Feb 11 '16 at 7:06
• @RustyStatistician There would be no point in taking the expectation if there was no $\beta$. My question was if I could just plug in the mean vector of $\beta$ even though it's inside both the log and the cdf... (I don't think it's possible since those functions are not linear.) – Daeyoung Lim Feb 11 '16 at 7:09
• I was confused because you said "I wonder if this is the same as $\log \Phi\left(x^{T}\mu\right)$" – RustyStatistician Feb 11 '16 at 7:11
• This may help stats.stackexchange.com/questions/57715/… – RustyStatistician Feb 11 '16 at 7:19
• Recall $g(E(X))\neq E(g(X))$ for nonlinear functions $g$, see en.wikipedia.org/wiki/Jensen%27s_inequality – Christoph Hanck Feb 11 '16 at 7:26

It should be noted that, in general, for any non-linear function $g$ of a random variable $X$, $\mathbb{E}[g(X)] \neq g(\mathbb{E}[X])$ . You can see this by expanding out the integrals; $$\mathbb{E}[g(X)] = \int_X g(x)f(x)dx;\;\;\;\;g(\mathbb{E}[X])=g\bigg(\int_X xf(x)dx\bigg)$$ You could see how the two would only be equal in the case $g$ was linear.
So given the above; $$\mathbb{E}_{\beta}\left[\log \Phi \left(x^{T}\beta\right)\right] \neq \log \Phi\left(x^{T}\mu\right)$$
You can estimate $\mathbb{E}_{\beta}\left[\log \Phi \left(x^{T}\beta\right)\right]$ via Monte Carlo
$$\mathbb{E}_{\beta}\left[\log \Phi \left(x^{T}\beta\right)\right]=\int_\beta \log \Phi \left(x^{T}\beta\right)f(\beta|\mu, \Sigma)d\beta \approx \frac{1}{G}\sum_{g=1}^G \log \Phi \left(x^{T}\beta^{(g)}\right)$$ where $\beta^{(1)},\beta^{(2)},...,\beta^{(G)} \sim \mathcal{N}(\mu,\Sigma)$.
If the dimension of $\beta$ is large, the above approximation may not work efficiently (it will take a very large $G$ for it to get close). Otherwise, this is a pretty standard and well accepted approach.