integrate a multivariate normal mixture

[Neddie]

Any advice on how to analytically evaluate this integral? It's a multivariate normal mixture, with the first element of the input vector integrated between specified limits, and the others integrated from -infinity to +infinity.

multivariate normal distribution:

\mathbf{x} = (x_1, ..., x_n)^T, \quad \boldsymbol{\mu} = (\mu_1, ..., \mu_n)^T, \quad \boldsymbol{\Sigma} \in R^{n \times n}, SPD, \quad N(\mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\Sigma}) = \left[ (2 \pi)^n | \boldsymbol{\Sigma} |\right]^{-\frac{1}{2}} exp \left[ -\dfrac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right]

mixture:

p(\mathbf{x}) = \sum_{m=1}^M w_m N(\mathbf{x}, \boldsymbol{\mu}_m, \boldsymbol{\Sigma}_m)

desired integral:

\int_a^b \int_{-\infty}^{\infty} ... \int_{-\infty}^{\infty} p(\mathbf{x}) \,dx_n ... dx_2 dx_1

Thanks!

[Neddie]

Of course, it's just the sum of the integrals over the mixture components. The integral over one normal distribution is really what I'm lacking.

[D.B.]

Given you marginalise out all the components except one, shouldn't this just boil down to integrating a single univariate normal distribution using erf? Or am I missing the point entirely?

[tcamolesi]

Hmmm, not really. This isn't a sum of normally distributed random variables, i.e.: X1 + X2, X1 ~ N(u1, s1), X2 ~ N(u2, s2). This is a mixture model.

As such, I believe you can just marginalize and integrate each component separately and do the weighted sum.

\int_{a}^{b}\int_{-\infty}^{\infty}\ldots \int_{-\infty}^{\infty}p(x)dx_{n}\ldots dx_{2} dx_{1} =
\int_{a}^{b}\int_{-\infty}^{\infty}\ldots \int_{-\infty}^{\infty}\sum_{m=1}^M w_m N(x,\mu_m, \Sigma_m) dx_{n}\ldots dx_{2} dx_{1} =
\sum_{m=1}^M w_m \left( \int_{a}^{b}\int_{-\infty}^{\infty}\ldots \int_{-\infty}^{\infty}N(x,\mu_m, \Sigma_m) dx_{n}\ldots dx_{2} dx_{1} \right)

[D.B.]

Yes, I know it's a mixture model and not a sum of variables, and I agree that you can marginalise each component on its own and then find the weighted sum . The OP though said that they're fine with the mixture model part, and that it's...

...The integral over one normal distribution is really what I'm lacking.

So the question seems to be, how do you integrate a single multi variate normal between limits? In this case though only one of the components is being integrated over particular limits a and b, and the rest are integrated between negative infinity and positive infinity. So to me it seems to me that you can discard all the components except the one we're interested in (x₁ in this case), and then just integrate that one between the limits a and b using the error function in the standard manner.