I just want to confirm one operation I am doing. If $\beta>0$ and $p(x|\theta)$ is a factorized multivariate Gaussian distribution with dimensionality $d$ and distribution given by:
$$p(x|\theta)=\Big[\frac{1}{2\pi\sigma^2}\exp\Big(-\frac{1}{2\sigma^2}(x-\mu)^T(x-\mu)\Big)\Big]^{d}$$
I would like to compute the are under the following curve,expressed through:
$$\int \Big[\frac{1}{2\pi\sigma^2}\exp\Big(-\frac{1}{2\sigma^2}(x-\mu)^T(x-\mu)\Big)\Big]^{d\beta}dx$$
Thank you
If I assume that you mean the integral of the distribution over $R^n$, then you can use a substitution of $$x_i = \frac{\sigma}{\sqrt{d\beta}}y_i+\mu$$ to transform the integral into a standard form:
$$ \left(\frac{1}{2\pi\sigma^2}\right)^{d\beta}\frac{\sigma^n}{(d\beta)^{n/2}}\int\exp(-\frac{1}{2}y^Ty)d^ny$$
The additional constant in the front comes from the Jacobian of the transformation. This integral has a solution (per Wikipedia: https://en.wikipedia.org/wiki/Gaussian_integral) of:
$$ \int\exp(-\frac{1}{2}y^Ty)d^ny = (2\pi)^{n/2}$$