I'm trying to get the log-likelihood of a Gaussian in the form
$\qquad p(\textbf{x}|u,\sigma^2)=\prod_{n=1}^{N}\mathcal{N}(x_n|\mu,\sigma^2) \quad (1)$$$ p(\textbf{x}|u,\sigma^2)=\prod_{n=1}^{N}\mathcal{N}(x_n|\mu,\sigma^2) \quad (1) $$
$\qquad ln\ p(\textbf{x}|\mu,\sigma^2)=-\frac{1}{2\sigma^2}\sum_{n=1}^N{(x_n-\mu)^2}-\frac{N}{2}ln\ \sigma^2-\frac{N}{2}ln\ (2\pi) \quad (2)$$$ \ln\ p(\textbf{x}|\mu,\sigma^2)=-\frac{1}{2\sigma^2}\sum_{n=1}^N{(x_n-\mu)^2}-\frac{N}{2}\ln\ \sigma^2-\frac{N}{2}\ln\ (2\pi) \quad (2) $$
I've have
Log[Product[PDF[NormalDistribution[μ, σ], Subscript[x, n]], {n, 1, bigN}]]
which outputs
$\qquad Log(\prod_{n=1}^{bigN}\frac{e^{\frac{(-\mu+x_n)^2}{2\sigma^2}}}{\sqrt{2\pi}\sigma})$.$$ \ln(\prod_{n=1}^{bigN}\frac{\mathrm e^{\frac{(-\mu+x_n)^2}{2\sigma^2}}}{\sqrt{2\pi}\sigma}). $$
I tried Expand, ExpandAll, PowerExpand, but I can't seem to get it to display like in Equation (2 RHS).
From
Product[Log[PDF[NormalDistribution[μ, σ], Subscript[x, n]]], {n, 1, bigN}] // PowerExpand`
I get closer with
$\qquad \prod_{n=1}^{bigN}(\frac{1}{2}(-Log[2]-Log[\pi])-Log[\sigma]-\frac{(-\mu+x_n)^2}{2\sigma^2}))$$$ \prod_{n=1}^{bigN}\left[\frac{1}{2}(-\ln 2-\ln\pi)-\ln\sigma-\frac{(-\mu+x_n)^2}{2\sigma^2}\right] $$
