Let's consider the Bayesian estimation of multivariate Normal distribution when the covariance matrix is known. Let $y_n$ be $$y_n \sim N(\mu, \Sigma), n=1,\ldots,N \qquad \mu \sim N(\mu_0 , \Sigma_{\mu_0})$$ and $\Sigma$ is known. To get the posterior probability distribution, I need to solve $$\log p(\mu|Y) = \log p(Y|\mu) + \log p(\mu) - \log p(Y) \\ \begin{aligned} \log p(\mu|Y) &= -\frac12 \sum_{n=1}^N(y_n - \mu)^T\Sigma^{-1}(y_n-\mu)-\frac{N}2 \log 2\pi^D |\Sigma | \\ &\quad - \frac{1}{2}(\mu-\mu_0)^T\Sigma_{\mu_0}^{-1}(\mu-\mu_0) - \frac12 \log 2\pi \left| \Sigma_{\mu_0} \right| + \text{cosnt} \end{aligned}$$ And my question is, how can I derive the mean vector of posterior $\overline{\mu}$? I have no idea what should be the next step from here. Any answers would be appreciated. Thank you in advance.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.