# Bayesian inference on partitioned multivariate Gaussian

My question is on Bayesian inference of partitioned multivariate Gaussian. To make things easier, suppose there is a 2-dimensional Guassian,

$$X_1 \sim N(\mu_1, \sigma^2_1) \\ X_2 \sim N(\mu_2, \sigma^2_2)$$

with covariance $\sigma_{1,2}$.

Suppose we know $\sigma_{i,j}$, $\sigma^2_1$ and $\sigma^2_2$; don't know $\mu_1$, $\mu_2$ but have priors for them as,

$$\mu_1 \sim N(\theta_1, \delta^2_1) \\ \mu_2 \sim N(\theta_2, \delta^2_2)$$

Now we have an observation $x_1$ for $X_1$. By Bayesian inference we get,

$$\theta'_1 | x_1 = \frac{\delta^2_1 x_1 + \sigma^2_1 \theta_1}{\delta^2_1 + \sigma^2_1} \\ \delta'^2_1 | x_1 = \frac{\delta^2_1 \sigma^2_1}{\delta^2_1 + \sigma^2_1}$$

and by partitioned Gaussian we have,

$$X_2 | X_1 \sim N \left(\mu_2 + \frac{\sigma_{1,2}}{\sigma^2_1}(x_1 - \mu_1), \sigma^2_2 - \frac{\sigma^2_{1, 2}}{\sigma^2_1} \right)$$

Finally my question is, how to update the correlated r.v using Bayesian inference, $$p(\mu_2 | x_1) = \frac{p(x_1 | \mu_2) p(\mu_2)}{p(x_1)}$$ since I don't know how to deal with $p(x_1 | \mu_2)$. Or maybe there's other ways around to get it? Hope you get the idea of what I'm trying to do.

Thanks!

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Or, from $X_2|X_1$ we have, $$\mu_2|x_1 = \mu_2 + \frac{\sigma_{1,2}}{\sigma^2_1}(x_1-\mu_1)$$. I'm not sure how to carry on from this to get p.d.f of $\mu_2|x_1$ either. Thanks! – shuaiyuancn Mar 16 '12 at 15:42
For your $2$-dimensional Gaussian, you neglected to specify the correlation between $X_1$ and $X_2$. – Michael Hardy Mar 16 '12 at 17:26
@MichaelHardy I said "with covariance $\sigma_{i,j}$". – shuaiyuancn Mar 16 '12 at 18:06
but you did not say "we know" the covariance – Henry Mar 16 '12 at 19:12
@Henry Sure. Thanks. Edited. – shuaiyuancn Mar 16 '12 at 20:58