Bayesian updating of a joint probability distribution based on the likelihood of one variable

I am struggling with the following question, as illustrated as well per below. I have a prior multivariate distribution, with correlation between the variables. I have obtained additional data on one variable. I would like to use the information to update my prior distribution.

I am struggling on what the best and most efficient way is to do this, and I would like to find a method that can be scaled to higher dimensional and non-conjugate distributions. I was thinking of

1. approximating each marginal,
2. calculating the correlation / co-variance matrix,
3. performing a standard Bayesian update on the marginal of interest, and then
4. using the updated marginal and the other un-changed marginals and the co-variance matrix to draw samples and approximate the updated posterior.

However, this feels quite laborious and sub-optimal. What are better ways to tackle this? • Can you write some equations so we can see better exactly what you mean? If you have a joint distribution and have observed partial data about one of the variables, then if the joint distribution is parametrized then you want to marginalize out the unseen variables and maximize the marginal likelihood. Nov 19 '16 at 0:41

As described, this should not be a problem: given a prior $\pi(\theta_1,\ldots,\theta_k)$ and an observation [or a sample] $X$ with density $f(x|\theta_1)$, the posterior distribution of the parameter is $$\pi(\theta_1,\ldots,\theta_k|x)\propto \pi(\theta_1,\ldots,\theta_k)f(x|\theta_1)$$which naturally and Bayesianly incorporates the information provided by $x$ into the posterior. No marginalisation is needed at any point.