1. Marginal of Gaussian

I have a slight problem with the marginal distribution of a Gaussian. First, assume
(d,r) are jointly Gaussian of dimension 2n. It is known that d = Wr + e, where W is a a convolution matrix and e is gaussian error. We define . Then, we marginalize to get .
Then the conditional distribution is
. Assume, further that .

Is the following then correct?
If, given that d only depends on kappa through r,

then is also Gaussian. However, I am not sure what the corresponding mean and covariance matrix is. I guess the mean is , but what about the covariance matrix?

2. Re: Marginal of Gaussian

Let see if I have get your point:

Now you assume jointly follows a multivariate Gaussian distribution. If you know the covariance matrix of this, then you can solve your problem.

Let

be the covariance matrix of .

You already known the the conditional distribution of and , so and are already known. I also assume that you know the "marginal" distributions of each of them, so the diagonals are also known.

The key question here is that:

given that d only depends on kappa through r,
So how do we interpret this?

If is a zero matrix, then will be independent. If this is not the case we need further assumption on the structure of covariance matrix I guess? Or do you mean is conditionally independent given ?

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Tosha (03-26-2015)

4. Re: Marginal of Gaussian

I will try to make myself more clear. I do apologize for my poor English.

We assume kappa to follow a first order Markov chain, thus the joint (d,r,kappa) is not Gaussian (if I am correct). I all try to formulate the model, which is a two level convolved model, below

1. Prior: , discrete first order Markov chain
2. Likelihood 1: where depends on which class belongs too.
3. Likelihood 2: , W is a fairly broad convolution matrix

I have attached a DAG of the model.

Our goal is to assess

where
,
however we are not able to assess it since the normalization constant is computational infeasible. Therefore, we use k-th order states . and look at the k-th order approximate posterior

As far as I have understood this is Gaussian since p(d,r) = p(d|r)p(r) where p(r) is a Gaussian approximation to r. Then, we can marginalize to find by extracting the appropriate rows and columns. Similarly, is also Gaussian.

Until now I understand what happens, but now I have note saying "combining the results above is also Gaussian. Marginalize wrt to obtain ." To be honest, I really don't see how to find the mean and covariance...

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