# how to derive a conditional covariance matrix (symbolically)

I have searched the database of the questions asked in this forum but to my surprise I could not find any satisfactory answer to my relatively simple question.

Given a vector of random variables (X1,X2,X3,X4) all having a normal distribution with mean (m1,m2,m3,m4) and variance (v1,v2,v3,v4). Each random variable is respectively conditioned to signals (s1,s2,s3,s4). I can compute manually covariance, say, between X1 and X2 as:

Cov(X1,X2|s1,s2)=E{(X1-m1s1)(X2-m2s2)|s1,s2}

where conditional mean of X1 given s1 is denoted by

m1s1 = E[X1|s1] and similarly, m2s2 = E[X2|s2] and E[X1X2|s1,s2] = Integrate[X1X2, p(X1X2|s1,s2), {X1, -inf, inf},{X2, -inf, inf}]

I can of course go ahead and derive all the elements of conditional covariance matrix. I thought that Mathematica can derive this conditional covariance matrix.

My ultimate objective is to calculate with a given set data the following ratio:

Det[Covariance matrix for X]/Det[Conditional covariance matrix X|S]

Any idea about how to achieve this?

• What does "conditioned to signals" mean? – JimB Nov 14 '18 at 23:01
• @JimB: consider X1 as an unknown true parameter and there is a signal about X1, i.e., S1=X1+e1 all is normally distributed. Other RVs are also in the same form like S2=X2+e2 where e1 and e2 are noises about the respective RVs. – Tugrul Temel Nov 14 '18 at 23:06
• Thanks for the explanation. I wonder if this post is helpful: stats.stackexchange.com/questions/161352/…. – JimB Nov 14 '18 at 23:11
• @JimB: I will take a look at the site you sent. I hope it is sufficient. Thanks. – Tugrul Temel Nov 14 '18 at 23:14
• @JimB: In fact, the very last part of the explanation given in the link you sent me (about conditional mean and conditional covariance) is what I am after. However, my problem is to derive it using Mathematica. In a simple case (say, X1 and s1), few days ago you have given me the answer in the context of mutual information. Now with current question, I want to move on to the multivariate case of mutual information, which requires the calculation of unconditional and conditional covariance matrices. I hope my question is clear. – Tugrul Temel Nov 14 '18 at 23:47