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Math Help - Multivariate Bayesian Updating

  1. #1
    Jan 2010

    Multivariate Bayesian Updating

    I have a question regarding the application of multivariate Bayesian Updating. I have a two-period model with two signals in each period. The expectation of the signals in the second period are updated using Bayes formula once the two signals from the first period are observed. I am using the following derivation of the multivariate Bayes formula E[x|y,z] = mean(x) + ((var(z)*cov(xy) - cov(xz)*cov(yz))/(var(y)*var(z) - cov(yz)^2))*(y-mean(y)) + ((var(y)*cov(xz) - cov(xy)*cov(yz))/(var(y)*var(z) - cov(yz)^2))*(z-mean(z))

    and var[x|y,z] = (var(x)*var(y)*var(z) - var(x)*cov(yz)^2 - var(y)*cov(xz)^2 - var(z)*cov(xy)^2 + 2*cov(xy)*cov(xz)*cov(yz))/(var(y)*var(z) - cov(yz)^2))

    To try to do the updating. Please bear with me as I am a novice to Bayesian updating and I apologize if I am not clear. My question is the following, and I am attaching my latex code for anyone with the time/desire/need for more information. I am having difficulty finding comfort that I am treating the variances correctly in equation (29) (an equation that deals strictly with variances of two larger equations, one of which is conditioned upon the information obtained in the first).

    Basic system of equations is as follows:
    W1 = X1 + P1
    W2 = X2 + P2
    Both X's and P's are essentially comprised of a mean and a variance term (which has a normal distribution zero mean). The variances differ accross time, and there are co-variances for all of these variables as well.

    The equation of interest is RP1 = 1/2r(var[W1] + var[W2|x1,P1] + 2cov[W1,[W2|x1,P1].

    Ok, so here are my questions.

    The variance of W1 is just var(x1) + var(p1) + 2cov(x1,P1).
    The variance of W2 is where I start to get lost. I believe that the structure of the variance of W2 is similar var(x2|x1,p1) + var(p2|x1,p1) + 2cov?

    Question 1: So is the covariance a conditional covariance or a simple covariance? I am not sure, I think that it is a conditional covariance, and I have computed one (to the best of my knowledge using the typical covariance formula, but I am not sure that I am right either in my assumption that a conditional covariance is required or that my calculations are correct (equation 28).

    Question 2: Finally, given the variance equation I am using above, I am not sure whether the third term in the RP1 equaiton is necessary or not. W1 is comprised of x1 and p1. W2 is already conditioned upon x1 and p1, so does the variance formula given above already take care of the covariance of W1 and W2? If it does not, then what is the appropriate way to go about computing this covariance.

    Thank you very much for your time and effort.


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