Bayesian Principal Component Analysis

We go through the following steps to produce a Bayesian principal component analysis.

1. Determine prior 2.

!!! in pca the model is a model of 3 principal components.... and we fit this to a distributions.... so the prior must be a parameter of that distribution.

!! the challenge as i see it !!

• we need to be able to formulate priors that indicate our measure of uncertainty
• a simple adjustment might be to say we expect rates to be lower at all terms by a constant amount (a downward shift)
• each principal component however is difficult to interpret because it is a vector in many directions
• so interpreting what each parameter of principal component model is non intuitive

• and yet bayesian framework IS setting priors for the parameters of the principal components

• it is possible to reverse engineer component scores that give our expected shift

• which in turn we could perhaps use as a mean value to set parameters
• but do we actually want to produce the set of possible parameters... in some ways no !!
• perhaps we can use it to answer question how likely expected outcome given historical data...
• but even then the different parameter values that give something close to expected outcome -- difficult to anticpate combinations
• and we havent even discussed combinations... does this need to be a joint probability distribution.

i think we should keep eigenvectors set at what they are...
so the challenge is reverse engineering parameter adjustments ....