Conditional Independence (III): conditionally UNcorrelated random variables (only multivariate normal distribution)

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 17:13

*Enlace a Spanish version

Summary:

This video complements the discussion on conditional independence of the video [condin1EN], particularizing it to the case where only linear predictions are desired and, therefore, only the variance-covariance matrix must be analyzed.

Therefore, the absence of covariance (that is, the non-correlation) between the prediction errors of two random variables given a third one is what will define the conditional non-correlation... well, sort of...

NOTE: you should take “informally” the concepts released here, since they would only be valid if we are sure that the random variables have been generated with a multidimensional normal distribution. Otherwise, the “conditional” distribution will not be the one given by the best linear prediction formulae, but some ”informal” conclusions can be drawn regarding capability of doing linear predictions. And, in fact, in the normal case, conditional uncorrelation would also mean conditional independence, so we are not discussing, in fact, any new concept from said independence if we wish to be more rigorous.

The point is that this conditional non-correlation allows making Bayesian networks that “chain” linear models with additive noise, as seen in the numerical examples.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.