Adjusted variance

We define the adjusted version of X given D, [X/D], to be the `residual' form

Adjusted quantities obey the following properties:

1.  

2. 

We write X as the sum of the two uncorrelated components

so that we can split as

The variance of the adjusted version of X, or the adjusted variance, , is defined to be

The value of is determined by our prior variances and covariances as

 

The variance of X resolved by D, , is defined as

We therefore write the variance partition for X as

 

In line with our various interpretations of belief adjustment, we may give corresponding interpretations to adjusted variance. We may view as:

• the `mean squared error' of the estimator ;
• a primitive expression, interpreted as we would a prior variance, but applied to the `residual variation' when we have extracted the variation in X `accounted for' by D;
• a simple, easily computable upper bound on full Bayes preposterior risk, under quadratic loss, for any full prior specification consistent with the given mean and variance specifications;
• within the more general view of the foundations, the adjustment variance attaches directly to our posterior beliefs.

We quantify the effect of an adjustment by evaluating the resolution, , defined as

 

If is near zero then either the collection D is not expected to be informative for X, relative to our prior knowledge about X, or our beliefs have not been specified in sufficient detail to exploit the information contained in D.

Finally, we define the adjusted covariance, to be