Bayes Linear Statistics - Example

Example

In Bayes linear statistics, the probability model is only partially specified, and it is not possible to calculate conditional probability by Bayes' rule. Instead Bayes linear suggests the calculation of an Adjusted Expectation.

To conduct a Bayes linear analysis it is necessary to identify some values that you expect to know shortly by making measurements D and some future value which you would like to know B. Here D refers to a vector containing data and B to a vector containing quantities you would like to predict. For the following example B and D are taken to be two-dimensional vectors i.e.

In order to specify a Bayes linear model it is necessary to supply expectations for the vectors B and D, and to also specify the correlation between each component of B and each component of D.

For example the expectations are specified as:

and the covariance matrix is specified as :

\begin{matrix} & X_1 & X_2 & Y_1 & Y_2 \\ X_1 & 1 & u & \gamma & \gamma \\ X_2 & u & 1 & \gamma & \gamma \\ Y_1 & \gamma & \gamma & 1 & v \\ Y_2 & \gamma & \gamma & v & 1 \\ \end{matrix}.

The repetition in this matrix, has some interesting implications to be discussed shortly.

An adjusted expectation is a linear estimator of the form

where and are chosen to minimise the prior expected loss for the observations i.e. in this case. That is for

where

are chosen in order to minimise the prior expected loss in estimating

In general the adjusted expectation is calculated with

Setting to minimise

From a proof provided in (Goldstein and Wooff 2007) it can be shown that:

For the case where Var(D) is not invertible the Moore–Penrose pseudoinverse should be used instead.

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