Procedure: Calculation of adjusted expectation and variance¶
Description and Background¶
In the context of Bayes linear methods, the Bayes linear adjustment is the appropriate method for updating prior second-order beliefs given observed data. The adjustment takes the form of linear fitting of our beliefs on the observed data quantities. Specifically, given two random vectors, \(B\), \(D\), the adjusted expectation for element \(B_i\), given \(D\), is the linear combination \(a_0 + a^T D\) minimising \(\textrm{E}[B_i - a_0 - a^T D)^2]\) over choices of \(\{a_0, a\}\).
Inputs¶
- \(\textrm{E}[B]\), \(\textrm{Var}[B]\) - prior expectation and variance for the vector \(B\)
- \(\textrm{E}[D]\), \(\textrm{Var}[D]\) - prior expectation and variance for the vector \(D\)
- \(\textrm{Cov}[B,D]\) - prior covariance between the vector \(B\) and the vector \(B\)
- \(D_{obs}\) - observed values of the vector \(D\)
Outputs¶
- \(\textrm{E}_D[B]\) - adjusted expectation for the uncertain quantity \(B\) given the observations \(D\)
- \(\textrm{Var}_D[B]\)- adjusted variance matrix for the uncertain quantity \(B\) given the observations \(D\)
Procedure¶
The adjusted expectation vector, \(\textrm{E}_D[B]\) is evaluated as
(If \(\textrm{Var}[D]\) is not invertible, then we use a generalised inverse such as Moore-Penrose).
The adjusted variance matrix for \(B\) given \(D\) is
Additional Comments¶
See DiscBayesLinearTheory for a full description of Bayes linear methods.