By Fearn T., Brown P.J., Besbeas P.
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Extra resources for A Bayesian decision theory approach to variable selection for discrimination
Take 0 < p, q < 1, two numbers such that p + q = 1. Pick two points a, b E R. Then 1. A Bernoulli distribution is a probability measure - the Bernoulli distribution corresponding to choosing points a and b with probabilities p and q respectively. The corresponding expectation on functions f is given by: (f) = p f( a) + q f( b). 2. A Poisson distribution is a distribution on the nonnegative integers determined by a parameter A > 0: with the corresponding expectation: 00 (f) = e- A ~ An - ~n! n=O fen) 46 Ch apter 3 3.
Then commute the L terms past Rb. he right. side commute first La past Rb. Comparing coefficients yields the formula. • Remark. Note that we can use any real number for 1', as long as we have a representation where the operators can act on a vector of the form Rrn, which is certainly the case for the HW and sl(2) algebras. he HW algebra, Rrn is effectively the function xr. IV. Formulas related to HW algebra For the HW algebra, Chapter 1, Prop. 2, gives the form of the coefficients . 1) (m,n) SCm, n,}; and the operator p = O.
ALGEBRAIC APPROACH TO INTEGRAL CALCULUS The study of representations of the HW algebra gives an algebraic approach to differential calculus. It is natural to wonder about integral calculus. Here we present an approach based on the commutation relations [J,x] = _J 2 which comes formally from the HW rule [J(D), x] some consequences of this commutation relation. = feD) for J = 1/ D. Let us see 1. Show that [x, In] = nJn+l. ) 2. Deduce that eax Je- ax J = J + aJ 2 + ... = - - I - aJ And hence the commutation rules a.
A Bayesian decision theory approach to variable selection for discrimination by Fearn T., Brown P.J., Besbeas P.