bgbb.PosteriorMeanLmProductMoment {BTYD}

BG/BB Posterior Mean (l,m)th Product Moment

Description

Computes the ⁠(l,m)⁠th product moment of the joint posterior distribution of P (the Bernoulli transaction process parameter) and Theta (the geometric dropout process parameter).

Usage

bgbb.PosteriorMeanLmProductMoment(params, l, m, x, t.x, n.cal)

Arguments

params

BG/BB parameters - a vector with alpha, beta, gamma, and delta, in that order. Alpha and beta are unobserved parameters for the beta-Bernoulli transaction process. Gamma and delta are unobserved parameters for the beta-geometric dropout process.

l

moment degree of P

m

moment degree of Theta

x

the number of repeat transactions made by the customer in the calibration period. Can also be vector of frequencies - see details.

t.x

recency - the transaction opportunity in which the customer made their last transaction. Can also be a vector of recencies - see details.

n.cal

number of transaction opportunities in the calibration period. Can also be a vector of calibration period transaction opportunities - see details.

Details

E((P)^l(Theta)^m | alpha, beta, gamma, delta, x, t.x, n)

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

The expected posterior ⁠(l,m)⁠th product moment.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See equation 17.