# bgbb.PosteriorMeanTransactionRate: BG/BB Posterior Mean Transaction Rate In BTYD: Implementing BTYD Models with the Log Sum Exp Patch

## Description

Computes the mean value of the marginal posterior value of P, the Bernoulli transaction process parameter.

## Usage

 `1` ```bgbb.PosteriorMeanTransactionRate(params, 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. `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 | alpha, beta, gamma, delta, x, t.x, n). This is calculated by setting `l = 1` and `m = 0` in `bgbb.PosteriorMeanLmProductMoment`.

`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 posterior mean transaction rate.

## 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.

`bgbb.rf.matrix.PosteriorMeanTransactionRate`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ``` data(donationsSummary) rf.matrix <- donationsSummary\$rf.matrix # donationsSummary\$rf.matrix already has appropriate column names # starting-point parameters startingparams <- c(1, 1, 0.5, 3) # estimated parameters est.params <- bgbb.EstimateParameters(rf.matrix, startingparams) # return the posterior mean transaction rate vector bgbb.rf.matrix.PosteriorMeanTransactionRate(est.params, rf.matrix) ```