equi.bg.LL: Beta-geometric log-likelihood function
In ghuiber/BTYD2: Implementing BTYD Models With The Log Sum Exp Patch

Description Usage Arguments Details Value References Examples

Log-likelihood function for the BG

1	equi.bg.LL(params, x, t, n)

`params`	A vector of the two Beta-Geometric parameters gamma and delta.
`x`	A vector with the number of transactions at discrete times t.
`t`	The timeline of transactions in vector x.
`n`	The total possible number of transactions at time t.

The log-likelihood function is a building block for estimating the value of params given a buying pattern x observed over time t. See the example.

The log likelihood of observing the combination of transactions x over timeline t given that x follows a Beta-Geometric distribution with parameters params.

Peter S. Fader, Bruce G. S. Hardie, Daniel McCarthy & Ramnath Vaidyanathan (2019) Exploring the Equivalence of Two Common Mixture Models for Duration Data, The American Statistician, 73:3, 288-295, DOI: 10.1080/00031305.2018.1543134 Web

# Get the calibration data. Remember that
# counts of households are cumulative, so 
# turn them to incremental:
library(tidyverse)
data("ace_snackfoods")
data <- ace_snackfoods$calibration %>% 
  rename(x = `Number of households`) %>% 
  mutate(bought = x - lag(x)) %>% 
  mutate(bought = if_else(is.na(bought), x, bought))
  
# Get maximum likelihood estimates:   
mlset <- optim(par = c(0, 3), 
               fn = equi.bg.LL, 
               x = data$bought, 
               t = data$Week, 
               n = 1499, 
               method="BFGS", 
               control=list(fnscale=-1))

# Log-likelihood maximum:
LL <- mlset$value

# Parameter estimates:
# Recall that bg.LL exponentiates par in
# order to enforce gamma > 0, delta > 0.
params <- exp(mlset$par)