Gamma-Gamma Model

The model assumes that the average transaction value for a customer remains stationary over time and is independent of the transaction frequency. It does not model the timing or count of purchases; it exclusively models the monetary value given that a transaction occurs. This separation of frequency and monetary value is a defining characteristic of the Buy-Till-You-Die (BTYD) framework.

The model employs a Bayesian hierarchical structure with a Gamma-Gamma conjugate pair:

• Individual Spend: A customer's unobserved average transaction value is assumed to follow a Gamma distribution.
• Observed Spend: The actual transaction amounts for a customer are assumed to follow a Gamma distribution with a shape parameter equal to the number of transactions.
• Conjugacy: This pairing allows for a mathematically tractable posterior distribution, enabling the model to update its belief about a customer's true average spend as more transactions are observed.

A primary purpose of the model is to account for heterogeneity in spending across the customer base. Without this, a simple arithmetic mean would be used, which fails to differentiate between a customer who makes many small purchases and one who makes a few large ones. The Gamma distribution over the mean spend parameter captures the natural variation in customer value, allowing the model to shrink extreme, noisy estimates for customers with few transactions toward the population mean.

The model's population-level parameters (shape and rate of the Gamma distribution on mean spend) are typically estimated using Maximum Likelihood Estimation (MLE) on historical transaction data. The inputs are:

• x: The total sum of a customer's historical transaction values.
• z: The total number of transactions for that customer. The likelihood function is derived from the compound Gamma-Gamma distribution, and optimization routines find the parameters that maximize the probability of observing the historical data.

The final output is the conditional expectation of a customer's average transaction value, given their observed spending history. The formula is a weighted average: E[M | x, z] = (p + x) / (q + z) Where:

• p and q are the estimated population-level Gamma parameters.
• x is the sum of observed spend.
• z is the number of observed transactions. This acts as a Bayesian shrinkage estimator, pulling the predicted mean spend toward p/q (the population mean) when z is small.

The Gamma-Gamma model is almost never used in isolation. It is the standard complement to the BG/NBD model (or its predecessor, the Pareto/NBD model) in the BTYD framework. The BG/NBD model predicts the expected number of future transactions for a customer. The Gamma-Gamma model then predicts the expected value per transaction. The final Customer Lifetime Value (CLV) is the product of these two independent predictions: E[CLV] = E[Transactions] * E[Monetary Value]