Beta-Geometric Model

At its core, the model assumes that in each discrete time period, a customer has a constant probability θ of churning, independent of tenure. This memoryless property means the probability of surviving to period t and then churning follows a Geometric distribution. The process models the 'die' component of buy-till-you-die frameworks, where churn is an absorbing state from which customers never return.

• Probability mass function: P(T = t) = θ(1-θ)^(t-1)
• Hazard rate remains constant at θ for each period
• Suitable for subscription businesses with regular renewal cycles

Not all customers share the same churn propensity. The model captures this unobserved heterogeneity by assuming individual churn probabilities θ are drawn from a Beta distribution with shape parameters α and β. This flexible distribution, bounded between 0 and 1, can take U-shaped, J-shaped, or bell-shaped forms to represent diverse retention patterns across the customer base.

• Beta(α, β) density: f(θ) ∝ θ^(α-1)(1-θ)^(β-1)
• α > 1, β > 1: unimodal distribution around a central tendency
• α < 1, β < 1: bimodal, capturing both loyalists and defectors

As renewal or cancellation data accumulates for a specific customer, the model applies Bayes' theorem to update the posterior distribution of that individual's churn probability. The Beta prior conjugacy with the Geometric likelihood yields a Beta posterior, enabling computationally efficient, closed-form updates without Markov Chain Monte Carlo sampling.

• Prior: θ ~ Beta(α, β)
• Likelihood: P(data | θ) based on observed renewals and cancellations
• Posterior: θ | data ~ Beta(α + cancellations, β + renewals)
• Shrinkage effect: sparse data pulls estimates toward the population mean

The model yields analytically tractable expressions for key business metrics. The survivor function S(t) gives the probability a randomly selected customer is still active after t periods, while the retention rate r(t) = S(t)/S(t-1) shows period-over-period stickiness. Unlike raw cohort tables, these functions produce smooth, monotonic decay curves.

• S(t) = B(α, β + t) / B(α, β) where B is the Beta function
• Retention rate increases over time as high-churn customers drop out
• Enables extrapolation beyond observed data for long-term CLV projections

The population-level parameters α and β are typically estimated via maximum likelihood estimation using historical cohort data. The likelihood function aggregates over all customers, accounting for both censored observations (still active) and uncensored observations (churned at a known period). Optimization is performed using numerical methods like the BFGS algorithm.

• Log-likelihood sums contributions from each customer's observed tenure
• Right-censoring handled naturally: active customers contribute S(t) to likelihood
• Standard errors derived from the inverse Hessian matrix at convergence

The Beta-Geometric model is designed specifically for contractual settings where churn events are directly observable—such as subscription cancellations, membership lapses, or service contract non-renewals. This contrasts with non-contractual models like the BG/NBD, where inactivity must be inferred from a lack of transactions rather than an explicit termination signal.

• Observable churn: event time is known with certainty
• No need to infer a 'dropout' state from transaction gaps
• Pairs with Gamma-Gamma model for monetary value prediction in full CLV estimation