Customer Lifetime Value - Index

Routing Summary

Customer-level probability models for customer-base analysis and customer lifetime value (CLV), anchored by the Fader–Hardie papers and technical notes: Pareto/NBD and BG/NBD for noncontractual transaction flow, gamma-gamma for spend, RFM iso-value curves for CLV, and the shifted-beta-geometric for contractual retention. Complements the aggregate response models (MMM, adstock, geo experiments) elsewhere in Market Response Models.

• Need the big picture, the CLV decomposition, or which model fits which business setting? → Customer Lifetime Value - Overview
• Need the original “counting your customers” model, its likelihood, $P(\text{alive})$ or conditional expectation? → Pareto-NBD Model
• Need the easy-to-fit alternative (Excel-level likelihood) and how it compares empirically? → BG-NBD Model
• Need expected spend per transaction with shrinkage toward the population mean? → Gamma-Gamma Model of Monetary Value
• Need to turn recency/frequency/monetary value into CLV, or to understand why more past purchases can mean lower value? → RFM Sufficient Statistics and Iso-Value Curves
• Need to project retention or the survivor curve for a subscription business? → Shifted-Beta-Geometric Model for Contractual Retention
• Need covariates, full-Bayes / PyMC-Marketing, cohort pooling, or the ML (ZILN) alternative? → Bayesian and Hierarchical Extensions of CLV Models
• Need the formula for discounted expected transactions? → DET closed form
• Need to explain why retention rates rise with tenure? → ruse of heterogeneity

Concept Map

Concept	Note	Type	Depends On	Key Result
CLV framing and model taxonomy	Customer Lifetime Value - Overview	overview	Single-Parameter Models; Hierarchical Models; Survival Analysis	CLV = margin × revenue/transaction × DET; contractual vs noncontractual × discrete vs continuous; validation by histogram, tracking plot, conditional expectations
Pareto/NBD	Pareto-NBD Model	method	Overview; Single-Parameter Models	Poisson purchases + exponential lifetime, gamma-mixed; $(x,t_x,T)$ sufficient; likelihood via ${}_2{F}_1$; $E[Y(t)\mid \cdot ]$ = $P(\text{alive})$ × updated-parameter mean
BG/NBD	BG-NBD Model	method	Pareto-NBD Model	Dropout w.p. $p$ after each purchase, $p\sim$ beta; likelihood in gamma/beta functions only; predictions correlate 0.996 with Pareto/NBD on CDNOW
Gamma-gamma spend	Gamma-Gamma Model of Monetary Value	method	Overview; Shrinkage	$E(Z\mid \bar{z},x)=\frac{q-1}{px+q-1}E(Z)+\frac{px}{px+q-1}\bar{z}$; spend independent of transaction process (corr. 0.06–0.11 on CDNOW)
RFM, DET and iso-value curves	RFM Sufficient Statistics and Iso-Value Curves	concept	Pareto-NBD Model; Gamma-Gamma	Closed-form DET with Tricomi $\Psi$; backward-bending iso-value curves (increasing frequency paradox); zero class ≈ 5% of CDNOW cohort value
sBG retention	Shifted-Beta-Geometric Model for Contractual Retention	method	Overview; Survival Analysis	$r_t=(\beta +t-1)/(\alpha +\beta +t-1)$ rises with tenure purely through heterogeneity; year-12 survival projected within ~4% from 7 years of data
Bayesian / hierarchical / ML extensions	Bayesian and Hierarchical Extensions of CLV Models	application	all of the above; Hierarchical Models	Covariates via $\alpha ={\alpha }_0{e}^{-{\gamma }_1^{\prime }{z}_1}$ etc.; PyMC-Marketing priors and cohort pooling; ZILN loss for new-customer LTV

Notes

• Customer Lifetime Value - Overview — CONTAINS: critique of RFM scoring regressions, CLV decomposition (margin × spend × DET), contractual CLV as discounted survivor sum, two-by-two taxonomy of settings and models, “buy till you die” template, validation standard, relevance to MMM / incrementality work, end-to-end numeric example.
• Pareto-NBD Model — CONTAINS: six assumptions, NBD and Pareto II marginals, individual-level likelihood and sufficiency of recency/frequency, population likelihood with $\alpha \gtrless \beta$ branches, mean $E[X(t)]$, $P(\text{alive})$ (individual and population), conditional expectation, estimation difficulties, CDNOW estimates, Python log-likelihood.
• BG-NBD Model — CONTAINS: five assumptions, likelihood (Eqs. 3, 6) and $A_1$–$A_4$ spreadsheet form, $E[X(t)]$, conditional expectation (Eq. 10) and implied $P(\text{alive})$, 81-world simulation (MAPE table), CDNOW comparison table vs Pareto/NBD, implementation caveats, worked rows of the Excel sheet, Python code.
• Gamma-Gamma Model of Monetary Value — CONTAINS: assumptions, why not normal or lognormal, marginal density of $\bar{z}$ (B2 distribution), inverse-gamma latent mean, conditional expectation as shrinkage, independence test on CDNOW, fit diagnostics and stability, shrinkage-weight table, Python MLE code.
• RFM Sufficient Statistics and Iso-Value Curves — CONTAINS: sufficiency argument, DET derivation and closed form, continuous discounting, CLV-from-RFM formula, increasing frequency paradox with explanation, two-stage holdout validation, CDNOW cohort valuation (Tables 2–3), author-stated limitations, DET grid example, Python code.
• Shifted-Beta-Geometric Model for Contractual Retention — CONTAINS: failure of curve-fitting extrapolation, coin-flip story, sBG pmf/survivor/retention formulas and recursion, ruse of heterogeneity, censored cohort-table MLE algorithm, Regular/High End estimates and replication (including the 0.688 vs 0.668 typo), limits, BdW and EG relatives, multi-cohort hierarchical proposal, Python code.
• Bayesian and Hierarchical Extensions of CLV Models — CONTAINS: time-invariant covariate theorem (note 019) and sign convention, beta-logistic rationale, endogeneity warning, empirical-Bayes vs full-Bayes, marginal-likelihood vs data-augmentation sampling, lifetimes and PyMC-Marketing classes / default priors / fit methods / covariates / cohort pooling, ZILN loss and evaluation metrics, model-choice table, covariate and API examples.

External / Cross-Folder Links

• Single-Parameter Models — gamma-Poisson and beta-binomial conjugate pairs underlying every model here.
• Hierarchical Models — population distributions over unit-level parameters; partial pooling across cohorts.
• Empirical Bayes - Overview, Empirical Bayes Interpretation of Shrinkage, Robbins Formula and Poisson Empirical Bayes, James-Stein Estimator — the plug-in prior logic and shrinkage form of all conditional expectations.
• Survival Analysis — survivor and hazard functions, censoring; latent vs observed churn.
• Delayed Feedback Model for Conversion Prediction, Delayed and Censored Feedback - Overview — related “not yet vs never” latent-state models.
• Monsters and Mixtures — continuous mixtures, zero-inflation and hurdle models.
• Heterogeneity in Agent Models — heterogeneity-driven aggregate dynamics; calibrated trait distributions for consumer agents.
• Discrete Choice Models — the other individual-level modelling tradition in marketing.
• Market Response Models - Overview, Bayesian Media Mix Modeling - Overview, Bayesian Estimation and Priors for MMM, ROAS, mROAS, and Optimal Media Mix, Optimal Marketing Decisions and Forecasting, Markets Data and Sales Drivers — aggregate response models and decision layers that can consume CLV as the value metric.
• Posterior Predictive Checking, MCMC Basics, Efficient MCMC, Generalized Linear Models, Metalearners for CATE, Partial Pooling as Multiple Comparisons Correction, Product Adoption and Diffusion Models — workflow, computation and adjacent methods.

Sources

• Fader Hardie Lee 2005 - Counting Your Customers the Easy Way BG-NBD — Fader, P. S., Hardie, B. G. S. & Lee, K. L. (2005), “‘Counting Your Customers’ the Easy Way: An Alternative to the Pareto/NBD Model,” Marketing Science 24(2), 275–284.
• Fader Hardie 2005 - A Note on Deriving the Pareto-NBD Model — Fader, P. S. & Hardie, B. G. S. (2005), “A Note on Deriving the Pareto/NBD Model and Related Expressions,” brucehardie.com/notes/009. (Derives the results of Schmittlein, Morrison & Colombo 1987, Management Science 33(1), 1–24, which is paywalled and not held.)
• Fader Hardie Lee 2005 - RFM and CLV Iso-Value Curves — Fader, P. S., Hardie, B. G. S. & Lee, K. L. (2005), “RFM and CLV: Using Iso-Value Curves for Customer Base Analysis,” Journal of Marketing Research 42(4), 415–430 (author preprint, Feb 2005).
• Fader Hardie 2013 - The Gamma-Gamma Model of Monetary Value — Fader, P. S. & Hardie, B. G. S. (2013), “The Gamma-Gamma Model of Monetary Value,” brucehardie.com/notes/025.
• Fader Hardie 2007 - How to Project Customer Retention — Fader, P. S. & Hardie, B. G. S. (2007), “How to Project Customer Retention,” Journal of Interactive Marketing 21(1), 76–90 (author preprint, May 2006).
• Fader Hardie 2007 - Incorporating Time-Invariant Covariates into the Pareto-NBD and BG-NBD Models — Fader, P. S. & Hardie, B. G. S. (2007), brucehardie.com/notes/019.
• Wang Liu Miao 2019 - A Deep Probabilistic Model for Customer Lifetime Value Prediction — Wang, X., Liu, T. & Miao, J. (2019), arXiv:1912.07753.
• Software read for grounding (not stored): PyMC-Marketing pymc_marketing/clv source (GitHub main, 2026-09-18); lifetimes README.