Additional Results for the Pareto/NBD Model Peter S. Fader www.petefader.com Bruce G. S. Hardie www.brucehardie.com January 24 Abstract This note derives expressions for i) the raw moments of the posterior distribution of Λ M, ii) the marginal posterior distributions of Λ M, iii) P(alive at T + t x, t x, T). Preliminaries Recall the basic Pareto/NBD model results (Fader Hardie 25): i) The individual-level likelihood function for someone with purchase history (x, t x, T) is L(λ, µ x, t x, T) λx µ λ + µ e (λ+µ)tx + λx+ λ + µ e (λ+µ)t. () ii) The likelihood function for a romly chosen individual with purchase history (x, t x, T) is Γ(r + x)αr {( ) ( ) } s r + x A + A 2 (2) r + s + x r + s + x A A 2 2F ( r + s + x, s + ; r + s + x + ; α β α+t x ) (α + t x ) r+s+x if α β 2F ( r + s + x, r + x; r + s + x + ; β α β+t x ) (β + t x ) r+s+x if α < β ( ) 2F r + s + x, s; r + s + x + ; α β α+t if α β (α + T) r+s+x ) 2F ( r + s + x, r + x + ; r + s + x + ; β α β+t (β + T) r+s+x if α < β iii) The probability that a customer with purchase history (x, t x, T) is alive at time T is the probability that the (unobserved) time at which he dies (ω) occurs after T, which is given by P(Ω > T λ, µ; x, t x, T) λx e (λ+µ)t L(λ, µ x, t x, T). (3) c 24 Peter S. Fader Bruce G.S. Hardie. This document can be found at < <http://brucehardie.com/notes/5/>.
iv) The joint posterior distribution of Λ M is g(λ, µ r, α, s, β; x, t x, T) L(λ, µ x, t x, T)g(λ r, α)g(µ s, β) g(λ r, α) αr λ r e λα, (4) is the gamma distribution that captures heterogeneity in transaction rates across customers, g(µ s, β) βs µ s e µβ (6) is the gamma distribution that captures heterogeneity in death rates across customers. One function we will come across is the confluent hypergeometric function of the second kind (also known as the Tricomi function), which has the following integral representation: Ψ(a, c; z) Γ(a) e zt t a ( + t) c a dt. A simple generalization of this is (Gradshteyn Ryzhik 994, equation 3.383 5) e px x q ( + ax) v dx a q Γ(q)Ψ(q, q + v; p/a). (7) 2 Raw Moments of the Posterior Distribution of Λ M For l, m,, 2,..., E[Λ l M m r, α, s, β; x, t x, T] is the (l, m)th raw moment of the joint posterior distribution of Λ M. We derive an expression for this quantity in the following manner. E[Λ l M m r, α, s, β; x, t x, T] λ l µ m g(λ, µ r, α, s, β; x, t x, T) λ l µ m L(λ, µ x, t x, T)g(λ r, α)g(µ s, β) λ l µ m L(λ, µ x, t x, T) αr λ r e λα µ s e µβ L(λ, µ x, t x, T) αr λ r+l e λα µ s+m e µβ Γ(r + l)γ(s + m) L(λ, µ x, t α l β m x, T) αr+l λ r+l e λα +m µ s+m e µβ Γ(r + l) Γ(s + m) Γ(r + l)γ(s + m) L(r + l, α, s + m, β x, t x, T), (8) α l β m L(r + l, α, s + m, β x, t x, T) is simply (2) evaluated using r + l in place of r s + m in place of s. (5) http://functions.wolfram.com/hypergeometricfunctions/hypergeometricu/ 2
3 Marginal Posterior Distributions of the Latent Variables 3. Marginal Posterior of Λ Given the expression for the joint posterior distribution of Λ M, (4), the marginal posterior distribution of Λ is given by: g(λ r, α, s, β; x, t x, T) B αr λ r+x e λ(α+tx) αr λ r+x 2 e λ(α+tx) g(λ, µ r, α, s, β; x, t x, T)dµ B + B 2, (9) µ s (λ + µ) e µ(β+tx) dµ µ s( + µ λ) e µ(β+t x) dµ which, recalling (7) (with p β + t x, q s +, v, a /λ), αr λ r+s+x e λ(α+tx) s Ψ ( s +, s + ; λ(β + t x ) ), () B 2 αr λ r+x e λ(α+t) αr λ r+x e λ(α+t) µ s (λ + µ) e µ(β+t) dµ which, recalling (7) (with p β + T, q s, v, a /λ), µ s ( + µ λ) e µ(β+t) dµ αr λ r+s+x e λ(α+t) Ψ ( s, s; λ(β + T) ). () The mean of this distribution is simply (8) evaluated with l m. 3.2 Marginal Posterior of M Given the expression for the joint posterior distribution of Λ M, (4), the marginal posterior distribution of the M is given by: g(µ r, α, s, β; x, t x, T) C βs µ s e µ(β+tx) βs µ s e µ(β+tx) g(λ, µ r, α, s, β; x, t x, T)dλ C + C 2, (2) λ r+x (λ + µ) e λ(α+tx) dλ λ r+x ( + λ µ ) e λ(α+tx) dλ 3
which, recalling (7) (with p α + t x, q r + x, v, a /µ), βs µ r+s+x e µ(β+tx) r Γ(r + x) α Ψ ( r + x, r + x; µ(α + t x ) ), (3) C 2 βs µ s e µ(β+t) βs µ s 2 e µ(β+t) λ r+x (λ + µ) e λ(α+t) dλ λ r+x ( + λ µ ) e λ(α+t) dλ which, recalling (7) (with p α + T, q r + x +, v, a /µ), βs µ r+s+x e µ(β+t) r Γ(r + x + ) α Ψ ( r + x +, r + x + ; µ(α + T) ). (4) The mean of this distribution is simply (8) evaluated with l m. 4 P(alive at T + t x, t x, T) Sting at time T, what is the probability that a customer with purchase history (x, t x, T) will still be alive at T +t? Let us first derive the conditional distribution of Ω, F(ω r, α, s, β; x, t x, T). Decomposing the structure of this quantity, F(ω r, α, s, β; x,t x, T) By definition, F(ω µ, Ω > T)P(Ω > T λ, µ; x, t x, T)g(λ, µ r, α, s, β; x, t x, T). (5) F(ω µ, Ω > T) S(ω µ, Ω > T) Substituting (3) (6) (6) in (5) gives us F(ω r, α, s, β; x, t x, T) Therefore, S(ω µ) S(T µ) e µ(ω T), ω > T. (6) e µ(ω T) λ x e (λ+µ)t g(λ r, α)g(µ s, β) { }{ } λ x e λt g(λ r, α)dλ e µω g(µ s, β)dµ ( ) r ( ) x ( ) s Γ(r + x) α β /, ω > T. (7) α + T α + T β + ω P(alive at T + t r, α, s, β; x, t x, T) F(T + t r, α, s, β; x, t x, T) ( ) r ( ) x ( ) s Γ(r + x) α β /. (8) α + T α + T β + T + t 4
References Fader, Peter S. Bruce G.S. Hardie (25), A Note on Deriving the Pareto/NBD Model Related Expressions, <http://brucehardie.com/notes/9/>. Gradshteyn, I. S. I. M. Ryzhik (994), Table of Integrals, Series, Products, 5th edition, San Diego, CA: Academic Press. 5