BetaGeoBetaBinom

class pymc_marketing.clv.distributions.BetaGeoBetaBinom(name, *args, **kwargs)

Population-level distribution class for a discrete, non-contractual, Beta-Geometric/Beta-Binomial process.

It is based on equation(5) from Fader, et al. in [1].

\[\begin{split}\mathbb{L}(\alpha, \beta, \gamma, \delta | x, t_x, n) &= \frac{B(\alpha+x,\beta+n-x)}{B(\alpha,\beta)} \frac{B(\gamma,\delta+n)}{B(\gamma,\delta)} \\ &+ \sum_{i=0}^{n-t_x-1}\frac{B(\alpha+x,\beta+t_x-x+i)}{B(\alpha,\beta)} \\ &\cdot \frac{B(\gamma+1,\delta+t_x+i)}{B(\gamma,\delta)}\end{split}\]

Support	\(t_j >= 0\) for \(j = 1, \dots,x\)
Mean	\(\mathbb{E}[X(n) | \alpha, \beta, \gamma, \delta] = (\frac{\alpha}{\alpha+\beta})(\frac{\delta}{\gamma-1}) \cdot{1-\frac{\Gamma(\gamma+\delta)}{\Gamma(\gamma+\delta+n)}\frac{\Gamma(1+\delta+n)}{\Gamma(1+ \delta)}}\)

References

[1]

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang (2010), “Customer-Base Analysis in a Discrete-Time Noncontractual Setting,” Marketing Science, 29 (6), 1086-1108. https://www.brucehardie.com/papers/020/fader_et_al_mksc_10.pdf

Methods

BetaGeoBetaBinom.__init__(*args, **kwargs)
BetaGeoBetaBinom.dist(alpha, beta, gamma, ...)	Get the distribution from the parameters.
BetaGeoBetaBinom.logp(alpha, beta, gamma, ...)	Log-likelihood of the distribution.

Attributes

rv_op