ModifiedBetaGeoNBD

class pymc_marketing.clv.distributions.ModifiedBetaGeoNBD(name: str, *args, rng=None, dims: str | Sequence[str | None] | None = None, initval=None, observed=None, total_size=None, transform=UNSET, default_transform=UNSET, **kwargs)

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

In MBG/NBD, a customer may drop out at time zero. This is in contrast with the BG/NBD model, which assumes all non-repeat customers are still active. Based on Batislam, et al. in [1], and Wagner & Hopper in [2] .

\[\begin{split}\mathbb{LL}(a, b, \alpha, r | x, t_x, T) = \ln \left[ A_1 * A_2 * (A_3 + \delta_{x>0} A_4) \right] \text{, where:} \\ \begin{align} A_1 &= \frac{\Gamma(r+x) \alpha^r)}{\Gamma(x)} \\ A_2 &= \frac{\Gamma(a+b) \Gamma(b+x+1)}{\Gamma(b) \Gamma(a+b+x+1)} \\ A_3 &= \left( \frac{1}{\alpha + T} \right)^(r+x) \\ A_4 &= \left( \frac{a}{b+x} \right) \left( \frac{1}{\alpha + t_x} \right)^(r+x) \\ \end{align}\end{split}\]

Support	\(t_j >= 0\) for \(j = 1, \dots,x\)
Mean	\(\mathbb{E}[X(n) | r, \alpha, a, b] = \frac{a+b-1}{a-1} \left[ 1 - \left(\frac{\alpha}{\alpha + T}\right)^r {_2}{F}{_1}(r,b;a+b-1;\frac{t}{\alpha + t}) \right]\)

References

[1]

Batislam, E.P., M. Denizel, A. Filiztekin (2007), “Empirical validation and comparison of models for customer base analysis,” International Journal of Research in Marketing, 24 (3), 201-209.

[2]

Wagner, U. and Hoppe D. (2008), “Erratum on the MBG/NBD Model,” International Journal of Research in Marketing, 25 (3), 225-226.

Methods

ModifiedBetaGeoNBD.__init__(*args, **kwargs)
ModifiedBetaGeoNBD.dist(a, b, r, alpha, T, ...)	Get the distribution from the parameters.
ModifiedBetaGeoNBD.logp(a, b, r, alpha, T)	Log-likelihood of the distribution.

Attributes

rv_op