Explore variants of the shifted Beta Geometric model

[ricardoV94]

From: Fader, P. S., & Hardie, B. G. (2007). How to project customer retention. Journal of Interactive Marketing, 21(1), 76-90. pdf

They mention this other model derived from the beta-binomial, which is conceptually equivalent:

Their model is based on assumptions simi-
lar to those behind the sBG model: (a) Each person
responds to a direct-mail solicitation with constant
probability p, and (b) p varies across the population
according to a beta distribution. While BM base their
framework on the beta-binomial model, it could have
been derived as an sBG model (e.g., the mailing on
which the prospect responds to the offer is character-
ized by the shifted-geometric distribution). As such, it
is possible to identify clear relationships between
some of the results in this article [e.g., rt and S(t)] and
some quantities of interest in a list-falloff setting.

Then extensions with cohort covariates:

The BM framework was extended by Rao and Steckel
(1995) to incorporate (time-invariant) descriptor
variables such as age, income, and sex. This is accom-
plished using the beta-logistic model (Heckman &
Willis, 1977),

Incorporating the effects of time-
varying covariates (e.g., marketing-mix effects, sea-
sonality) is more complicated. The key is to bring in
all of these factors at the right level; that is, at the
level of the latent parameter of interest (in this case,
�) instead of just “jamming” different covariate effects
into a regression-like model (see Schweidel, Fader, &
Bradlow, 2006, for a discussion of how to do this in a
continuous-time contractual setting.)

And extensions with time effets:

Both the sBG model and its continuous-time analog
(i.e., the EG model) are based on the assumption that
the commonly observed phenomenon of increasing
retention rates is due entirely to heterogeneity;
individual-customer-level retention rates are assumed
to be constant. If we wish to allow for the possibility of
time dynamics at the level of the individual customer,
we can no longer characterize the duration of an indi-
vidual’s relationship with the firm using either the
shifted-geometric or exponential distributions, both of
which have the “memoryless” property (i.e., the proba-
bility of survival to s � t, given survival to t , is the
same as the initial probability of survival to s ). In a
continuous-time setting, we can accommodate this
effect by assuming that individual lifetimes can be
characterized by the Weibull distribution, which allows
for an individual’s risk of canceling a contract to
increase or decrease as the length of the relationship
with the firm increases. In a discrete-time contractual
setting, this leads to the beta-discrete-Weibull (BdW)
model (Fader & Hardie, 2006), which is a generaliza-
tion of the sBG model, while in a continuous-time con-
tractual setting, this leads to a generalization of the EG
model, the Weibull-gamma (WG) model (Hardie et al.,
1998; Morrison & Schmittlein, 1980).

[ColtAllen]

There's a PDF related to this in Bruce Hardie's notes:

https://www.brucehardie.com/notes/037/time-varying_covariates_in_BG.pdf

[ColtAllen]

Closing to create two related, albeit separate issues for this.