I don’t think so because the BTYDplus implementation uses exp(-mu*x) for when the same integral appears in the expression for calculating P(alive) (which I intend to be in generated quantities once I get everything else working). However I acknowledge that it is totally possible that I messed up the marginal likelihood.
If it helps, here’s the math I did to arrive at that particular integral. Note that \tau \sim Exp(\mu), however there obviously has to be some truncation here because the lifetime \tau must be after the customer’s last transaction time t_x.
L(k, \lambda, \mu) = \int_{t_x}^{T} L(k, \lambda, \mu | \tau)L(\tau) d\tau + \int_{T}^{\infty} L(k, \lambda, \mu | \tau)L(\tau) d\tau
= K * [\int_{t_x}^{T} S_{\Gamma}(\tau - t_x | k, k\lambda) * \mu e^{-\mu\tau}d\tau + \int_{T}^\infty S_{\Gamma}(T - t_x | k, k\lambda) * \mu e^{-\mu\tau}d\tau].
The second term has a closed form, and the first term is what’s above.