In the article "Bayesian variable selection for Poisson regression with underreported responses" the author defines $t_i^0$ as the number of actual occurences in a study in the $i$th covariate pattern. Let $\lambda_i$ be the actual estimated rate of events, $\pi_i$ the probability of underreporting, $y_i$ the actual reported events, then the author assumes for $t_i^0$:
\begin{gather} t_i^0 \text{~ shifted Poisson}(\lambda_i (1-\pi_i), y_i), \text{ where } t_i^0 = y_i, y_i+1,y_i+2, \ldots \end{gather}
I was trying to work out a closed form for this equation like:
\begin{gather} \displaystyle\frac{(\lambda_i(1-\pi_i))^{t_i^0}}{t_i^0!e^{\lambda_i(1-\pi_i)}}, \end{gather}
but when I use this formula to work out the Bayesian likelihood, I don't find the correct answer. Does anybody have an idea what I am doing wrong?