Observed function of hidden random variables

Question

Let's say a worker can perform 4 types of tasks in a day: A,B,C,D.

Each of which tasks takes time that is distributed according to some probability distribution, say

$$ T_A \sim Gamma(\alpha_A, \beta_A)\\ T_B \sim Gamma(\alpha_B, \beta_B)\\ T_C \sim Gamma(\alpha_C, \beta_C)\\ T_D \sim Gamma(\alpha_D, \beta_D) $$ The data that we have is the total number of tasks per category performed by a person as well as the total time. In other words: $$ T_{total} = T_A\times n_A + T_B\times n_B +T_C\times n_C + T_D\times n_D $$ The problem: infer $\alpha_A, \beta_B, \ldots, \alpha_D, \beta_D$, given $n_A, n_B, n_C, n_D, T_{total}$.

Is there any way to do this in pymc3 or stan or winbugs (or anything else)? Or do I have to derive the probability distribution of $T_total$ in terms of all of the parameters?

Answer 1

A Gamma-distributed random variable that is multiplied by a constant is Gamma-distributed with the same shape and a new scale (see Wikipedia). So $T_j \times n_j$ is Gamma-distributed with shape $\alpha_j$ and scale $\beta_j \times n_j$. Furthermore, the distribution of the sum of Gamma-distributed random variables is known and has several forms (for a review, see this paper by Nadarajah, which uses the alternate inverse-scale parameterization of the Gamma distribution).

In principle, such a distribution could be utilized for the likelihood of $T_{\mbox{Total}}$ in Stan (via its increment_log_prob() and log_sum_exp() functions) but in practice, you would have to make some decision about how to truncate the infinite summation in the PDF.