In a Bayesian analysis, I came across the following distribution that results ends up looking like a re-scaled Dirichlet distribution. The motivation comes from looking at probabilities $x_1, \ldots, x_J$, where we condition on an event that occurs with probability $\sum_{j=1}^{J} w_j x_j$ (and weights $w_1, \ldots, w_J$ are known). Its density function looks like
$$g(\mathbf{x} \mid \mathbf{w}, \mathbf{a}, b) \propto \left(\frac{\prod_{j=1}^{J} x_j^{\beta_j}}{\sum_{j=1}^{J} w_j x_j} \right)^{b} \times \prod_{j=1}^{J} x_j^{\alpha_j - 1} \propto \left(\sum_{j=1}^{J} w_j x_j \right)^{-b} \times \prod_{j=1}^{J} x_j^{a_j - 1}$$
where $\sum_{j=1}^J x_j = 1$ and I defined $a_j = \alpha_j + b \beta_j$. I am particularly interested in the following case with joint distribution
$$g(\mathbf{x} \mid \{\mathbf{w}_i\}_{i=1}^N, \{\mathbf{a}_i\}_{i=1}^N, \{b_i\}_{i=1}^N) \propto \prod_{i=1}^N \left(\sum_{j=1}^{J} w_{ij} x_j \right)^{-b_i} \times \prod_{j=1}^{J} x_j^{a_{ij} - 1} \propto \prod_{i=1}^N \left(\sum_{j=1}^{J} w_{ij} x_j \right)^{-b_i} \times \prod_{j=1}^{J} x_j^{\sum_{i=1}^N a_{ij} - 1}$$
Question 1: Is anyone aware if this distribution is similar to anything known in the literature? This reference on multivariate distributions (https://onlinelibrary.wiley.com/doi/10.1002/0471722065.ch49) leads to extensions that look somewhat similar, like Lochner's or Louiville distributions, or Wikipedia suggests the Generalized Dirichlet distribution (https://en.wikipedia.org/wiki/Generalized_Dirichlet_distribution). However, I do not see a clear equivalence to these.
Question 2: Do you see a simple way to sample from this distribution? The case I am interested in would have large $N$ but small number of probabilities $J$. I could always do an Markov Chain Monte Carlo (MCMC) approach given that I am able to evaluate the kernel of the distribution. Importance sampling seems like it could be a good candidate as well. Ideas on this are welcome.
