Corner case in `dirichlet_lpdf`: error when both alpha - 1 = theta = 0

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cpp
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调研方向

从 stan/math/prim/prob/dirichlet_lpdf.hpp 中链接的 theta_log 计算开始，然后检查相应的 partials_vec<1> 处理。确定 alpha - 1 = theta = 0 情况是否应在不产生错误的情况下贡献零，并确保相关的 autodiff 行为一致。

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描述

In the code there is currently:

  const auto& theta_log
      = to_ref_if<!is_constant_all<T_prior_size>::value>(theta_dbl.log());

  if (include_summand<propto, T_prob, T_prior_size>::value) {
    lp += (theta_log * alpha_m_1).sum();
  }

I understand that by the convention (and also by Stan documentation) for Dirichlet distribution, we usually require theta > 0. But when you think about it, for the corner case when alpha -1 == theta == 0, it is reasonable to define the lpdf to be 0 by the same logic why we define pow(0,0)=1. This is relevant in my application, where I may have on input an information that certain index of theta is a forbidden state and may thus be represented as hard constrained to be equal to 0, but then the Dirichlet prior crashes no matter what. This would be very inefficient for me to workaround by deleting the offending array element as I need to do this over a matrix with many rows each of which may have different constraints, so currently I needed to make my custom implementation of dirichlet_lpdf just to deal with that. I thus propose we change the code to something like:

  if (include_summand<propto, T_prob, T_prior_size>::value) {
    lp += lmultiply(alpha_m_1, theta).sum();
  }

The partials_vec<1> may need a corresponding adjustment.

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