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I would like to use the following kind of prior in a Stan simulation

$$ f_{K \mid G} (k \mid g) = \frac{1}{I_g(b,D)}|k|^{\frac{b-2}{2}} \exp \biggl \{ -\frac{1}{2} \text{tr} (Dk) \biggr \}\mathbb{1}_{P_g}(k)$$

where, for $p \in \mathbb{N}$, $k$ is a real valued $p \times p$ matrix, $b >2$, $b \in \mathbb{N}$, $D$ is a real valued $p \times p$ positive definite matrix and $g =(V_g,E_g)$ is an undirected graph with $V_g= \{1, \dots, p\}$ vertexes and $E_g \subset V_g \times V_g$ edges. The set $P_g$ is defined as follows: $$ P_g := \{ A \in S_+(p) \mid A(i,j)=0 \Leftarrow (i,j) \in V_g\times V_g \setminus E_g \}$$ where $S_+(p)$ is the set of real valued $p \times p$ positive definite matrices. Here $I_g(b,D)$ is the normalization constant for the distribution. $b$ and $D$ are known parameters but $G$ is distributed as the uniform on the space of graphs with $p$ nodes. This kind of prior is taken from the article "Mohammadi and Wit, Bayesian Structure Learning in Sparse Gaussian Graphical Models".

The problem is the calculation of the normalizing constant. I can obtain simulation of this prior from an other software but how to put it into stan?

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If $b$ and $D$ are known, then you can do without the normalizing constant in Stan and just do target += 0.5 * (b - 2) * log_determinant(k) - 0.5 * sum(diagonal(D * k)); If $D$ is known but not $b$, then you can possibly marginalize $b$ over integers greater than 2 (up to some large but finite limit). If $D$ is not known, then omitting the normalizing constant would invalidate the resulting posterior draws. You need some sort of a deterministic approximation for the log normalizing constant.

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  • $\begingroup$ Thank you for your answer. I'm sorry if it was not clear from my question but $b$ and $D$ are known parameters. On the other hand $G$ is a graph distributed as the uniform on the space of graphs with $p$-nodes. Then I cannot omit the normalization constant. Am I wrong? $\endgroup$
    – Bremen000
    Jan 4, 2019 at 17:52
  • $\begingroup$ I believe you are right, but I haven't read the Mohammadi and Wit paper. The first challenge would be constructing a $k$ matrix from the admissible region defined by $P_g$. If the volume of that set depends on unknowns, then you need to account for its logarithm in Stan. $\endgroup$ Jan 4, 2019 at 19:00

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