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Joint distribution

In probability a joint distribution of the variable set $\Omega$ is a probability distribution of $\Omega$. In ProBT it is rather a probability measure of $\Omega$ (see definition), that is, it is not necessarely normalized.

Definition 23   Let $S_{\Omega}=\{\Omega_{s_1},\Omega_{s_2},...,\Omega_{s_n}\}$ be a partition of a set of variables $\Omega$ and $S_{\gamma}=\{\gamma_1,\gamma_2,...,\gamma_n\}$ where $\gamma_{i=1,2,...,n}$ is a computable object on $\Omega_i=\Omega_{s_i}\cup\Omega_{k_i}$. $S_{\gamma}$ is said to be a description of $\Omega$ if and only if $\Omega_{k_i} \subseteq \bigcup_{j=1}^{i-1}\Omega_{s_i}$.

The prevoius definition restricts

Definition 24   Let $S_{\gamma}=\{\gamma_1,\gamma_2,...,\gamma_n\}$ be a description of $\Omega$ and $\Omega_s,\Omega_k\subset\Omega$ with $\Omega_s \cap \Omega_k = \emptyset$. A joint distribution on $\Omega$ given $S_{\gamma}$ is a kernel where the compute function is defined as follows:

$$compute(\omega) = \prod_{i=1}^{n}\gamma_i.compute(\omega_i)$$

A joint distribution is provided of the method $ask(\Omega_s,\Omega_k)$ returning:

• a conditional kernel $\rho$ on $\Omega_s$ given $\Omega_k$ if $\Omega_k \neq \emptyset$ and
• a kernel $\varphi$ on $\Omega_s$ if $\Omega_k = \emptyset$.

In both cases the compute function is defined as:

$$compute(\omega_s \cup \omega_k) = %\frac{ \sum_{\omega_f\in\Omega_f}\prod_{i=1}^n\gamma_i.compute(\omega_i) %}$$ (3.20)

where $\Omega_f=\Omega\backslash(\Omega_s\cup\Omega_k)$ (i.e. $\Omega_f$ is the complement of $(\Omega_s\cup\Omega_k)$ on $\Omega$) and $\omega_i$ is the projections of $(\omega_s\cup\omega_k\cup\omega_f) \in\Omega$ on $\Omega_i$.