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Computable objects

Definition 2   Given a set of variables $\Omega$ a function $f$ is said to be a probability measure of $\Omega$ if $f$ is a positive real-valued function over $\Omega$ (i.e. $f:\Omega \to I\!\!R^+$).

Note that, a probability measure can be used to define a density probability function $f^*$ on $\Omega$ as follow :

$$f^*(\Omega) = \frac{f(\Omega)}{K} \mbox{ with } K = \sum_{\Omega_d}\int_{\Omega_c} f(\Omega)$$ (1.1)

where $\Omega_d,\Omega_c\subset \Omega$ are the subsets of discrete and continuous variables of $\Omega$. Probability measures are used to define computable objects: the basic probabilistic program primitive.

Definition 3   A computable object on $\Omega$ is defined as an abstract object provided of a probability measure function $compute(\Omega)$.

Computable objects are dived into two main groups built-in computable objects and inferred computable objects. A built-in computable object is provided by ProBT and its $compute(\cdot)$ function is predefined. In contrast, inferred computable object are obtained by means of another or others computable objects. Two object classes are derived from computable object, unconditional kernel and conditional kernel. For simplicity in the follow we will call kernel an unconditional kernel. Therefore, we have four main types of computable objects:

• built-in kernels
• built-in conditional kernels
• inferred kernels
• inferred conditional kernels

In the following sections we show that while inferred kernels are generated by a conditional kernel an inferred conditional kernel is generated by a joint distribution: a kernel composed by a set of computable objects.