We call PF a basic distribution. 13 The choice of P [n] is free as long as the compatibility condition is met. If we want all interpretations to be equiprobable, we should set P ( n )( A 1 = x 1 , . . . , A n = x n ) = 1/2 n for ery (x 1 , . . ,x n ). The resulting P F is a uniform distribution over Ω F just like the one over the unit interval [0, 1]. If, on the other hand, we stipulate no interpretation except ω 0 = {c 1 ,c 2 ,...} should be possible, we put, for each n,

$$P_{F}^{\{n\}}(A_{1}=\,x_{1},\ldots,\,A_{n}\,=\,x_{n}\,)=\left\{\begin{array}{l l}{{1}}&{{\mathrm{if}\quad\quad\forall i\,\,x_{i}\,=\,c_{i}\,\,(1\,\leq\,i\leq\,n)}}\\ {{0}}&{{\mathrm{o.w.}}}\end{array}\right.$$

Then P F places all probability mass on ω 0 and gives probability 0 to the rest.

Define a parameterized logic program as a definite clause program 14 DB = FU R where F is a set of unit clauses, R is a set of non-unit clauses such that no clause head in R is unifiable with a unit clause in F and a parameterized basic distribution P F is associated with F. A parameterized PF is obtained from a collection of parameterized joint distributions ssent of the state of the state of the state of the state of the state of the state of the state of the state of the state of the state.

flexible P F is, but at the cost of tractability. The choice of parameterized finite distributions made by Sato (1995) was simple:

$$P_{F}^{(2n)}(ON_{1}=x_{1},OFF_{2}=x_{2},\ldots,OFF_{2n}=x_{2n}\ \mid\theta_{1},\ldots,\theta_{n}\,)$$ $$=\prod_{i\,=\,1}^{n}P_{bs}(ON_{2i-1}=x_{2i-1},OFF_{2i}=x_{2i}\ \mid\theta_{i})$$  re

$$\mathrm{where}$$
$$P_{bs}(ON_{2i-1}=x_{2i-1},\,OFF_{2i}=x_{2i}\mid\theta_{i}\,)$$ $$=\left\{\begin{array}{rl}0&\mbox{if$x_{2i-1}=x_{2i}$}\\ \theta_{i}&\mbox{if$x_{2i-1}=1,\ x_{2i}=0$}\\ 1-\theta_{i}&\mbox{if$x_{2i-1}=0,\ x_{2i}=1$}.\end{array}\right.\tag{4}$$

P bs (ON 2i-1 = x2i-1, OFF 2i = x2i | 0i) (1 < i < n) represents a probabilistic binary switch, i.e. a Bernoulli trial, using two exclusive atoms ON 2 i -1 and OFF 2 in such a way that either one of them is true on each trial but never both. Bi is a parameter specifying the probability that the switch i is on.  The resulting PF is a probability measure over the infinite product of independent binary outcomes. It might look too simple but expressive enough for Bayesian networks, Markov chains and HMMs (Sato, 1995; Sato & Kameya, 1997).

## 3.2 Extending Pf To Pdb

In this subsection, we extend PF to a probability measure PDB over the possible worlds for L, i.e. the set of all possible truth assignments to ground atoms in L through the least