Probabilistic compositional models: Solution of an equivalence problem

Equivalence of compositional expressions and independence relations in compositional models

“…Note that a sequential compositional expression is uniquely determined by its base sequence; accordingly, as in [16] and [17] the base sequence of a sequential compositional expression may be called its "structure". A further property of a sequential compositional expression is that, by Fact 4.3, each intermediate table is always an extension of the distribution function f 1 in f on X 1 (which is the key of θ).…”

“…Note that Kratochvíl (see Remark 4.1 in [17]) calls "equivalent" two (sequential) compositional expressions θ 1 and θ 2 if M θ1 = M θ2 . The following simple example shows that our notion of equivalence is stronger than Kratochvíl's equivalence.…”

“…Structural properties of persegrams are stated by Kratochvíl [16], who in [17] provides a method to decide, given two generating sequences of probability distributions, whether or not the conditional independences in the two compositional models are the same.…”

“…Next, we define the notion of equivalence between compositional expressions having the same base scheme and the same key. (Note that our notion of equivalence is stronger than that studied in [16,17].) Thus, Theorem 1.1 can be re-stated as follows: If X 2 ∩ X 3 ⊆ X 1 , then the four compositional expressions…”

Compositional models, Bayesian models and recursive factorization models

“…In some cases, the composition scheme has a closed form and a typical case is the composition scheme associated with a sequential expression (see "formal ratios" introduced by Kratochvíl [13,14]). …”