Semantics of Probabilistic Programs

• Approaches discussed in literature– E1pE2 => E1 (with prob. p)

=> E2 (with prob. (1-p))• Probability of a specific output is not explicit.

– [E1pE2]s = (p)*[E1]s + (1-p)*[E2]s• [E]s is a measure function from events to probabilities.• Forward or Backward?

• Practical Issues– Backward implementation is difficult. (inverses,

representation of sets).– Need to be able to compute probabilities, expectations

inside of the program.

Haskell Implementation

• Using type classes to realize a general parameterized type (Prob a).– Has capability to generalize to product types (a b).

• data (DomainClass a) => (Prob a) = …– (DomainClass a) is an assertion that ‘a’ must be an

instance of DomainClass.– Still must provide ‘instance DomainClass T’ declaration,

which defines functions to operate on values of type T.

• Underlying representation– Dimensions: exact/interpolated countable/uncountable– Sample sets and functions considered.