Distribution Expressions

Caesar supports a limited number of probability distributions as built-ins. They are allowed as the right-hand side of an assignment, e.g. x = ber(1, 1);. Distribution expressions are not allowed to occur nested inside expressions.

All distribution expressions except for flip take literal arguments, i.e. numbers. Expressions such as 1+x or even 1+1 are not supported as arguments.

Bernoulli

The Bernoulli distribution.

Constant-only with Odds

proc ber(pa: UInt, pb: UInt) -> (r: Bool)

This version takes two odds: ber(pa, pb) returns true with probability pa/(pa+pb) and false with probability pb/(pa+pb).

Formally: vc[x = ber(pa, pb)](φ) = (pa/(pa+pb)) * φ[x/true] + (pb/(pa+pb)) * φ[x/false].

Note that calls with pa+pb = 0 will result in a constant zero expectation.

Symbolic with Probabilities

proc flip(p: UReal) -> (r: Bool)

Returns true with probability p and false with probability 1-p. Note: if p is not a valid probability (not in the range [0,1]), then the result of this distribution is undefined!

This distribution accepts symbolic parameters (not just constants).

Uniform

proc unif(a: UInt, b: UInt) -> (r: UInt)

The uniform distribution returns the values in the closed interval [a,b] with uniform probability.

Note that calls with a <= b will result in a constant zero expectation.

Binomial

proc binom(n: UInt, pa: UInt, pb: UInt) -> (r: UInt)

Returns values r according to the binomial distribution where n is the number of trials, pa are the odds of success and pb are the odds of failure.

Hypergeometric

proc hyper(pN: UInt, k: UInt, pn: UInt) -> (r: UInt)

Return values according to the hypergeometric distribution where pN is the population size, k is the number of success states in the population and pn is the number of draws. The result r is the number of observed successes, weighted by its probability.