Probability. Should be between 0 and 1.

Distribution over values of type a.

Due to their internal representations, Distribution can not have Functor or Monad instances. However, select is the equivalent of fmap for distributions and always and andThen are respectively the equivalent of return and >>=.

Converts the distribution to a mapping from values to their probability. Values with probability 0 are not included in the resulting mapping.

Converts the distribution to a list of increasing values whose probability is greater than 0. To each value is associated its probability.

 Returns the number of elements with non-zero probability in the distribution.

Values in the distribution with non-zero probability.

Distribution that assigns to each value from the given (value, weight) pairs a probability proportional to weight.

>>> fromList [('A', 1), ('B', 2), ('C', 1)]
fromList [('A',1 % 4),('B',1 % 2),('C',1 % 4)]

Values may appear multiple times in the list. In this case, their total weight is the sum of the different associated weights. Values whose total weight is zero or negative are ignored.

Distribution that assigns to x the probability of 1.

>>> always 0
fromList [(0,1 % 1)]

>>> always 42
fromList [(42,1 % 1)]

Uniform distribution over the values. The probability of each element is proportional to its number of appearance in the list.

>>> uniform [1 .. 6]
fromList [(1,1 % 6),(2,1 % 6),(3,1 % 6),(4,1 % 6),(5,1 % 6),(6,1 % 6)]

True with given probability and False with complementary probability.

Applies a function to the values in the distribution.

>>> select abs $ uniform [-1, 0, 1]
fromList [(0,1 % 3),(1,2 % 3)]

Returns a new distribution conditioning on the predicate holding on the value.

>>> assuming (> 2) $ uniform [1 .. 6]
fromList [(3,1 % 4),(4,1 % 4),(5,1 % 4),(6,1 % 4)]

Note that the resulting distribution will be invalid if the predicate does not hold on any of the values.

>>> assuming (> 7) $ uniform [1 .. 6]
fromList []

Returns a new distribution using the Bayesian update rule.

Using this example: https://en.wikipedia.org/wiki/Bayesian_inference#Probability_of_a_hypothesis

data CookieBowl = Bowl1 | Bowl2 deriving (Eq,Ord)
data CookieType = Plain | ChocolateChip deriving (Eq,Ord)

assumption :: Distribution CookieBowl
assumption = uniform [Bowl1,Bowl2]

update :: Cookie -> Distribution CookieBowl -> Distribution CookieBowl
update c = observing f where
  f b = case b of
    -- Bowl #1 contains 10 chocolate chip cookies and 30 plain cookies
    Bowl1 -> fromList [(c == ChocolateChip,10),(c == Plain,30)]
    -- Bowl #2 contains 20 of each flavour of cookie
    Bowl2 -> fromList [(c == ChocolateChip,20),(c == Plain,20)]

The "update" function in this example can be used to update the probability distribution of which bowl you have based on observing a random cookie inside the bowl.

Binomial distribution. Assigns to each number of successes its probability.

>>> trials 2 $ uniform [True, False]
fromList [(0,1 % 4),(1,1 % 2),(2,1 % 4)]

Takes n samples from the distribution and returns the distribution of their sum.

>>> times 2 $ uniform [1 .. 3]
fromList [(2,1 % 9),(3,2 % 9),(4,1 % 3),(5,2 % 9),(6,1 % 9)]

This function makes use of the more efficient trials functions for input distributions of size 2.

>>> size $ times 10000 $ uniform [1, 10]
10001

Computes for each value in the distribution a new distribution, and then combines those distributions, giving each the weight of the original value.

>>> uniform [1 .. 3] `andThen` (\ n -> uniform [1 .. n])
fromList [(1,11 % 18),(2,5 % 18),(3,1 % 9)]

See the Experiment data type in the Monadic module for a more "natural" monadic interface.

 Determines if a distribution is valid.

A distribution is valid if and only if its domain is non-empty. Invalid distributions may arise from the use of assuming for instance.