Safe Haskell	Safe
Language	Haskell98

Numeric.Probability.Distribution

Contents

Events
Distributions
- Auxiliary functions for constructing and working with distributions
- Normalization = grouping
Spread: functions to convert a list of values into a distribution

Description

Deterministic and probabilistic values

Synopsis

type Event a = a -> Bool
oneOf :: Eq a => [a] -> Event a
just :: Eq a => a -> Event a
newtype T prob a = Cons {
- decons :: [(a, prob)]
}
certainly :: Num prob => a -> T prob a
errorMargin :: RealFloat prob => prob
approx :: (RealFloat prob, Ord a) => T prob a -> T prob a -> Bool
lift :: Num prob => ([(a, prob)] -> [(a, prob)]) -> T prob a -> T prob a
size :: T prob a -> Int
check :: (RealFloat prob, Show prob) => T prob a -> T prob a
cons :: (RealFloat prob, Show prob) => [(a, prob)] -> T prob a
sumP :: Num prob => [(a, prob)] -> prob
sortP :: Ord prob => [(a, prob)] -> [(a, prob)]
sortPDesc :: Ord prob => [(a, prob)] -> [(a, prob)]
sortElem :: Ord a => [(a, prob)] -> [(a, prob)]
norm :: (Num prob, Ord a) => T prob a -> T prob a
norm' :: (Num prob, Ord a) => [(a, prob)] -> [(a, prob)]
norm'' :: (Num prob, Ord a) => [(a, prob)] -> [(a, prob)]
pretty :: (Ord a, Show a, Num prob, Ord prob) => (prob -> String) -> T prob a -> String
(//%) :: (Ord a, Show a) => T Rational a -> () -> IO ()
equal :: (Num prob, Eq prob, Ord a) => T prob a -> T prob a -> Bool
type Spread prob a = [a] -> T prob a
choose :: Num prob => prob -> a -> a -> T prob a
enum :: Fractional prob => [Int] -> Spread prob a
relative :: Fractional prob => [prob] -> Spread prob a
shape :: Fractional prob => (prob -> prob) -> Spread prob a
linear :: Fractional prob => Spread prob a
uniform :: Fractional prob => Spread prob a
negExp :: Floating prob => Spread prob a
normal :: Floating prob => Spread prob a
extract :: T prob a -> [a]
map :: (Num prob, Ord b) => (a -> b) -> T prob a -> T prob b
unfold :: (Num prob, Ord a) => T prob (T prob a) -> T prob a
cond :: Num prob => T prob Bool -> T prob a -> T prob a -> T prob a
truth :: Num prob => T prob Bool -> prob
(?=<<) :: Fractional prob => (a -> Bool) -> T prob a -> T prob a
(>>=?) :: Fractional prob => T prob a -> (a -> Bool) -> T prob a
data Select a
- = Case a
- | Other
above :: (Num prob, Ord prob, Ord a) => prob -> T prob a -> T prob (Select a)
below :: (Num prob, Ord prob, Ord a) => prob -> T prob a -> T prob (Select a)
select :: (Num prob, Ord prob, Ord a) => (prob -> Bool) -> T prob a -> T prob (Select a)
fromFreqs :: Fractional prob => [(a, prob)] -> T prob a
filter :: Fractional prob => (a -> Bool) -> T prob a -> T prob a
mapMaybe :: Fractional prob => (a -> Maybe b) -> T prob a -> T prob b
selectP :: (Num prob, Ord prob) => T prob a -> prob -> a
scanP :: (Num prob, Ord prob) => prob -> [(a, prob)] -> a
(??) :: Num prob => Event a -> T prob a -> prob
expected :: Num a => T a a -> a
variance :: Num a => T a a -> a
stdDev :: Floating a => T a a -> a

Events

type Event a = a -> Bool #

oneOf :: Eq a => [a] -> Event a #

just :: Eq a => a -> Event a #

Distributions

newtype T prob a #

Probability disribution

The underlying data structure is a list. Unfortunately we cannot use a more efficient data structure because the key type must be of class Ord, but the Monad class does not allow constraints for result types. The Monad instance is particularly useful because many generic monad functions make sense here, monad transformers can be used and the monadic design allows to simulate probabilistic games in an elegant manner.

We have the same problem like making Data.Set an instance of Monad, see http://www.randomhacks.net/articles/2007/03/15/data-set-monad-haskell-macros

If you need efficiency, you should remove redundant elements by norm. norm converts to Map and back internally and you can hope that the compiler fuses the lists with the intermediate Map structure.

The defined monad is equivalent to WriterT (Product prob) [] a. See http://www.randomhacks.net/articles/2007/02/21/refactoring-probability-distributions.

Constructors

Cons
Fields decons :: [(a, prob)]

Instances

Fractional prob => C prob (T prob) #
Instance details Defined in Numeric.Probability.Object Methods fromFrequencies :: [(a, prob)] -> T prob a #
Num prob => Monad (T prob) #
Instance details Defined in Numeric.Probability.Distribution Methods (>>=) :: T prob a -> (a -> T prob b) -> T prob b # (>>) :: T prob a -> T prob b -> T prob b # return :: a -> T prob a # fail :: String -> T prob a #
Functor (T prob) #
Instance details Defined in Numeric.Probability.Distribution Methods fmap :: (a -> b) -> T prob a -> T prob b # (<$) :: a -> T prob b -> T prob a #
Num prob => Applicative (T prob) #
Instance details Defined in Numeric.Probability.Distribution Methods pure :: a -> T prob a # (<>) :: T prob (a -> b) -> T prob a -> T prob b # liftA2 :: (a -> b -> c) -> T prob a -> T prob b -> T prob c # (>) :: T prob a -> T prob b -> T prob b # (<*) :: T prob a -> T prob b -> T prob a #
(Num prob, Ord prob, Random prob) => C (T prob) #
Instance details Defined in Numeric.Probability.Simulation Methods (~.) :: (Fractional prob0, Ord prob0, Random prob0, Ord a) => Int -> (a -> T prob a) -> Transition prob0 a # (~..) :: (Fractional prob0, Ord prob0, Random prob0, Ord a) => (Int, Int) -> (a -> T prob a) -> RExpand prob0 a # (~*.) :: (Fractional prob0, Ord prob0, Random prob0, Ord a) => (Int, Int) -> (a -> T prob a) -> Transition prob0 a #
Eq (T prob a) #	We would like to have an equality test of type (==) :: T prob a -> T prob a -> T prob Bool that is consistent with the `Num` instance. We would certainly define x==y = norm (liftM2 (==) x y) . However the `Eq` class enforces the type T prob a -> T prob a -> Bool . We could implement this as check for equal distributions. This would be inconsistent with the `Num` instance because it compares entire distributions, not only individual outcomes. Thus we provide this function as `equal`. I would prefer to omit the `Eq` instance completely, but unfortunately the `Num` instance requires `Eq` as superclass.
Instance details Defined in Numeric.Probability.Distribution Methods (==) :: T prob a -> T prob a -> Bool # (/=) :: T prob a -> T prob a -> Bool #
(Num prob, Ord prob, Ord a, Fractional a) => Fractional (T prob a) #
Instance details Defined in Numeric.Probability.Distribution Methods (/) :: T prob a -> T prob a -> T prob a # recip :: T prob a -> T prob a # fromRational :: Rational -> T prob a #
(Num prob, Ord prob, Ord a, Num a) => Num (T prob a) #
Instance details Defined in Numeric.Probability.Distribution Methods (+) :: T prob a -> T prob a -> T prob a # (-) :: T prob a -> T prob a -> T prob a # (*) :: T prob a -> T prob a -> T prob a # negate :: T prob a -> T prob a # abs :: T prob a -> T prob a # signum :: T prob a -> T prob a # fromInteger :: Integer -> T prob a #
(Num prob, Ord prob, Show prob, Ord a, Show a) => Show (T prob a) #
Instance details Defined in Numeric.Probability.Distribution Methods showsPrec :: Int -> T prob a -> ShowS # show :: T prob a -> String # showList :: [T prob a] -> ShowS #
(ToFloat prob, Expected a) => Expected (T prob a) #
Instance details Defined in Numeric.Probability.Expectation Methods expected :: T prob a -> Float #

certainly :: Num prob => a -> T prob a #

errorMargin :: RealFloat prob => prob #

approx :: (RealFloat prob, Ord a) => T prob a -> T prob a -> Bool #

Check whether two distributions are equal when neglecting rounding errors. We do not want to put this into an Eq instance, since it is not exact equivalence and it seems to be too easy to mix it up with liftM2 (==) x y.

Auxiliary functions for constructing and working with distributions

lift :: Num prob => ([(a, prob)] -> [(a, prob)]) -> T prob a -> T prob a #

size :: T prob a -> Int #

check :: (RealFloat prob, Show prob) => T prob a -> T prob a #

cons :: (RealFloat prob, Show prob) => [(a, prob)] -> T prob a #

can fail because of rounding errors, better use fromFreqs

sumP :: Num prob => [(a, prob)] -> prob #

sortP :: Ord prob => [(a, prob)] -> [(a, prob)] #

sortPDesc :: Ord prob => [(a, prob)] -> [(a, prob)] #

sortElem :: Ord a => [(a, prob)] -> [(a, prob)] #

Normalization = grouping

norm :: (Num prob, Ord a) => T prob a -> T prob a #

norm' :: (Num prob, Ord a) => [(a, prob)] -> [(a, prob)] #

norm'' :: (Num prob, Ord a) => [(a, prob)] -> [(a, prob)] #

pretty :: (Ord a, Show a, Num prob, Ord prob) => (prob -> String) -> T prob a -> String #

pretty printing

(//%) :: (Ord a, Show a) => T Rational a -> () -> IO () infix 0 #

equal :: (Num prob, Eq prob, Ord a) => T prob a -> T prob a -> Bool #

Spread: functions to convert a list of values into a distribution

type Spread prob a = [a] -> T prob a #

distribution generators

choose :: Num prob => prob -> a -> a -> T prob a #

enum :: Fractional prob => [Int] -> Spread prob a #

relative :: Fractional prob => [prob] -> Spread prob a #

Give a list of frequencies, they do not need to sum up to 1.

shape :: Fractional prob => (prob -> prob) -> Spread prob a #

linear :: Fractional prob => Spread prob a #

uniform :: Fractional prob => Spread prob a #

negExp :: Floating prob => Spread prob a #

normal :: Floating prob => Spread prob a #

extract :: T prob a -> [a] #

extracting and mapping the domain of a distribution

map :: (Num prob, Ord b) => (a -> b) -> T prob a -> T prob b #

fmap with normalization

unfold :: (Num prob, Ord a) => T prob (T prob a) -> T prob a #

unfold a distribution of distributions into one distribution, this is join with normalization.

cond #

Arguments

:: Num prob
=> T prob Bool
-> T prob a	True
-> T prob a	False
-> T prob a

conditional distribution

truth :: Num prob => T prob Bool -> prob #

(?=<<) :: Fractional prob => (a -> Bool) -> T prob a -> T prob a infixr 1 #

conditional probability, identical to filter

(>>=?) :: Fractional prob => T prob a -> (a -> Bool) -> T prob a infixl 1 #

filter in infix form. Can be considered an additional monadic combinator, which can be used where you would want guard otherwise.

data Select a #

filtering distributions

Constructors

Case a
Other

Instances

Eq a => Eq (Select a) #
Instance details Defined in Numeric.Probability.Distribution Methods (==) :: Select a -> Select a -> Bool # (/=) :: Select a -> Select a -> Bool #
Ord a => Ord (Select a) #
Instance details Defined in Numeric.Probability.Distribution Methods compare :: Select a -> Select a -> Ordering # (<) :: Select a -> Select a -> Bool # (<=) :: Select a -> Select a -> Bool # (>) :: Select a -> Select a -> Bool # (>=) :: Select a -> Select a -> Bool # max :: Select a -> Select a -> Select a # min :: Select a -> Select a -> Select a #
Show a => Show (Select a) #
Instance details Defined in Numeric.Probability.Distribution Methods showsPrec :: Int -> Select a -> ShowS # show :: Select a -> String # showList :: [Select a] -> ShowS #

above :: (Num prob, Ord prob, Ord a) => prob -> T prob a -> T prob (Select a) #

below :: (Num prob, Ord prob, Ord a) => prob -> T prob a -> T prob (Select a) #

select :: (Num prob, Ord prob, Ord a) => (prob -> Bool) -> T prob a -> T prob (Select a) #

fromFreqs :: Fractional prob => [(a, prob)] -> T prob a #

filter :: Fractional prob => (a -> Bool) -> T prob a -> T prob a #

mapMaybe :: Fractional prob => (a -> Maybe b) -> T prob a -> T prob b #

selectP :: (Num prob, Ord prob) => T prob a -> prob -> a #

selecting from distributions

scanP :: (Num prob, Ord prob) => prob -> [(a, prob)] -> a #

(??) :: Num prob => Event a -> T prob a -> prob infixr 1 #

expected :: Num a => T a a -> a #

expectation value

variance :: Num a => T a a -> a #

statistical analyses

stdDev :: Floating a => T a a -> a #