commiteb9408953e098ffb4f3592f851c89e65f40d770fparente893a3d0be6933bc957c969bdec004085f5ab702Author:Jared Tobin <jared@jtobin.ca>Date:Tue, 19 Nov 2013 20:33:51 +1300 Remove old core code.Diffstat:

D | src/Measurable.hs | | | 161 | ------------------------------------------------------------------------------- |

1 file changed, 0 insertions(+), 161 deletions(-)diff --git a/src/Measurable.hs b/src/Measurable.hs@@ -1,161 +0,0 @@ -{-# LANGUAGE BangPatterns #-} - -module Measurable where - -import Control.Applicative -import Control.Monad -import Data.List -import Data.Monoid -import Numeric.Integration.TanhSinh - --- | A measure is a set function from some sigma-algebra to the extended real --- line. In practical terms we define probability in terms of measures; for a --- probability measure /P/ and some measurable set /A/, the measure of the set --- is defined to be that set's probability. --- --- For any sigma field, there is a one-to-one correspondence between measures --- and increasing linear functionals on its associated space of positive --- measurable functions. That is, --- --- P(A) = f(I_A) --- --- For A a measurable set and I_A its indicator function. So we can generally --- abuse notation to write something like P(I_A), even though we don't --- directly apply the measure to a function. --- --- So we can actually deal with measures in terms of measurable functions, --- rather than via the sigma algebra or measurable sets. Instead of taking --- a set and returning a probability, we can take a function and return a --- probability. --- --- Once we have a measure, we use it by integrating against it. Take a --- real-valued random variable (i.e., measurable function) /X/. The mean of X --- is E X = int_R X d(Px), for Px the distribution (image measure) of X. --- --- We can generalize this to work with arbitrary measurable functions and --- measures. Expectation can be defined by taking a measurable function and --- applying a measure to it - i.e., just function application. --- --- So really, a Measure in this sense is an expression of a particular --- computational process - expectation. We leave a callback to be --- plugged in - a measurable function - and from there, we can finish the --- computation and return a value. A measure is actually represented as a --- *program* that, given a measurable function, integrates that function. --- It's thus completely equivalent to the Continuation monad, albeit with --- a restricted result type. --- --- The Functor instance provides the ability to create pushforward/image --- measures. That's handled by good ol' fmap. The strength of the Monad --- instance is that it allows us to do conditional probability. That is, --- --- parameterModel >>= dataModel == model --- --- Ex, given 'betaMeasure a b' and 'binomMeasure n p' functions that create --- the obvious measures, we can express a beta-binomial model like so: --- --- betaBinomialConjugate :: Double -> Double -> Int -> Measure Double --- betaBinomialConjugate a b n = do --- p <- betaMeasure a b --- binomMeasure n p --- - -newtype Measure a = Measure { measure :: (a -> Double) -> Double } - -instance Num a => Num (Measure a) where - (+) = liftA2 (+) - (-) = liftA2 (-) - (*) = liftA2 (*) - abs = id - signum mu = error "fromInteger: not supported for Measures" - fromInteger = error "fromInteger: not supported for Measures" - -instance Fractional a => Monoid (Measure a) where - mempty = identityMeasure - mappend = (+) - -instance Functor Measure where - fmap f mu = Measure $ \g -> measure mu $ g . f -- pushforward/image measure - -instance Applicative Measure where - pure = return - (<*>) = ap - -instance Monad Measure where - return x = Measure (\f -> f x) - mu >>= f = Measure $ \d -> - measure mu $ \g -> - measure (f g) d - --- | The volume is obtained by integrating against a constant. This is '1' for --- any probability measure. -volume :: Measure a -> Double -volume mu = measure mu (const 1) - --- | The expectation is obtained by integrating against the identity function. --- --- This is just just (`runCont` id). -expectation :: Measure Double -> Double -expectation mu = measure mu id - --- | The variance is obtained by integrating against the usual function. -variance :: Measure Double -> Double -variance mu = measure mu (^ 2) - expectation mu ^ 2 - --- | Create a measure from a collection of observations from some distribution. -fromObservations :: Fractional a => [a] -> Measure a -fromObservations xs = Measure (`weightedAverage` xs) - --- | Create a measure from a density function. -fromDensity :: (Double -> Double) -> Measure Double -fromDensity d = Measure $ \f -> quadratureTanhSinh $ liftA2 (*) f d - where quadratureTanhSinh = result . last . everywhere trap - --- | Create a measure from a mass function. -fromMassFunction :: (a -> Double) -> [a] -> Measure a -fromMassFunction p support = Measure $ \f -> - sum . map (liftA2 (*) f p) $ support - --- | The (sum) identity measure. -identityMeasure :: Fractional a => Measure a -identityMeasure = fromObservations [] - --- | Simple average. -average :: Fractional a => [a] -> a -average xs = fst $ foldl' - (\(!m, !n) x -> (m + (x - m) / fromIntegral (n + 1), n + 1)) (0, 0) xs -{-# INLINE average #-} - --- | Weighted average. -weightedAverage :: Fractional c => (a -> c) -> [a] -> c -weightedAverage f = average . map f -{-# INLINE weightedAverage #-} - --- | Integrate from a to b. -to :: (Num a, Ord a) => a -> a -> a -> a -to a b x | x >= a && x <= b = 1 - | otherwise = 0 - --- | Cumulative distribution function for a measure. --- --- Really cool; works perfectly in both discrete and continuous cases. --- --- > let f = cdf (fromObservations [1..10]) --- > cdf 0 --- 0.0 --- > cdf 1 --- 0.1 --- > cdf 10 --- 1.0 --- > cdf 11 --- 1.0 --- --- > let g = cdf (fromDensity standardNormal) --- > cdf 0 --- 0.504 --- > cdf 1 --- 0.838 --- > cdf (1 / 0) --- 0.999 -cdf :: Measure Double -> Double -> Double -cdf mu b = expectation $ negate (1 / 0) `to` b <$> mu -