Copyright | (c) 2015 Jared Tobin |
---|---|

License | MIT |

Maintainer | Jared Tobin <jared@jtobin.ca> |

Stability | unstable |

Portability | ghc |

Safe Haskell | None |

Language | Haskell2010 |

This module presents a simple combinator language for Markov transition operators that are useful in MCMC.

Any transition operators sharing the same stationary distribution and obeying the Markov and reversibility properties can be combined in a couple of ways, such that the resulting operator preserves the stationary distribution and desirable properties amenable for MCMC.

We can deterministically concatenate operators end-to-end, or sample from a collection of them according to some probability distribution. See Geyer, 2005 for details.

The result is a simple grammar for building composite, property-preserving transition operators from existing ones:

transition ::= primitive transition | concatT transition transition | sampleT transition transition

In addition to the above, this module provides a number of combinators for
building composite transition operators. It re-exports a number of
production-quality transition operators from the *mighty-metropolis*,
*speedy-slice*, and *hasty-hamiltonian* libraries.

Markov chains can then be run over arbitrary `Target`

s using whatever
transition operator is desired.

import Numeric.MCMC import Data.Sampling.Types target :: [Double] -> Double target [x0, x1] = negate (5 *(x1 - x0 ^ 2) ^ 2 + 0.05 * (1 - x0) ^ 2) rosenbrock :: Target [Double] rosenbrock = Target target Nothing transition :: Transition IO (Chain [Double] b) transition = concatT (sampleT (metropolis 0.5) (metropolis 1.0)) (sampleT (slice 2.0) (slice 3.0)) main :: IO () main = withSystemRandom . asGenIO $ mcmc 10000 [0, 0] transition rosenbrock

See the attached test suite for other examples.

- concatT :: Monad m => Transition m a -> Transition m a -> Transition m a
- concatAllT :: Monad m => [Transition m a] -> Transition m a
- sampleT :: PrimMonad m => Transition m a -> Transition m a -> Transition m a
- sampleAllT :: PrimMonad m => [Transition m a] -> Transition m a
- bernoulliT :: PrimMonad m => Double -> Transition m a -> Transition m a -> Transition m a
- frequency :: PrimMonad m => [(Int, Transition m a)] -> Transition m a
- anneal :: (Monad m, Functor f) => Double -> Transition m (Chain (f Double) b) -> Transition m (Chain (f Double) b)
- mcmc :: (MonadIO m, PrimMonad m, Show (t a)) => Int -> t a -> Transition m (Chain (t a) b) -> Target (t a) -> Gen (PrimState m) -> m ()
- chain :: (MonadIO m, PrimMonad m) => Int -> t a -> Transition m (Chain (t a) b) -> Target (t a) -> Gen (PrimState m) -> m [Chain (t a) b]
- module Data.Sampling.Types
- metropolis :: (Traversable f, PrimMonad m) => Double -> Transition m (Chain (f Double) b)
- hamiltonian :: (Num (IxValue (t Double)), Traversable t, FunctorWithIndex (Index (t Double)) t, Ixed (t Double), PrimMonad m, (~) * (IxValue (t Double)) Double) => Double -> Int -> Transition m (Chain (t Double) b)
- slice :: (PrimMonad m, FoldableWithIndex (Index (t a)) t, Ixed (t a), Num (IxValue (t a)), Variate (IxValue (t a))) => IxValue (t a) -> Transition m (Chain (t a) b)
- create :: PrimMonad m => m (Gen (PrimState m))
- createSystemRandom :: IO GenIO
- withSystemRandom :: PrimBase m => (Gen (PrimState m) -> m a) -> IO a
- asGenIO :: (GenIO -> IO a) -> GenIO -> IO a
- class Monad m => PrimMonad (m :: * -> *) where
- type family PrimState (m :: * -> *) :: *
- data RealWorld :: *

# Documentation

concatT :: Monad m => Transition m a -> Transition m a -> Transition m a Source #

Deterministically concat transition operators together.

concatAllT :: Monad m => [Transition m a] -> Transition m a Source #

Deterministically concat a list of transition operators together.

sampleT :: PrimMonad m => Transition m a -> Transition m a -> Transition m a Source #

Probabilistically concat transition operators together.

sampleAllT :: PrimMonad m => [Transition m a] -> Transition m a Source #

Probabilistically concat transition operators together via a uniform distribution.

bernoulliT :: PrimMonad m => Double -> Transition m a -> Transition m a -> Transition m a Source #

Probabilistically concat transition operators together using a Bernoulli distribution with the supplied success probability.

This is just a generalization of sampleT.

frequency :: PrimMonad m => [(Int, Transition m a)] -> Transition m a Source #

Probabilistically concat transition operators together using the supplied frequency distribution.

This function is more-or-less an exact copy of `frequency`

,
except here applied to transition operators.

anneal :: (Monad m, Functor f) => Double -> Transition m (Chain (f Double) b) -> Transition m (Chain (f Double) b) Source #

An annealing transformer.

When executed, the supplied transition operator will execute over the parameter space annealed to the supplied inverse temperature.

let annealedTransition = anneal 0.30 (slice 0.5)

mcmc :: (MonadIO m, PrimMonad m, Show (t a)) => Int -> t a -> Transition m (Chain (t a) b) -> Target (t a) -> Gen (PrimState m) -> m () Source #

Trace `n`

iterations of a Markov chain and stream them to stdout.

`>>>`

-0.48939312153007863,0.13290702689491818 1.4541485365128892e-2,-0.4859905564050404 0.22487398491619448,-0.29769783186855125`withSystemRandom . asGenIO $ mcmc 3 [0, 0] (metropolis 0.5) rosenbrock`

chain :: (MonadIO m, PrimMonad m) => Int -> t a -> Transition m (Chain (t a) b) -> Target (t a) -> Gen (PrimState m) -> m [Chain (t a) b] Source #

Trace `n`

iterations of a Markov chain and collect them in a list.

`>>>`

`results <- withSystemRandom . asGenIO $ chain 3 [0, 0] (metropolis 0.5) rosenbrock`

# Re-exported

module Data.Sampling.Types

metropolis :: (Traversable f, PrimMonad m) => Double -> Transition m (Chain (f Double) b) #

A generic Metropolis transition operator.

hamiltonian :: (Num (IxValue (t Double)), Traversable t, FunctorWithIndex (Index (t Double)) t, Ixed (t Double), PrimMonad m, (~) * (IxValue (t Double)) Double) => Double -> Int -> Transition m (Chain (t Double) b) #

A Hamiltonian transition operator.

slice :: (PrimMonad m, FoldableWithIndex (Index (t a)) t, Ixed (t a), Num (IxValue (t a)), Variate (IxValue (t a))) => IxValue (t a) -> Transition m (Chain (t a) b) #

A slice sampling transition operator.

createSystemRandom :: IO GenIO #

Seed a PRNG with data from the system's fast source of pseudo-random
numbers. All the caveats of `withSystemRandom`

apply here as well.

withSystemRandom :: PrimBase m => (Gen (PrimState m) -> m a) -> IO a #

Seed a PRNG with data from the system's fast source of
pseudo-random numbers ("`/dev/urandom`

" on Unix-like systems or
`RtlGenRandom`

on Windows), then run the given action.

This is a somewhat expensive function, and is intended to be called
only occasionally (e.g. once per thread). You should use the `Gen`

it creates to generate many random numbers.

asGenIO :: (GenIO -> IO a) -> GenIO -> IO a #

Constrain the type of an action to run in the `IO`

monad.

class Monad m => PrimMonad (m :: * -> *) #

Class of monads which can perform primitive state-transformer actions

PrimMonad IO | |

PrimMonad (ST s) | |

PrimMonad m => PrimMonad (MaybeT m) | |

PrimMonad m => PrimMonad (Prob m) | |

PrimMonad m => PrimMonad (ListT m) | |

PrimMonad m => PrimMonad (IdentityT * m) | |

PrimMonad m => PrimMonad (ExceptT e m) | |

(Error e, PrimMonad m) => PrimMonad (ErrorT e m) | |

PrimMonad m => PrimMonad (StateT s m) | |

PrimMonad m => PrimMonad (StateT s m) | |

(Monoid w, PrimMonad m) => PrimMonad (WriterT w m) | |

(Monoid w, PrimMonad m) => PrimMonad (WriterT w m) | |

PrimMonad m => PrimMonad (ReaderT * r m) | |

(Monoid w, PrimMonad m) => PrimMonad (RWST r w s m) | |

(Monoid w, PrimMonad m) => PrimMonad (RWST r w s m) | |

type family PrimState (m :: * -> *) :: * #

State token type

type PrimState IO | |

type PrimState (ST s) | |

type PrimState (MaybeT m) | |

type PrimState (Prob m) | |

type PrimState (ListT m) | |

type PrimState (IdentityT * m) | |

type PrimState (ExceptT e m) | |

type PrimState (ErrorT e m) | |

type PrimState (StateT s m) | |

type PrimState (StateT s m) | |

type PrimState (WriterT w m) | |

type PrimState (WriterT w m) | |

type PrimState (ReaderT * r m) | |

type PrimState (RWST r w s m) | |

type PrimState (RWST r w s m) | |