{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE OverloadedStrings #-}
module Mcmc.Algorithm.Metropolis
( MHG (..),
mhg,
mhgSave,
mhgLoad,
mhgAccept,
)
where
import Codec.Compression.GZip
import Control.Monad
import Control.Monad.IO.Class
import Data.Aeson
import qualified Data.ByteString.Lazy.Char8 as BL
import Data.Time
import Mcmc.Algorithm
import Mcmc.Chain.Chain
import Mcmc.Chain.Link
import Mcmc.Chain.Save
import Mcmc.Chain.Trace
import Mcmc.Monitor
import Mcmc.Proposal
import Mcmc.Settings
import Numeric.Log
import System.Random.MWC
import Text.Printf
import Prelude hiding (cycle)
newtype MHG a = MHG {MHG a -> Chain a
fromMHG :: Chain a}
instance ToJSON a => Algorithm (MHG a) where
aName :: MHG a -> String
aName = String -> MHG a -> String
forall a b. a -> b -> a
const String
"Metropolis-Hastings-Green (MHG)"
aIteration :: MHG a -> Int
aIteration = Chain a -> Int
forall a. Chain a -> Int
iteration (Chain a -> Int) -> (MHG a -> Chain a) -> MHG a -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. MHG a -> Chain a
forall a. MHG a -> Chain a
fromMHG
aIterate :: ParallelizationMode -> MHG a -> IO (MHG a)
aIterate = ParallelizationMode -> MHG a -> IO (MHG a)
forall a. ParallelizationMode -> MHG a -> IO (MHG a)
mhgIterate
aAutoTune :: MHG a -> MHG a
aAutoTune = MHG a -> MHG a
forall a. MHG a -> MHG a
mhgAutoTune
aResetAcceptance :: MHG a -> MHG a
aResetAcceptance = MHG a -> MHG a
forall a. MHG a -> MHG a
mhgResetAcceptance
aSummarizeCycle :: MHG a -> ByteString
aSummarizeCycle = MHG a -> ByteString
forall a. MHG a -> ByteString
mhgSummarizeCycle
aOpenMonitors :: AnalysisName -> ExecutionMode -> MHG a -> IO (MHG a)
aOpenMonitors = AnalysisName -> ExecutionMode -> MHG a -> IO (MHG a)
forall a. AnalysisName -> ExecutionMode -> MHG a -> IO (MHG a)
mhgOpenMonitors
aExecuteMonitors :: Verbosity -> UTCTime -> Int -> MHG a -> IO (Maybe ByteString)
aExecuteMonitors = Verbosity -> UTCTime -> Int -> MHG a -> IO (Maybe ByteString)
forall a.
Verbosity -> UTCTime -> Int -> MHG a -> IO (Maybe ByteString)
mhgExecuteMonitors
aStdMonitorHeader :: MHG a -> ByteString
aStdMonitorHeader = MHG a -> ByteString
forall a. MHG a -> ByteString
mhgStdMonitorHeader
aCloseMonitors :: MHG a -> IO (MHG a)
aCloseMonitors = MHG a -> IO (MHG a)
forall a. MHG a -> IO (MHG a)
mhgCloseMonitors
aSave :: AnalysisName -> MHG a -> IO ()
aSave = AnalysisName -> MHG a -> IO ()
forall a. ToJSON a => AnalysisName -> MHG a -> IO ()
mhgSave
mhg ::
PriorFunction a ->
LikelihoodFunction a ->
Cycle a ->
Monitor a ->
a ->
GenIO ->
IO (MHG a)
mhg :: PriorFunction a
-> PriorFunction a
-> Cycle a
-> Monitor a
-> a
-> GenIO
-> IO (MHG a)
mhg PriorFunction a
pr PriorFunction a
lh Cycle a
cc Monitor a
mn a
i0 GenIO
g = do
Trace a
tr <- Int -> Link a -> IO (Trace a)
forall a. Int -> Link a -> IO (Trace a)
replicateT Int
traceLength Link a
l0
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG (Chain a -> MHG a) -> Chain a -> MHG a
forall a b. (a -> b) -> a -> b
$ Int
-> Link a
-> Int
-> Trace a
-> Acceptance (Proposal a)
-> GenIO
-> Int
-> PriorFunction a
-> PriorFunction a
-> Cycle a
-> Monitor a
-> Chain a
forall a.
Int
-> Link a
-> Int
-> Trace a
-> Acceptance (Proposal a)
-> GenIO
-> Int
-> PriorFunction a
-> PriorFunction a
-> Cycle a
-> Monitor a
-> Chain a
Chain Int
0 Link a
l0 Int
0 Trace a
tr Acceptance (Proposal a)
ac GenIO
g Int
0 PriorFunction a
pr PriorFunction a
lh Cycle a
cc Monitor a
mn
where
l0 :: Link a
l0 = a -> Log Double -> Log Double -> Link a
forall a. a -> Log Double -> Log Double -> Link a
Link a
i0 (PriorFunction a
pr a
i0) (PriorFunction a
lh a
i0)
ac :: Acceptance (Proposal a)
ac = [Proposal a] -> Acceptance (Proposal a)
forall k. Ord k => [k] -> Acceptance k
emptyA ([Proposal a] -> Acceptance (Proposal a))
-> [Proposal a] -> Acceptance (Proposal a)
forall a b. (a -> b) -> a -> b
$ Cycle a -> [Proposal a]
forall a. Cycle a -> [Proposal a]
ccProposals Cycle a
cc
batchMonitorSizes :: [Int]
batchMonitorSizes = (MonitorBatch a -> Int) -> [MonitorBatch a] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map MonitorBatch a -> Int
forall a. MonitorBatch a -> Int
getMonitorBatchSize ([MonitorBatch a] -> [Int]) -> [MonitorBatch a] -> [Int]
forall a b. (a -> b) -> a -> b
$ Monitor a -> [MonitorBatch a]
forall a. Monitor a -> [MonitorBatch a]
mBatches Monitor a
mn
traceLength :: Int
traceLength = [Int] -> Int
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum ([Int] -> Int) -> [Int] -> Int
forall a b. (a -> b) -> a -> b
$ Int
1 Int -> [Int] -> [Int]
forall a. a -> [a] -> [a]
: [Int]
batchMonitorSizes
mhgFn :: AnalysisName -> FilePath
mhgFn :: AnalysisName -> String
mhgFn (AnalysisName String
nm) = String
nm String -> String -> String
forall a. [a] -> [a] -> [a]
++ String
".mhg"
mhgSave ::
ToJSON a =>
AnalysisName ->
MHG a ->
IO ()
mhgSave :: AnalysisName -> MHG a -> IO ()
mhgSave AnalysisName
nm (MHG Chain a
c) = do
SavedChain a
savedChain <- Chain a -> IO (SavedChain a)
forall a. Chain a -> IO (SavedChain a)
toSavedChain Chain a
c
String -> ByteString -> IO ()
BL.writeFile (AnalysisName -> String
mhgFn AnalysisName
nm) (ByteString -> IO ()) -> ByteString -> IO ()
forall a b. (a -> b) -> a -> b
$ ByteString -> ByteString
compress (ByteString -> ByteString) -> ByteString -> ByteString
forall a b. (a -> b) -> a -> b
$ SavedChain a -> ByteString
forall a. ToJSON a => a -> ByteString
encode SavedChain a
savedChain
mhgLoad ::
FromJSON a =>
PriorFunction a ->
LikelihoodFunction a ->
Cycle a ->
Monitor a ->
AnalysisName ->
IO (MHG a)
mhgLoad :: PriorFunction a
-> PriorFunction a
-> Cycle a
-> Monitor a
-> AnalysisName
-> IO (MHG a)
mhgLoad PriorFunction a
pr PriorFunction a
lh Cycle a
cc Monitor a
mn AnalysisName
nm = do
Either String (SavedChain a)
savedChain <- ByteString -> Either String (SavedChain a)
forall a. FromJSON a => ByteString -> Either String a
eitherDecode (ByteString -> Either String (SavedChain a))
-> (ByteString -> ByteString)
-> ByteString
-> Either String (SavedChain a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ByteString -> ByteString
decompress (ByteString -> Either String (SavedChain a))
-> IO ByteString -> IO (Either String (SavedChain a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> String -> IO ByteString
BL.readFile (AnalysisName -> String
mhgFn AnalysisName
nm)
Chain a
chain <- (String -> IO (Chain a))
-> (SavedChain a -> IO (Chain a))
-> Either String (SavedChain a)
-> IO (Chain a)
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either String -> IO (Chain a)
forall a. HasCallStack => String -> a
error (PriorFunction a
-> PriorFunction a
-> Cycle a
-> Monitor a
-> SavedChain a
-> IO (Chain a)
forall a.
PriorFunction a
-> PriorFunction a
-> Cycle a
-> Monitor a
-> SavedChain a
-> IO (Chain a)
fromSavedChain PriorFunction a
pr PriorFunction a
lh Cycle a
cc Monitor a
mn) Either String (SavedChain a)
savedChain
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG Chain a
chain
mhgRatio :: Log Double -> Log Double -> Log Double -> Log Double -> Log Double
mhgRatio :: Log Double -> Log Double -> Log Double -> Log Double -> Log Double
mhgRatio Log Double
fX Log Double
fY Log Double
q Log Double
j = Log Double
fY Log Double -> Log Double -> Log Double
forall a. Fractional a => a -> a -> a
/ Log Double
fX Log Double -> Log Double -> Log Double
forall a. Num a => a -> a -> a
* Log Double
q Log Double -> Log Double -> Log Double
forall a. Num a => a -> a -> a
* Log Double
j
{-# INLINE mhgRatio #-}
mhgAccept :: Log Double -> GenIO -> IO Bool
mhgAccept :: Log Double -> GenIO -> IO Bool
mhgAccept Log Double
r GenIO
g
| Log Double -> Double
forall a. Log a -> a
ln Log Double
r Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
>= Double
0.0 = Bool -> IO Bool
forall (m :: * -> *) a. Monad m => a -> m a
return Bool
True
| Bool
otherwise = do
Double
b <- GenIO -> IO Double
forall a (m :: * -> *).
(Variate a, PrimMonad m) =>
Gen (PrimState m) -> m a
uniform GenIO
g
Bool -> IO Bool
forall (m :: * -> *) a. Monad m => a -> m a
return (Bool -> IO Bool) -> Bool -> IO Bool
forall a b. (a -> b) -> a -> b
$ Double
b Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
< Double -> Double
forall a. Floating a => a -> a
exp (Log Double -> Double
forall a. Log a -> a
ln Log Double
r)
mhgPropose :: MHG a -> Proposal a -> IO (MHG a)
mhgPropose :: MHG a -> Proposal a -> IO (MHG a)
mhgPropose (MHG Chain a
c) Proposal a
p = do
(!a
y, !Log Double
q, !Log Double
j) <- IO (a, Log Double, Log Double) -> IO (a, Log Double, Log Double)
forall (m :: * -> *) a. MonadIO m => IO a -> m a
liftIO (IO (a, Log Double, Log Double) -> IO (a, Log Double, Log Double))
-> IO (a, Log Double, Log Double) -> IO (a, Log Double, Log Double)
forall a b. (a -> b) -> a -> b
$ a -> Gen RealWorld -> IO (a, Log Double, Log Double)
s a
x Gen RealWorld
g
let !pY :: Log Double
pY = PriorFunction a
pF a
y
!lY :: Log Double
lY = PriorFunction a
lF a
y
!r :: Log Double
r = Log Double -> Log Double -> Log Double -> Log Double -> Log Double
mhgRatio (Log Double
pX Log Double -> Log Double -> Log Double
forall a. Num a => a -> a -> a
* Log Double
lX) (Log Double
pY Log Double -> Log Double -> Log Double
forall a. Num a => a -> a -> a
* Log Double
lY) Log Double
q Log Double
j
Bool
accept <- Log Double -> GenIO -> IO Bool
mhgAccept Log Double
r Gen RealWorld
GenIO
g
if Bool
accept
then do
let !ac' :: Acceptance (Proposal a)
ac' = Proposal a
-> Bool -> Acceptance (Proposal a) -> Acceptance (Proposal a)
forall k. Ord k => k -> Bool -> Acceptance k -> Acceptance k
pushA Proposal a
p Bool
True Acceptance (Proposal a)
ac
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG (Chain a -> MHG a) -> Chain a -> MHG a
forall a b. (a -> b) -> a -> b
$ Chain a
c {link :: Link a
link = a -> Log Double -> Log Double -> Link a
forall a. a -> Log Double -> Log Double -> Link a
Link a
y Log Double
pY Log Double
lY, acceptance :: Acceptance (Proposal a)
acceptance = Acceptance (Proposal a)
ac'}
else do
let !ac' :: Acceptance (Proposal a)
ac' = Proposal a
-> Bool -> Acceptance (Proposal a) -> Acceptance (Proposal a)
forall k. Ord k => k -> Bool -> Acceptance k -> Acceptance k
pushA Proposal a
p Bool
False Acceptance (Proposal a)
ac
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG (Chain a -> MHG a) -> Chain a -> MHG a
forall a b. (a -> b) -> a -> b
$ Chain a
c {acceptance :: Acceptance (Proposal a)
acceptance = Proposal a
-> Bool -> Acceptance (Proposal a) -> Acceptance (Proposal a)
forall k. Ord k => k -> Bool -> Acceptance k -> Acceptance k
pushA Proposal a
p Bool
False Acceptance (Proposal a)
ac'}
where
s :: ProposalSimple a
s = Proposal a -> ProposalSimple a
forall a. Proposal a -> ProposalSimple a
pSimple Proposal a
p
(Link a
x Log Double
pX Log Double
lX) = Chain a -> Link a
forall a. Chain a -> Link a
link Chain a
c
pF :: PriorFunction a
pF = Chain a -> PriorFunction a
forall a. Chain a -> PriorFunction a
priorFunction Chain a
c
lF :: PriorFunction a
lF = Chain a -> PriorFunction a
forall a. Chain a -> PriorFunction a
likelihoodFunction Chain a
c
ac :: Acceptance (Proposal a)
ac = Chain a -> Acceptance (Proposal a)
forall a. Chain a -> Acceptance (Proposal a)
acceptance Chain a
c
g :: GenIO
g = Chain a -> GenIO
forall a. Chain a -> GenIO
generator Chain a
c
mhgPush :: MHG a -> IO (MHG a)
mhgPush :: MHG a -> IO (MHG a)
mhgPush (MHG Chain a
c) = do
Trace a
t' <- Link a -> Trace a -> IO (Trace a)
forall a. Link a -> Trace a -> IO (Trace a)
pushT Link a
i Trace a
t
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG Chain a
c {trace :: Trace a
trace = Trace a
t', iteration :: Int
iteration = Int -> Int
forall a. Enum a => a -> a
succ Int
n}
where
i :: Link a
i = Chain a -> Link a
forall a. Chain a -> Link a
link Chain a
c
t :: Trace a
t = Chain a -> Trace a
forall a. Chain a -> Trace a
trace Chain a
c
n :: Int
n = Chain a -> Int
forall a. Chain a -> Int
iteration Chain a
c
mhgIterate :: ParallelizationMode -> MHG a -> IO (MHG a)
mhgIterate :: ParallelizationMode -> MHG a -> IO (MHG a)
mhgIterate ParallelizationMode
_ MHG a
a = do
[Proposal a]
ps <- Cycle a -> GenIO -> IO [Proposal a]
forall a. Cycle a -> GenIO -> IO [Proposal a]
orderProposals Cycle a
cc Gen RealWorld
GenIO
g
MHG a
a' <- (MHG a -> Proposal a -> IO (MHG a))
-> MHG a -> [Proposal a] -> IO (MHG a)
forall (t :: * -> *) (m :: * -> *) b a.
(Foldable t, Monad m) =>
(b -> a -> m b) -> b -> t a -> m b
foldM MHG a -> Proposal a -> IO (MHG a)
forall a. MHG a -> Proposal a -> IO (MHG a)
mhgPropose MHG a
a [Proposal a]
ps
MHG a -> IO (MHG a)
forall a. MHG a -> IO (MHG a)
mhgPush MHG a
a'
where
c :: Chain a
c = MHG a -> Chain a
forall a. MHG a -> Chain a
fromMHG MHG a
a
cc :: Cycle a
cc = Chain a -> Cycle a
forall a. Chain a -> Cycle a
cycle Chain a
c
g :: GenIO
g = Chain a -> GenIO
forall a. Chain a -> GenIO
generator Chain a
c
mhgAutoTune :: MHG a -> MHG a
mhgAutoTune :: MHG a -> MHG a
mhgAutoTune (MHG Chain a
c) = Chain a -> MHG a
forall a. Chain a -> MHG a
MHG (Chain a -> MHG a) -> Chain a -> MHG a
forall a b. (a -> b) -> a -> b
$ Chain a
c {cycle :: Cycle a
cycle = Acceptance (Proposal a) -> Cycle a -> Cycle a
forall a. Acceptance (Proposal a) -> Cycle a -> Cycle a
autoTuneCycle Acceptance (Proposal a)
ac Cycle a
cc}
where
ac :: Acceptance (Proposal a)
ac = Chain a -> Acceptance (Proposal a)
forall a. Chain a -> Acceptance (Proposal a)
acceptance Chain a
c
cc :: Cycle a
cc = Chain a -> Cycle a
forall a. Chain a -> Cycle a
cycle Chain a
c
mhgResetAcceptance :: MHG a -> MHG a
mhgResetAcceptance :: MHG a -> MHG a
mhgResetAcceptance (MHG Chain a
c) = Chain a -> MHG a
forall a. Chain a -> MHG a
MHG (Chain a -> MHG a) -> Chain a -> MHG a
forall a b. (a -> b) -> a -> b
$ Chain a
c {acceptance :: Acceptance (Proposal a)
acceptance = Acceptance (Proposal a) -> Acceptance (Proposal a)
forall k. Ord k => Acceptance k -> Acceptance k
resetA Acceptance (Proposal a)
ac}
where
ac :: Acceptance (Proposal a)
ac = Chain a -> Acceptance (Proposal a)
forall a. Chain a -> Acceptance (Proposal a)
acceptance Chain a
c
mhgSummarizeCycle :: MHG a -> BL.ByteString
mhgSummarizeCycle :: MHG a -> ByteString
mhgSummarizeCycle (MHG Chain a
c) = Acceptance (Proposal a) -> Cycle a -> ByteString
forall a. Acceptance (Proposal a) -> Cycle a -> ByteString
summarizeCycle Acceptance (Proposal a)
ac Cycle a
cc
where
cc :: Cycle a
cc = Chain a -> Cycle a
forall a. Chain a -> Cycle a
cycle Chain a
c
ac :: Acceptance (Proposal a)
ac = Chain a -> Acceptance (Proposal a)
forall a. Chain a -> Acceptance (Proposal a)
acceptance Chain a
c
mhgOpenMonitors :: AnalysisName -> ExecutionMode -> MHG a -> IO (MHG a)
mhgOpenMonitors :: AnalysisName -> ExecutionMode -> MHG a -> IO (MHG a)
mhgOpenMonitors AnalysisName
nm ExecutionMode
em (MHG Chain a
c) = do
Monitor a
m' <- String -> String -> ExecutionMode -> Monitor a -> IO (Monitor a)
forall a.
String -> String -> ExecutionMode -> Monitor a -> IO (Monitor a)
mOpen String
pre String
suf ExecutionMode
em Monitor a
m
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG Chain a
c {monitor :: Monitor a
monitor = Monitor a
m'}
where
m :: Monitor a
m = Chain a -> Monitor a
forall a. Chain a -> Monitor a
monitor Chain a
c
pre :: String
pre = AnalysisName -> String
fromAnalysisName AnalysisName
nm
suf :: String
suf = String -> Int -> String
forall r. PrintfType r => String -> r
printf String
"%02d" (Int -> String) -> Int -> String
forall a b. (a -> b) -> a -> b
$ Chain a -> Int
forall a. Chain a -> Int
chainId Chain a
c
mhgExecuteMonitors ::
Verbosity ->
UTCTime ->
Int ->
MHG a ->
IO (Maybe BL.ByteString)
mhgExecuteMonitors :: Verbosity -> UTCTime -> Int -> MHG a -> IO (Maybe ByteString)
mhgExecuteMonitors Verbosity
vb UTCTime
t0 Int
iTotal (MHG Chain a
c) = Verbosity
-> Int
-> Int
-> UTCTime
-> Trace a
-> Int
-> Monitor a
-> IO (Maybe ByteString)
forall a.
Verbosity
-> Int
-> Int
-> UTCTime
-> Trace a
-> Int
-> Monitor a
-> IO (Maybe ByteString)
mExec Verbosity
vb Int
i Int
i0 UTCTime
t0 Trace a
tr Int
iTotal Monitor a
m
where
i :: Int
i = Chain a -> Int
forall a. Chain a -> Int
iteration Chain a
c
i0 :: Int
i0 = Chain a -> Int
forall a. Chain a -> Int
start Chain a
c
tr :: Trace a
tr = Chain a -> Trace a
forall a. Chain a -> Trace a
trace Chain a
c
m :: Monitor a
m = Chain a -> Monitor a
forall a. Chain a -> Monitor a
monitor Chain a
c
mhgStdMonitorHeader :: MHG a -> BL.ByteString
(MHG Chain a
c) = MonitorStdOut a -> ByteString
forall a. MonitorStdOut a -> ByteString
msHeader (Monitor a -> MonitorStdOut a
forall a. Monitor a -> MonitorStdOut a
mStdOut (Monitor a -> MonitorStdOut a) -> Monitor a -> MonitorStdOut a
forall a b. (a -> b) -> a -> b
$ Chain a -> Monitor a
forall a. Chain a -> Monitor a
monitor Chain a
c)
mhgCloseMonitors :: MHG a -> IO (MHG a)
mhgCloseMonitors :: MHG a -> IO (MHG a)
mhgCloseMonitors (MHG Chain a
c) = do
Monitor a
m' <- Monitor a -> IO (Monitor a)
forall a. Monitor a -> IO (Monitor a)
mClose Monitor a
m
MHG a -> IO (MHG a)
forall (m :: * -> *) a. Monad m => a -> m a
return (MHG a -> IO (MHG a)) -> MHG a -> IO (MHG a)
forall a b. (a -> b) -> a -> b
$ Chain a -> MHG a
forall a. Chain a -> MHG a
MHG (Chain a -> MHG a) -> Chain a -> MHG a
forall a b. (a -> b) -> a -> b
$ Chain a
c {monitor :: Monitor a
monitor = Monitor a
m'}
where
m :: Monitor a
m = Chain a -> Monitor a
forall a. Chain a -> Monitor a
monitor Chain a
c