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199 lines
7.5 KiB
199 lines
7.5 KiB
// <copyright file="MCMC.fsx" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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// http://mathnetnumerics.codeplex.com
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//
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// Copyright (c) 2009-2013 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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#r "../../out/lib/Net40/MathNet.Numerics.dll"
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#r "../../out/lib/Net40/MathNet.Numerics.FSharp.dll"
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open MathNet.Numerics
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open MathNet.Numerics.Random
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open MathNet.Numerics.Statistics
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open MathNet.Numerics.Distributions
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open MathNet.Numerics.Statistics.Mcmc
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/// The number of samples to gather for each sampler.
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let N = 10000
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/// The random number generator we use for the examples.
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let rnd = new MersenneTwister()
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//
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// Example 1: Sampling a Beta distributed variable through rejection sampling.
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//
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// Target Distribution: Beta(2.7, 6.3)
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//
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// -----------------------------------------------------------------------------
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do
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printfn "Rejection Sampling Example"
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/// The target distribution.
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let beta = new Beta(2.7, 6.3)
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/// Samples uniform distributed variables.
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let uniform = new ContinuousUniform(0.0, 1.0, RandomSource = rnd)
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/// Implements the rejection sampling procedure.
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let rs = new RejectionSampler<float>( ( fun x -> x**(beta.A-1.0) * (1.0 - x)**(beta.B-1.0) ),
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( fun x -> 0.021 ),
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( fun () -> uniform.Sample()) )
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/// An array of samples from the rejection sampler.
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let arr = rs.Sample(N)
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/// The true distribution.
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printfn "\tEmpirical Mean = %f (should be %f)" (Statistics.Mean(arr)) beta.Mean
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printfn "\tEmpirical StdDev = %f (should be %f)" (Statistics.StandardDeviation(arr)) beta.StdDev
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printfn "\tAcceptance rate = %f" rs.AcceptanceRate
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printfn ""
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//
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// Example 2: Sampling a normal distributed variable through Metropolis sampling.
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//
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// Target Distribution: Normal(1.0, 3.5)
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//
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// -----------------------------------------------------------------------------
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do
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printfn "Metropolis Sampling Example"
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let mean, stddev = 1.0, 3.5
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let normal = new Normal(mean, stddev)
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/// Implements the rejection sampling procedure.
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let ms = new MetropolisSampler<float>( 0.1, (fun x -> log(normal.Density(x))),
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(fun x -> Normal.Sample(rnd, x, 0.3)), 20,
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RandomSource = rnd )
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/// An array of samples from the rejection sampler.
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let arr = ms.Sample(N)
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/// The true distribution.
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printfn "\tEmpirical Mean = %f (should be %f)" (Statistics.Mean(arr)) normal.Mean
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printfn "\tEmpirical StdDev = %f (should be %f)" (Statistics.StandardDeviation(arr)) normal.StdDev
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printfn "\tAcceptance rate = %f" ms.AcceptanceRate
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printfn ""
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//
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// Example 3: Sampling a normal distributed variable through Metropolis-Hastings sampling
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// with a symmetric proposal distribution.
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//
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// Target Distribution: Normal(1.0, 3.5)
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//
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// -----------------------------------------------------------------------------------------
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do
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printfn "Metropolis Hastings Sampling Example (Symmetric Proposal)"
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let mean, stddev = 1.0, 3.5
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let normal = new Normal(mean, stddev)
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/// Evaluates the log normal distribution.
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let npdf x m s = -0.5*(x-m)*(x-m)/(s*s) - 0.5 * log(Constants.Pi2 * s * s)
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/// Implements the rejection sampling procedure.
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let ms = new MetropolisHastingsSampler<float>( 0.1, (fun x -> log(normal.Density(x))),
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(fun x y -> npdf x y 0.3), (fun x -> Normal.Sample(rnd, x, 0.3)), 10,
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RandomSource = rnd )
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/// An array of samples from the rejection sampler.
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let arr = ms.Sample(N)
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/// The true distribution.
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printfn "\tEmpirical Mean = %f (should be %f)" (Statistics.Mean(arr)) normal.Mean
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printfn "\tEmpirical StdDev = %f (should be %f)" (Statistics.StandardDeviation(arr)) normal.StdDev
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printfn "\tAcceptance rate = %f" ms.AcceptanceRate
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printfn ""
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//
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// Example 4: Sampling a normal distributed variable through Metropolis-Hastings sampling
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// with a asymmetric proposal distribution.
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//
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// Target Distribution: Normal(1.0, 3.5)
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//
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// -----------------------------------------------------------------------------------------
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do
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printfn "Metropolis Hastings Sampling Example (Assymetric Proposal)"
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let mean, stddev = 1.0, 3.5
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let normal = new Normal(mean, stddev)
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/// Evaluates the logarithm of the normal distribution function.
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let npdf x m s = -0.5*(x-m)*(x-m)/(s*s) - 0.5 * log(Constants.Pi2 * s * s)
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/// Samples from a mixture that is biased towards samples larger than x.
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let mixSample x =
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if Bernoulli.Sample(rnd, 0.5) = 1 then
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Normal.Sample(rnd, x, 0.3)
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else
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Normal.Sample(rnd, x + 0.1, 0.3)
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/// The transition kernel for the proposal above.
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let krnl xnew x = log (0.5 * exp(npdf xnew x 0.3) + 0.5 * exp(npdf xnew (x+0.1) 0.3))
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/// Implements the rejection sampling procedure.
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let ms = new MetropolisHastingsSampler<float>( 0.1, (fun x -> log(normal.Density(x))),
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(fun xnew x -> krnl xnew x), (fun x -> mixSample x), 10,
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RandomSource = rnd )
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/// An array of samples from the rejection sampler.
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let arr = ms.Sample(N)
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/// The true distribution.
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printfn "\tEmpirical Mean = %f (should be %f)" (Statistics.Mean(arr)) normal.Mean
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printfn "\tEmpirical StdDev = %f (should be %f)" (Statistics.StandardDeviation(arr)) normal.StdDev
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printfn "\tAcceptance rate = %f" ms.AcceptanceRate
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printfn ""
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//
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// Example 5: Slice sampling a normal distributed random variable.
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//
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// Target Distribution: Normal(1.0, 3.5)
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//
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// -----------------------------------------------------------------------------------------
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do
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printfn "Slice Sampling Example"
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let mean, stddev = 1.0, 3.5
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let normal = new Normal(mean, stddev)
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/// Evaluates the unnormalized logarithm of the normal distribution function.
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let npdf x m s = -0.5*(x-m)*(x-m)/(s*s)
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/// Implements the rejection sampling procedure.
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let ms = new UnivariateSliceSampler( 0.1, (fun x -> npdf x mean stddev), 5, 1.0, RandomSource = rnd )
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/// An array of samples from the rejection sampler.
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let arr = ms.Sample(N)
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/// The true distribution.
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printfn "\tEmpirical Mean = %f (should be %f)" (Statistics.Mean(arr)) normal.Mean
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printfn "\tEmpirical StdDev = %f (should be %f)" (Statistics.StandardDeviation(arr)) normal.StdDev
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printfn ""
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