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https://github.com/jgorham/SteinDiscrepancy.jl

Practical tools for quantifying how well a sample approximates a target distribution
https://github.com/jgorham/SteinDiscrepancy.jl

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Practical tools for quantifying how well a sample approximates a target distribution

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README 

        
# SteinDiscrepancy.jl



## What's a Stein discrepancy?



To improve the efficiency of Monte Carlo estimation, practitioners are

turning to biased Markov chain Monte Carlo procedures that trade off

asymptotic exactness for computational speed. The reasoning is sound: a

reduction in variance due to more rapid sampling can outweigh the bias

introduced. However, the inexactness creates new challenges for sampler and

parameter selection, since standard measures of sample quality like

effective sample size do not account for asymptotic bias. To address these

challenges, we introduced new computable quality measures, based on Stein's

method, that quantify the maximum discrepancy between sample and target

expectations over large classes of test functions. We call these measures

Stein discrepancies.



For a more detailed explanation, take a peek at the latest papers:



[Measuring Sample Quality with Diffusions](https://arxiv.org/abs/1611.06972),

[Measuring Sample Quality with Kernels](https://arxiv.org/abs/1703.01717).



These build on previous work from



[Measuring Sample Quality with Stein's Method](http://arxiv.org/abs/1506.03039)



and its companion paper



[Multivariate Stein Factors for a Class of Strongly Log-concave

Distributions](http://arxiv.org/abs/1512.07392).



These latter two papers are a more gentle introduction describing how the

Stein discrepancy bounds standard probability metrics like the

[Wasserstein distance](https://en.wikipedia.org/wiki/Wasserstein_metric).



Code built on `SteinDiscrepancy.jl` recreating all experiments in the above

papers can be found at the repo

[stein_discrepancy](https://github.com/jgorham/stein_discrepancy).

Please note that experiments in aforementioned repo all run on v0.0.2 of

this package with Julia v0.5.



## Where has it been used?



Since its introduction in [Measuring Sample Quality with Stein's

Method](http://arxiv.org/abs/1506.03039), the Stein discrepancy has been

incorporated into a variety of applications including:



1. Hypothesis testing

   * [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://arxiv.org/abs/1602.03253)

   * [A Kernel Test of Goodness of Fit](https://arxiv.org/abs/1602.02964)

2. Variational inference

   * [Operator Variational Inference](https://arxiv.org/abs/1610.09033)

   * [Two Methods For Wild Variational Inference](https://arxiv.org/abs/1612.00081)

   * [Approximate Inference with Amortised MCMC](https://arxiv.org/abs/1702.08343)

   * [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://arxiv.org/abs/1608.04471)

3. Importance sampling

   * [Black-box Importance Sampling](https://arxiv.org/abs/1610.05247)

   * [Stein Variational Adaptive Importance Sampling](https://arxiv.org/abs/1704.05201)

4. Training generative adversarial networks (GANs)

   * [Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning](https://arxiv.org/abs/1611.01722)

5. Training variational autoencoders (VAEs)

   * [Stein Variational Autoencoder](https://arxiv.org/abs/1704.05155)

6. Sample quality measurement

   * [Measuring Sample Quality with Stein's Method](http://arxiv.org/abs/1506.03039)

   * [Measuring Sample Quality with Diffusions](https://arxiv.org/abs/1611.06972)

   * [Measuring Sample Quality with Kernels](https://arxiv.org/abs/1703.01717)



## So how do I use it?



This software has been tested on Julia v0.6. This release implements two

classes of Stein discrepancies: graph Stein discrepancies and kernel Stein

discrepancies.



### Graph Stein discrepancies



Computing the graph Stein discrepancy requires solving a linear program

(LP), and thus you'll need some kind of LP solver installed to use this

software. We use JuMP ([Julia for Mathematical

Programming](https://jump.readthedocs.org/en/latest/)) to interface with

these solvers; any of the supported JuMP LP solvers with do just fine.



Once you have an LP solver installed, computing our measure is easy.

Below we'll first show how to compute the Langevin graph Stein discrepancy

for a univariate Gaussian target:



```julia

# import the gsd function (graph Stein discrepancy)

using SteinDiscrepancy: gsd

# define the grad log density of univariate Gaussian (we always expect vector inputs!)

function gradlogp(x::Array{Float64,1})

    -x

end

# generates 100 points from N(0,1)

X = randn(100)

# use a solver covered by JuMP

using Clp

solver = Clp.ClpSolver(LogLevel=4)

result = gsd(points=X, gradlogdensity=gradlogp, solver=solver)

graph_stein_discrepancy = result.objectivevalue[1]

```



Here's another example that will compute the Langevin graph Stein

discrepancy for a bivariate uniform sample (notice this target distribution

has a bounded support so these bounds become part of the input parameters):



```julia

# do the necessary imports

using SteinDiscrepancy: gsd

# define the grad log density of bivariate uniform target

function gradlogp(x::Array{Float64,1})

    zeros(size(x))

end

# generates 100 points from Unif([0,1]^2)

X = rand(100,2)

# use a solver covered by JuMP

using Clp

solver = Clp.ClpSolver(LogLevel=4)

result = gsd(points=X,

             gradlogdensity=gradlogp,

             solver=solver,

             supportlowerbounds=zeros(2),

             supportupperbounds=ones(2))

discrepancy = vec(result.objectivevalue)

```



The variable `discrepancy` here will encode the Stein discrepancy along each

dimension. The final discrepancy is just the sum of this vector.



### Kernel Stein discrepancies



Computing the kernel Stein discrepancies does not require a LP solver. In

lieu of a solver, you will need to specify a kernel function. Many common

kernels are already implemented in `src/kernels`, but if yours is not there

for some reason, feel free to inherit from the `SteinKernel` type and roll

your own.



With a kernel in hand, computing the kernel Stein discrepancy is easy:



```julia

# import the kernel stein discrepancy function and kernel to use

using SteinDiscrepancy: SteinInverseMultiquadricKernel, ksd

# define the grad log density of standard normal target

function gradlogp(x::Array{Float64,1})

    -x

end

# grab sample

X = randn(500, 3)

# create the kernel instance

kernel = SteinInverseMultiquadricKernel()

# compute the KSD2

result = ksd(points=X, gradlogdensity=gradlogp, kernel=kernel)

# get the final ksd

kernel_stein_discrepancy = sqrt(result.discrepancy2)

```



If your target has constrained support, you should simply use a kernel that 

respects these constraints (no `supportlowerbounds` and `supportupperbounds` 

arguments are needed). In the case you have a rectangular domain, the

`SteinRectangularDomainMetaKernel` should be useful. Here's an example

for a bivariate uniform target:



```julia

using SteinDiscrepancy:

    SteinRectangularDomainMetaKernel,

    SteinInverseMultiquadricKernel,

    ksd

# define the grad log density of standard normal target

function gradlogp(x::Array{Float64,1})

    zeros(length(d))

end

# grab sample

X = rand(500, 2)

# create the base kernel instance

imqkernel = SteinInverseMultiquadricKernel()

# clip the kernel to have the target distribution's domain

imqrectkernel = SteinRectangularDomainMetaKernel(imqkernel, [0., 0.], [1., 1.])

# compute the KSD2

result = ksd(points=X, gradlogdensity=gradlogp, kernel=imqrectkernel)

# get the final ksd

kernel_stein_discrepancy = sqrt(result.discrepancy2)

```



## Summary of the Code



All code is available in the src directory of the repo. Many examples for

computing the stein_discrepancy are in the test directory.



### Contents of src



* discrepancy - Code for computing Stein discrepancy

* kernels - Commonly used kernels



### Compiling Code in discrepancy/spanner directory



**Our C++ code should be compiled automatically when the package is

built**. However, if this doesn't work for some reason, you can issue the

following commands to compile the code in `src/discrepancy/spanner`:



```

cd /src/discrepancy/spanner

make

make clean

```



The last step isn't necessary, but it will remove some superfluous

files. If you want to kill everything made in the build process, just run



```

make distclean

```