DIMESampler.jl

Differential-Independence Mixture Ensemble ("DIME") MCMC sampling for Julia

This is a standalone Julia implementation of the DIME sampler proposed in DIME MCMC: A Swiss Army Knife for Bayesian Inference (Gregor Boehl, 2022, SSRN No. 4250395).

The sampler has a series of advantages over conventional samplers:

1. DIME MCMC is a (very fast) gradient-free global multi-start optimizer and, at the same time, a MCMC sampler that converges to the posterior distribution. This makes any posterior mode density maximization prior to MCMC sampling superfluous.
2. The DIME sampler is pretty robust for odd shaped, multimodal distributions.
3. DIME MCMC is parallelizable: many chains can run in parallel, and the necessary number of draws decreases almost one-to-one with the number of chains.
4. DIME proposals are generated from an endogenous and adaptive proposal distribution, thereby providing close-to-optimal proposal distributions for black box target distributions without the need for manual fine-tuning.

There is a nice set of slides on my website which explains the DIME principle.

Figure: A trimodal example distribution in 35 dimensions

Installation

Just get the package from the official Julia registry:

using Pkg; Pkg.add("DIMESampler")

The package should work with Julia versions starting from v1.6. There exist complementary implementations for Python and for matlab.

Usage

The core functionality is included in the function RunDIME:

# import package
using DIMESampler

# define your density function
function LogProb(x):
    ...
    return lprob
end

# define the initial ensemble
initchain = ...

# define the number of iterations to run
niter = ...

# off you go sampling
chains, lprobs, propdist = RunDIME(LogProb, initchain, niter)
...

The LogProb function returning the log-density must be vectorized, i.e. able to evaluate inputs with shape [ndim, :]. When using bounded priors it is recommended to use parameter transformations to maintain high acceptance rates.

The ensemble can be evaluated in parallel, which is one of the central advantages of ensemble MCMC. To have LogProb evaluate its vectorized input in parallel you can e.g. use pmap from Distributed

LogProbParallel(x) = pmap(LogProb, eachslice(x, dims=2))

and then pass this function to RunDIME instead.

Tutorial

Define a challenging example distribution with three separate modes (the distribution from the figure above):

# some imports
using DIMESampler, Distributions, Random, LinearAlgebra, Plots

# make it reproducible
Random.seed!(1)

# define distribution
m = 2
cov_scale = 0.05
weight = (0.33, 0.1)
ndim = 35

LogProb = CreateDIMETestFunc(ndim, weight, m, cov_scale)

LogProb will now return the log-PDF of a 35-dimensional Gaussian mixture.

Important: the function returning the log-density must be vectorized, i.e. able to evaluate inputs with shape [ndim, :]. If you want to make use of parallelization (which is one of the central advantages of ensemble MCMC), you may want to ensure that this function evaluates its vectorized input in parallel, i.e. using pmap from Distributed:

LogProbParallel(x) = pmap(LogProb, eachslice(x, dims=2))

For this example this is overkill since the overhead from parallelization is huge. Just using the vectorized LogProb is perfect.

Next, define the initial ensemble. In a Bayesian setup, a good initial ensemble would be a sample from the prior distribution. Here, we will go for a sample from a rather flat Gaussian distribution.

initvar = 2
nchain = ndim*5 # a sane default
initcov = I(ndim)*initvar
initmean = zeros(ndim)
initchain = rand(MvNormal(initmean, initcov), nchain)

Setting the number of parallel chains to 5*ndim is a sane default. For highly irregular distributions with several modes you should use more chains. Very simple distributions can go with less.

Now let the sampler run for 5000 iterations.

niter = 5000
chains, lprobs, propdist = RunDIME(LogProb, initchain, niter, progress=true, aimh_prob=0.1)

[ll/MAF:  12.187(4e+00)/19% | -5e-04] 100.0%┣███████████████████████████████┫ 5.0k/5.0k [00:15<00:00, 198it/s]

The setting of aimh_prob is the actual default value. For less complex distributions (e.g. distributions closer to Gaussian) a higher value can be chosen, which accelerates burn-in. The information in the progress bar has the structure [ll/MAF: <maximum log-prob>(<standard deviation of log-prob>)/<mean acceptance fraction> | <log state weight>]..., where <log state weight> is the current log-weight on the history of the proposal distribution. The closer this value is to zero (i.e. the actual weight to one), the less relevant are current ensembles for the estimated proposal distribution. It can hence be seen as a measure of convergence.

The following code creates the figure above, which is a plot of the marginal distribution along the first dimension (remember that this actually is a 35-dimensional distribution).

# analytical marginal distribution in first dimension
x = range(-4,4,1000)
mpdf = DIMETestFuncMarginalPDF(x, cov_scale, m, weight)

plot(x, mpdf, label="Target", lw=2, legend_position=:topleft)
plot!(x, pdf.(Normal(0, sqrt(initvar)), x), label="Initialization")
plot!(x, pdf.(TDist(10), (x .- propdist.μ[1])./sqrt(propdist.Σ[1,1]*10/8)), label="Final proposal")
# histogram of the actual sample
histogram!(chains[end-niter÷2:end,:,1][:], normalize=true, alpha=.5, label="Sample", color="black", bins=100)

To ensure proper mixing, let us also have a look at the MCMC traces, again focussing on the first dimension:

plot(chains[:,:,1], color="cyan4", alpha=.1, legend=false, size=(900,600))

Note how chains are also switching between the three modes because of the global proposal kernel.

While DIME is a MCMC sampler, it can straightforwardly be used as a global optimization routine. To this end, specify some broad starting region (in a non-Bayesian setup there is no prior) and let the sampler run for an extended number of iterations. Finally, assess whether the maximum value per ensemble did not change much in the last few hundred iterations. In a normal Bayesian setup, plotting the associated log-likelihood over time also helps to assess convergence to the posterior distribution.

plot(lprobs[:,:], color="orange4", alpha=.05, legend=false, size=(900,300))
plot!(maximum(lprobs)*ones(niter), color="blue3")

References

If you are using this software in your research, please cite

@techreport{boehl2022mcmc,
author={Gregor Boehl},
title={Ensemble MCMC Sampling for Robust Bayesian Inference},
journal={Available at SSRN 4250395},
year={2022}
}

Contributors

Many thanks go to DominikHe262!