# AdaptiveMCMC.jl

This package provides implementations of some general-purpose random-walk based adaptive MCMC algorithms, including the following:

- Adaptive Metropolis, proposal covariance adaptation, (Haario, Saksman and Tamminen, 2001, and Andrieu and Moulines, 2006)
- Adaptive scaling Metropolis, acceptance rate adaptation for scale (e.g. as in Andrieu and Thoms, 2008, and Atchadé and Fort, 2010)
- Robust Adaptive Metropolis, acceptance rate adaptation for shape (Vihola, 2012)
- Adaptive Parallel Tempering, acceptance rate adaptation for temperature levels (Miasojedow, Moulines and Vihola, 2013)

The aim of the package is to provide a simple and modular general-purpose implementation, which may be easily used to sample from a log-target density, but also used in a variety of custom settings.

See also AdaptiveParticleMCMC.jl which uses this package with SequentialMonteCarlo.jl for adaptive particle MCMC.

## Getting the package

```
using Pkg
Pkg.add("AdaptiveMCMC")
```

## Quick start

```
# Load the package
using AdaptiveMCMC
# Define a function which returns log-density values:
log_p(x) = -.5*sum(x.^2)
# Run 10k iterations of the Adaptive Metropolis:
out = adaptive_rwm(zeros(2), log_p, 10_000; algorithm=:am)
# Calculate '95% credible intervals':
using Statistics
mapslices(x->"$(mean(x)) ± $(1.96std(x))", out.X, dims=2)
```

The function `adaptive_rwm`

('Adaptive Random walk Metropolis') is a simple implenentation
which does sampling for a given log-target density with the chosen method.

## Adaptive parallel tempering

The `adaptive_rwm`

also implements tempering, which is used if an optional argument `L≥2`

(number of temperature levels) is supplied. Here is a simple multimodal distribution sampled with APT:

```
# Multimodal target of dimension d.
function multimodalTarget(d::Int, sigma2=0.1^2, sigman=sigma2)
# The means of mixtures
m = [2.18 5.76; 3.25 3.47; 5.41 2.65; 4.93 1.50; 8.67 9.59;
1.70 0.50; 2.70 7.88; 1.83 0.09; 4.24 8.48; 4.59 5.60;
4.98 3.70; 2.26 0.31; 8.41 1.68; 6.91 5.81; 1.14 2.39;
5.54 6.86; 3.93 8.82; 6.87 5.40; 8.33 9.50; 1.69 8.11]'
n_m = size(m,2)
@assert d>=2 "Dimension should be >= 2"
let m=m, n_m=size(m,2), d=d
function log_p(x::Vector{Float64})
l_dens = -0.5*(mapslices(sum, (m.-x[1:2]).^2, dims=1)/sigma2)
if d>2
l_dens .-= 0.5*mapslices(sum, x[3:d].^2, dims=1)/sigman
end
l_max = maximum(l_dens) # Prevent underflow by log-sum trick
l_max + log(sum(exp.(l_dens.-l_max)))
end
end
end
using AdaptiveMCMC
n = 100_000; L = 2
rwm = adaptive_rwm(zeros(2), multimodalTarget(2), n; thin=10)
apt = adaptive_rwm(zeros(2), multimodalTarget(2), div(n,L); L = L, thin=10)
# Assuming you have 'Plots' installed:
using Plots
plot(scatter(rwm.X[1,:], rwm.X[2,:], title="w/o tempering", legend=:none),
scatter(apt.X[1,:], apt.X[2,:], title="w/ tempering", legend=:none), layout=(1,2))
```

## Using with Distributions and LabelledArrays

MCMC is often useful with hierarchical models. These may be conveniently built using `Distributions`

and `LabelledArrays`

packages. The following example assumes these packages to be installed.

```
using Distributions, LabelledArrays, AdaptiveMCMC
# Define convenience log-transform for continuous univariate distributions
struct LogTransformedDistribution{Dist <: ContinuousUnivariateDistribution}
d::Dist
end
import Distributions.logpdf
logpdf(d::LogTransformedDistribution, x) = logpdf(d.d, exp(x)) + x
import Base.log
log(d::ContinuousUnivariateDistribution) = LogTransformedDistribution(d)
# This example is modified from Turing Getting Started:
# https://turing.ml/dev/docs/using-turing/get-started
function buildModel(x=0.0, y=1.0)
let x=x, y=y
function(v)
p = 0.0
p += logpdf(log(InverseGamma(2,3)), v.log_s)
ss = exp(.5*v.log_s)
p += logpdf(Normal(0, ss), v.m)
p += logpdf(Normal(v.m, ss), x)
p += logpdf(Normal(v.m, ss), y)
p
end
end
end
# Initial state vector (labelled with keys `s` and `m`)
x0 = LVector(log_s=1.0, m=0.0); log_p = buildModel(3.3, 4.14)
# Hint: If you do not have a good guess of the mode of log_p (which is
# a good initial value for MCMC), you may use optimisation:
#using Optim; o = optimize(x -> -log_p(x), x0); x0 = o.minimizer
out = adaptive_rwm(x0, log_p, 1_000_000; thin=20)
using StatsPlots # Assuming installed
corrplot(out.X', labels=[keys(x0)...])
```

## Custom sampler

The package provides also simple building blocks which you can use within a 'custom' MCMC sampler. Here is an example:

```
using AdaptiveMCMC
# Sampler in R^d
function mySampler(log_p, n, x0)
# Initialise random walk sampler state: r.x current state, r.y proposal
r = RWMState(x0)
# Initialise Adaptive Metropolis state (with default parameters)
s = AdaptiveMetropolis(x0)
# Other adaptations are: AdaptiveScalingMetropolis,
# AdaptiveScalingWithinAdaptiveMetropolis, and RobustAdaptiveMetropolis
X = zeros(eltype(x0), length(x0), n) # Allocate output storage
p_x = log_p(r.x) # = log_p(x0); the initial log target
for k = 1:n
# Draw new proposal r.x -> r.y:
draw!(r, s)
p_y = log_p(r.y) # Calculate log target at proposal
alpha = min(one(p_x), exp(p_y - p_x)) # The Metropolis acceptance probability
if rand() <= alpha
p_x = p_y
# This 'accepts', or interchanges r.x <-> r.y:
# (NB: do not do r.x = r.y; these are (pointers to) vectors!)
accept!(r)
end
# Do the adaptation update:
adapt!(s, r, alpha, k)
X[:,k] = r.x # Save the current sample
end
X
end
# Standard normal target for testing
normal_log_p(x) = -mapreduce(e->e*e, +, x)/2
# Run 1M iterations of the sampler targetting 30d standard Normal:
X = mySampler(normal_log_p, 1_000_000, zeros(30))
```

## More information & citations

The algorithms implemented in the package are discussed in the following reference:

- M. Vihola. Ergonomic and reliable Bayesian inference with adaptive Markov chain Monte Carlo. In W. W. Piegrorsch, R. Levine, H. H. Zhang, and T. C. M. Lee, editors,
*Handbook of Computational Statistics and Data Science*, Wiley, to appear.

(Electronic version available in Wiley StatsRef: https://doi.org/10.1002/9781118445112.stat08286)

If you use the package, please cite this publication.

Details of the implementation, such as about arguments and options, there are help fields written to the functions, which you may access, for instance, by typing `? adaptive_mcmc`

in the Julia REPL.