Robust implementation for random-walk Metropolis-Hastings algorithms
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67 Stars
Updated Last
1 Year Ago
Started In
September 2019

AdvancedMH.jl currently provides a robust implementation of random walk Metropolis-Hastings samplers.

Further development aims to provide a suite of adaptive Metropolis-Hastings implementations.

AdvancedMH works by allowing users to define composable `Proposal` structs in different formats.

Usage

First, construct a `DensityModel`, which is a wrapper around the log density function for your inference problem. The `DensityModel` is then used in a `sample` call.

```# Import the package.
using Distributions
using MCMCChains

using LinearAlgebra

# Generate a set of data from the posterior we want to estimate.
data = rand(Normal(0, 1), 30)

# Define the components of a basic model.
insupport(θ) = θ[2] >= 0
dist(θ) = Normal(θ[1], θ[2])
density(θ) = insupport(θ) ? sum(logpdf.(dist(θ), data)) : -Inf

# Construct a DensityModel.
model = DensityModel(density)

# Set up our sampler with a joint multivariate Normal proposal.
spl = RWMH(MvNormal(zeros(2), I))

# Sample from the posterior.
chain = sample(model, spl, 100000; param_names=["μ", "σ"], chain_type=Chains)```

Output:

```Object of type Chains, with data of type 100000×3×1 Array{Float64,3}

Iterations        = 1:100000
Thinning interval = 1
Chains            = 1
Samples per chain = 100000
internals         = lp
parameters        = μ, σ

2-element Array{ChainDataFrame,1}

Summary Statistics

│ Row │ parameters │ mean     │ std      │ naive_se    │ mcse       │ ess     │ r_hat   │
│     │ Symbol     │ Float64  │ Float64  │ Float64     │ Float64    │ Any     │ Any     │
├─────┼────────────┼──────────┼──────────┼─────────────┼────────────┼─────────┼─────────┤
│ 1   │ μ          │ 0.156152 │ 0.19963  │ 0.000631285 │ 0.00323033 │ 3911.73 │ 1.00009 │
│ 2   │ σ          │ 1.07493  │ 0.150111 │ 0.000474693 │ 0.00240317 │ 3707.73 │ 1.00027 │

Quantiles

│ Row │ parameters │ 2.5%     │ 25.0%     │ 50.0%    │ 75.0%    │ 97.5%    │
│     │ Symbol     │ Float64  │ Float64   │ Float64  │ Float64  │ Float64  │
├─────┼────────────┼──────────┼───────────┼──────────┼──────────┼──────────┤
│ 1   │ μ          │ -0.23361 │ 0.0297006 │ 0.159139 │ 0.283493 │ 0.558694 │
│ 2   │ σ          │ 0.828288 │ 0.972682  │ 1.05804  │ 1.16155  │ 1.41349  │
```

Usage with `LogDensityProblems.jl`

Alternatively, you can define your model with the `LogDensityProblems.jl` interface:

```using LogDensityProblems

struct LogTargetDensity end

LogDensityProblems.logdensity(p::LogTargetDensity, θ) = density(θ)  # standard multivariate normal
LogDensityProblems.dimension(p::LogTargetDensity) = 2
LogDensityProblems.capabilities(::LogTargetDensity) = LogDensityProblems.LogDensityOrder{0}()

sample(LogTargetDensity(), spl, 100000; param_names=["μ", "σ"], chain_type=Chains)```

Proposals

AdvancedMH offers various methods of defining your inference problem. Behind the scenes, a `MetropolisHastings` sampler simply holds some set of `Proposal` structs. AdvancedMH will return posterior samples in the "shape" of the proposal provided -- currently supported methods are `Array{Proposal}`, `Proposal`, and `NamedTuple{Proposal}`. For example, proposals can be created as:

```# Provide a univariate proposal.
m1 = DensityModel(x -> logpdf(Normal(x,1), 1.0))
p1 = StaticProposal(Normal(0,1))
c1 = sample(m1, MetropolisHastings(p1), 100; chain_type=Vector{NamedTuple})

# Draw from a vector of distributions.
m2 = DensityModel(x -> logpdf(Normal(x[1], x[2]), 1.0))
p2 = StaticProposal([Normal(0,1), InverseGamma(2,3)])
c2 = sample(m2, MetropolisHastings(p2), 100; chain_type=Vector{NamedTuple})

# Draw from a `NamedTuple` of distributions.
m3 = DensityModel(x -> logpdf(Normal(x.a, x.b), 1.0))
p3 = (a=StaticProposal(Normal(0,1)), b=StaticProposal(InverseGamma(2,3)))
c3 = sample(m3, MetropolisHastings(p3), 100; chain_type=Vector{NamedTuple})

# Draw from a functional proposal.
m4 = DensityModel(x -> logpdf(Normal(x,1), 1.0))
p4 = StaticProposal((x=1.0) -> Normal(x, 1))
c4 = sample(m4, MetropolisHastings(p4), 100; chain_type=Vector{NamedTuple})```

Static vs. Random Walk

Currently there are only two methods of inference available. Static MH simply draws from the prior, with no conditioning on the previous sample. Random walk will add the proposal to the previously observed value. If you are constructing a `Proposal` by hand, you can determine whether the proposal is a `StaticProposal` or a `RandomWalkProposal` using

```static_prop = StaticProposal(Normal(0,1))
rw_prop = RandomWalkProposal(Normal(0,1))```

Different methods are easily composeable. One parameter can be static and another can be a random walk, each of which may be drawn from separate distributions.

Multiple chains

```# Sample 4 chains from the posterior serially, without thread or process parallelism.
chain = sample(model, RWMH(init_params), MCMCSerial(), 100000, 4; param_names=["μ","σ"], chain_type=Chains)

# Sample 4 chains from the posterior using multiple threads.
chain = sample(model, RWMH(init_params), MCMCThreads(), 100000, 4; param_names=["μ","σ"], chain_type=Chains)

# Sample 4 chains from the posterior using multiple processes.
chain = sample(model, RWMH(init_params), MCMCDistributed(), 100000, 4; param_names=["μ","σ"], chain_type=Chains)```

AdvancedMH.jl also offers an implementation of MALA if the `ForwardDiff` and `DiffResults` packages are available.

A `MALA` sampler can be constructed by `MALA(proposal)` where `proposal` is a function that takes the gradient computed at the current sample. It is required to specify an initial sample `init_params` when calling `sample`.

```# Import the package.
using Distributions
using MCMCChains
using ForwardDiff
using StructArrays

using LinearAlgebra

# Generate a set of data from the posterior we want to estimate.
data = rand(Normal(0, 1), 30)

# Define the components of a basic model.
insupport(θ) = θ[2] >= 0
dist(θ) = Normal(θ[1], θ[2])
density(θ) = insupport(θ) ? sum(logpdf.(dist(θ), data)) : -Inf

# Construct a DensityModel.
model = DensityModel(density)

# Set up the sampler with a multivariate Gaussian proposal.
σ² = 0.01
spl = MALA(x -> MvNormal((σ² / 2) .* x, σ² * I))

# Sample from the posterior.
chain = sample(model, spl, 100000; init_params=ones(2), chain_type=StructArray, param_names=["μ", "σ"])```

Usage with `LogDensityProblems.jl`

As above, we can define the model with the LogDensityProblems.jl interface. We can implement the gradient of the log density function manually, or use `LogDensityProblemsAD.jl` to provide us with the gradient computation used in MALA. Using our implementation of the `LogDensityProblems.jl` interface above:

```using LogDensityProblemsAD