Using External Samplers on Turing Models
Turing provides several wrapped samplers from external sampling libraries, e.g., HMC samplers from AdvancedHMC.
These wrappers allow new users to seamlessly sample statistical models without leaving Turing
However, these wrappers might only sometimes be complete, missing some functionality from the wrapped sampling library.
Moreover, users might want to use samplers currently not wrapped within Turing.
For these reasons, Turing also makes running external samplers on Turing models easy without any necessary modifications or wrapping!
Throughout, we will use a 10-dimensional Neal's funnel as a running example::
# Import libraries.
using Turing, Random, LinearAlgebra
d = 10
@model function funnel()
θ ~ Truncated(Normal(0, 3), -3, 3)
z ~ MvNormal(zeros(d - 1), exp(θ) * I)
return x ~ MvNormal(z, I)
end
funnel (generic function with 2 methods)
Now we sample the model to generate some observations, which we can then condition on.
(; x) = rand(funnel() | (θ=0,))
model = funnel() | (; x);
Users can use any sampler algorithm to sample this model if it follows the AbstractMCMC API.
Before discussing how this is done in practice, giving a high-level description of the process is interesting.
Imagine that we created an instance of an external sampler that we will call spl such that typeof(spl)<:AbstractMCMC.AbstractSampler.
In order to avoid type ambiguity within Turing, at the moment it is necessary to declare spl as an external sampler to Turing espl = externalsampler(spl), where externalsampler(s::AbstractMCMC.AbstractSampler) is a Turing function that types our external sampler adequately.
An excellent point to start to show how this is done in practice is by looking at the sampling library AdvancedMH ((AdvancedMH's GitHub)[[https://github.com/TuringLang/AdvancedMH.jl]) for Metropolis-Hastings (MH) methods.
Let's say we want to use a random walk Metropolis-Hastings sampler without specifying the proposal distributions.
The code below constructs an MH sampler using a multivariate Gaussian distribution with zero mean and unit variance in d dimensions as a random walk proposal.
# Importing the sampling library
using AdvancedMH
rwmh = AdvancedMH.RWMH(d)
AdvancedMH.MetropolisHastings{AdvancedMH.RandomWalkProposal{false, Distribu
tions.ZeroMeanIsoNormal{Tuple{Base.OneTo{Int64}}}}}(AdvancedMH.RandomWalkPr
oposal{false, Distributions.ZeroMeanIsoNormal{Tuple{Base.OneTo{Int64}}}}(Ze
roMeanIsoNormal(
dim: 10
μ: Zeros(10)
Σ: [1.0 0.0 … 0.0 0.0; 0.0 1.0 … 0.0 0.0; … ; 0.0 0.0 … 1.0 0.0; 0.0 0.0 …
0.0 1.0]
)
))
Sampling is then as easy as:
chain = sample(model, externalsampler(rwmh), 10_000)
Chains MCMC chain (10000×11×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 1.35 seconds
Compute duration = 1.35 seconds
parameters = θ, z[1], z[2], z[3], z[4], z[5], z[6], z[7], z[8], z[9]
internals = lp
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat
e ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64
⋯
θ 0.0121 0.8650 0.1314 47.4626 86.6241 1.0210
⋯
z[1] -0.5324 0.7682 0.0892 79.9027 142.8232 1.0157
⋯
z[2] 1.4455 0.9096 0.1159 65.5889 170.0869 1.0017
⋯
z[3] 0.0219 0.7492 0.0668 128.3552 146.8813 1.0227
⋯
z[4] 0.8745 0.7209 0.0684 108.6626 194.7976 1.0117
⋯
z[5] 0.2145 0.7323 0.0834 81.6500 224.5874 1.0042
⋯
z[6] -0.9286 0.8387 0.0970 68.8226 62.2614 1.0112
⋯
z[7] 0.0349 0.6947 0.0546 153.0607 206.9673 1.0096
⋯
z[8] -0.6871 0.7771 0.0929 70.7733 73.1976 1.0010
⋯
z[9] -0.4666 0.7502 0.0709 107.7177 150.2779 1.0320
⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -1.4451 -0.6533 0.1445 0.6074 1.5776
z[1] -2.0639 -1.0141 -0.5672 0.0405 0.8580
z[2] -0.1706 0.8554 1.3782 2.0158 3.3755
z[3] -1.6737 -0.4554 0.0411 0.5822 1.3927
z[4] -0.3173 0.3945 0.7433 1.3393 2.5053
z[5] -1.1105 -0.3785 0.1465 0.7622 1.6893
z[6] -2.7768 -1.4725 -0.8207 -0.3002 0.3896
z[7] -1.4561 -0.3197 -0.0252 0.4072 1.5296
z[8] -2.2797 -1.1979 -0.5583 -0.2634 0.8090
z[9] -1.9283 -1.0024 -0.4508 0.0157 1.0232
Going beyond the Turing API
As previously mentioned, the Turing wrappers can often limit the capabilities of the sampling libraries they wrap.
AdvancedHMC[^1] ((AdvancedHMC's GitHub)[https://github.com/TuringLang/AdvancedHMC.jl]) is a clear example of this. A common practice when performing HMC is to provide an initial guess for the mass matrix.
However, the native HMC sampler within Turing only allows the user to specify the type of the mass matrix despite the two options being possible within AdvancedHMC.
Thankfully, we can use Turing's support for external samplers to define an HMC sampler with a custom mass matrix in AdvancedHMC and then use it to sample our Turing model.
We will use the library Pathfinder[^2] ((Pathfinder's GitHub)[https://github.com/mlcolab/Pathfinder.jl]) to construct our estimate of mass matrix.
Pathfinder is a variational inference algorithm that first finds the maximum a posteriori (MAP) estimate of a target posterior distribution and then uses the trace of the optimization to construct a sequence of multivariate normal approximations to the target distribution.
In this process, Pathfinder computes an estimate of the mass matrix the user can access.
The code below shows this can be done in practice.
using AdvancedHMC, Pathfinder
# Running pathfinder
draws = 1_000
result_multi = multipathfinder(model, draws; nruns=8)
# Estimating the metric
inv_metric = result_multi.pathfinder_results[1].fit_distribution.Σ
metric = DenseEuclideanMetric(Matrix(inv_metric))
# Creating an AdvancedHMC NUTS sampler with the custom metric.
n_adapts = 1000 # Number of adaptation steps
tap = 0.9 # Large target acceptance probability to deal with the funnel structure of the posterior
nuts = AdvancedHMC.NUTS(tap; metric=metric)
# Sample
chain = sample(model, externalsampler(nuts), 10_000; n_adapts=1_000)
Chains MCMC chain (10000×23×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 7.86 seconds
Compute duration = 7.86 seconds
parameters = θ, z[1], z[2], z[3], z[4], z[5], z[6], z[7], z[8], z[9]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error,
tree_depth, numerical_error, step_size, nom_step_size, is_adapt
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
θ -0.2492 1.0673 0.0270 1671.7391 1194.4760 1.000
5 ⋯
z[1] -0.4565 0.7999 0.0427 696.1796 223.1810 1.000
8 ⋯
z[2] 1.2484 0.9075 0.0195 1738.0874 6160.8376 1.000
9 ⋯
z[3] -0.0604 0.6995 0.0169 1831.5252 227.1630 1.000
5 ⋯
z[4] 0.7984 0.7731 0.0113 5020.0991 6377.8484 1.000
3 ⋯
z[5] 0.1271 0.6803 0.0149 2271.3501 6945.0708 1.000
4 ⋯
z[6] -0.8429 0.7960 0.0160 2097.4126 6229.0701 1.000
8 ⋯
z[7] 0.0519 0.6792 0.0112 3494.3136 6492.5774 1.001
5 ⋯
z[8] -0.6428 0.7585 0.0166 1713.6610 275.4449 1.000
2 ⋯
z[9] -0.5094 0.7169 0.0090 6606.0620 6527.5550 0.999
9 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -2.5807 -0.9079 -0.1165 0.5004 1.5692
z[1] -2.1186 -0.9179 -0.4120 0.0161 1.0303
z[2] -0.1721 0.5453 1.1648 1.8331 3.2340
z[3] -1.5168 -0.4808 -0.0554 0.3698 1.3044
z[4] -0.4726 0.2400 0.6981 1.2735 2.4939
z[5] -1.1811 -0.3058 0.0980 0.5481 1.5113
z[6] -2.6191 -1.3479 -0.7414 -0.2526 0.4340
z[7] -1.3363 -0.3714 0.0424 0.4862 1.3870
z[8] -2.2896 -1.1139 -0.5708 -0.1191 0.6331
z[9] -2.0732 -0.9451 -0.4368 -0.0097 0.7970
Using new inference methods
So far we have used Turing's support for external samplers to go beyond the capabilities of the wrappers. We want to use this support to employ a sampler not supported within Turing's ecosystem yet. We will use the recently developed Micro-Cannoncial Hamiltonian Monte Carlo (MCHMC) sampler to showcase this. MCHMC[^3,^4] ((MCHMC's GitHub)[https://github.com/JaimeRZP/MicroCanonicalHMC.jl]) is HMC sampler that uses one single Hamiltonian energy level to explore the whole parameter space. This is achieved by simulating the dynamics of a microcanonical Hamiltonian with an additional noise term to ensure ergodicity.
Using this as well as other inference methods outside the Turing ecosystem is as simple as executing the code shown below:
using MicroCanonicalHMC
# Create MCHMC sampler
n_adapts = 1_000 # adaptation steps
tev = 0.01 # target energy variance
mchmc = MCHMC(n_adapts, tev; adaptive=true)
# Sample
chain = sample(model, externalsampler(mchmc), 10_000)
Chains MCMC chain (10000×11×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 1.38 seconds
Compute duration = 1.38 seconds
parameters = θ, z[1], z[2], z[3], z[4], z[5], z[6], z[7], z[8], z[9]
internals = lp
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat
e ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64
⋯
θ -0.1569 0.9784 0.0598 269.6519 340.0034 1.0155
⋯
z[1] -0.5781 0.7430 0.0405 348.2620 605.1068 1.0128
⋯
z[2] 1.3363 0.8683 0.0456 372.5555 780.0139 1.0060
⋯
z[3] -0.0674 0.6713 0.0335 410.2338 550.3580 1.0045
⋯
z[4] 0.8433 0.7846 0.0423 360.3244 695.7552 1.0092
⋯
z[5] 0.0755 0.6543 0.0296 494.4099 756.8816 1.0012
⋯
z[6] -0.8784 0.7778 0.0332 580.4709 900.7821 1.0006
⋯
z[7] 0.0237 0.6999 0.0313 507.0552 844.3985 1.0049
⋯
z[8] -0.6734 0.7208 0.0353 437.0096 627.6718 1.0053
⋯
z[9] -0.5416 0.6778 0.0286 584.2623 845.4894 1.0020
⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -2.3225 -0.7729 -0.0732 0.5369 1.5595
z[1] -2.1870 -1.0472 -0.5105 -0.0661 0.7538
z[2] -0.1247 0.6855 1.2825 1.9083 3.1700
z[3] -1.4243 -0.4847 -0.0793 0.3526 1.3199
z[4] -0.4886 0.2799 0.7717 1.3260 2.5832
z[5] -1.2468 -0.3372 0.0697 0.4793 1.4003
z[6] -2.5755 -1.3640 -0.8194 -0.3205 0.4669
z[7] -1.4323 -0.4014 0.0350 0.4669 1.3771
z[8] -2.2817 -1.1065 -0.6062 -0.1848 0.5883
z[9] -2.0221 -0.9544 -0.4811 -0.0792 0.6479
The only requirement to work with externalsampler is that the provided sampler must implement the AbstractMCMC.jl-interface [INSERT LINK] for a model of type AbstractMCMC.LogDensityModel [INSERT LINK].
As previously stated, in order to use external sampling libraries within Turing they must follow the AbstractMCMC API.
In this section, we will briefly dwell on what this entails.
First and foremost, the sampler should be a subtype of AbstractMCMC.AbstractSampler.
Second, the stepping function of the MCMC algorithm must be made defined using AbstractMCMC.step and follow the structure below:
# First step
function AbstractMCMC.step{T<:AbstractMCMC.AbstractSampler}(
rng::Random.AbstractRNG,
model::AbstractMCMC.LogDensityModel,
spl::T;
kwargs...,
)
[...]
return transition, sample
end
# N+1 step
function AbstractMCMC.step{T<:AbstractMCMC.AbstractSampler}(
rng::Random.AbstractRNG,
model::AbstractMCMC.LogDensityModel,
sampler::T,
state;
kwargs...,
)
[...]
return transition, sample
end
There are several characteristics to note in these functions:
-
There must be two
stepfunctions:- A function that performs the first step and initializes the sampler.
- A function that performs the following steps and takes an extra input,
state, which carries the initialization information.
-
The functions must follow the displayed signatures.
-
The output of the functions must be a transition, the current state of the sampler, and a sample, what is saved to the MCMC chain.
The last requirement is that the transition must be structured with a field θ, which contains the values of the parameters of the model for said transition.
This allows Turing to seamlessly extract the parameter values at each step of the chain when bundling the chains.
Note that if the external sampler produces transitions that Turing cannot parse, the bundling of the samples will be different or fail.
For practical examples of how to adapt a sampling library to the AbstractMCMC interface, the readers can consult the following libraries:
Refences
[^1]: Xu et al., AdvancedHMC.jl: A robust, modular and efficient implementation of advanced HMC algorithms, 2019 [^2]: Zhang et al., Pathfinder: Parallel quasi-Newton variational inference, 2021 [^3]: Robnik et al, Microcanonical Hamiltonian Monte Carlo, 2022 [^4]: Robnik and Seljak, Langevine Hamiltonian Monte Carlo, 2023