# 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)
endfunnel (generic function with 2 methods)Using External Samplers
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::
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) 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)MetropolisHastings{RandomWalkProposal{false, ZeroMeanIsoNormal{Tuple{Base.OneTo{Int64}}}}}(RandomWalkProposal{false, ZeroMeanIsoNormal{Tuple{Base.OneTo{Int64}}}}(ZeroMeanIsoNormal(
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]
)
))setprogress!(false)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 = 3.88 seconds
Compute duration = 3.88 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 0.9264 0.6140 0.0585 119.0781 82.5694 1.0009 ⋯
z[1] 0.6612 0.7724 0.0717 118.5779 178.4398 1.0319 ⋯
z[2] -0.0080 0.8370 0.0809 110.6860 173.4552 1.0043 ⋯
z[3] -0.4514 0.8066 0.0669 146.0654 212.6612 1.0143 ⋯
z[4] -1.7583 0.8414 0.0920 81.9007 111.0115 1.0273 ⋯
z[5] 2.2715 0.8622 0.0894 95.9043 197.5817 1.0032 ⋯
z[6] -0.0598 0.7841 0.0614 160.6928 155.1497 1.0031 ⋯
z[7] 0.8957 0.8253 0.0676 154.3165 239.9771 1.0026 ⋯
z[8] -2.5790 0.8973 0.1078 69.3281 155.2039 1.0048 ⋯
z[9] -0.6353 0.7829 0.0644 147.6962 181.3845 1.0063 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -0.5252 0.5149 0.9316 1.3449 2.0749
z[1] -0.7963 0.0697 0.6718 1.1875 2.1769
z[2] -1.9009 -0.5184 0.0216 0.6027 1.5897
z[3] -2.0123 -1.0084 -0.4415 0.1386 1.1082
z[4] -3.6289 -2.3445 -1.6652 -1.2400 -0.2874
z[5] 0.7487 1.7268 2.3243 2.8719 3.9474
z[6] -1.5351 -0.6105 -0.0350 0.4457 1.5754
z[7] -0.5591 0.3563 0.8718 1.4141 2.5839
z[8] -4.3843 -3.1687 -2.5405 -2.0369 -0.5343
z[9] -2.3505 -1.1315 -0.6194 -0.1538 0.9901Going beyond the Turing API
As previously mentioned, the Turing wrappers can often limit the capabilities of the sampling libraries they wrap. AdvancedHMC1 (AdvancedHMC’s GitHub) 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 Pathfinder2 ((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)┌ Warning: Pareto shape k = 0.74 > 0.7. Resulting importance sampling estimates are likely to be unstable.
└ @ PSIS ~/.julia/packages/PSIS/fU76x/src/core.jl:364
[ Info: Found initial step size 0.8Chains MCMC chain (10000×23×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 5.8 seconds
Compute duration = 5.8 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, hamiltonian_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 rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
θ 0.8869 0.7524 0.0147 4356.1701 2310.2037 1.0000 ⋯
z[1] 0.6551 0.8366 0.0079 11319.7280 7629.5344 1.0000 ⋯
z[2] -0.0127 0.8237 0.0072 13091.9387 7650.0905 1.0000 ⋯
z[3] -0.4417 0.8434 0.0076 12336.6258 7393.2199 0.9999 ⋯
z[4] -1.8052 0.9136 0.0099 8421.2618 4566.9553 1.0000 ⋯
z[5] 2.1879 0.9524 0.0111 7202.3284 3861.0426 1.0003 ⋯
z[6] 0.0190 0.8381 0.0076 12205.9131 7894.0347 1.0000 ⋯
z[7] 0.7681 0.8497 0.0087 9569.6699 7919.5154 1.0003 ⋯
z[8] -2.5229 0.9895 0.0115 7124.7803 3079.1234 1.0001 ⋯
z[9] -0.5790 0.8434 0.0076 12217.2213 7944.0191 0.9999 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -0.7030 0.4490 0.9166 1.3806 2.2612
z[1] -0.9594 0.0822 0.6437 1.2045 2.3153
z[2] -1.6380 -0.5479 -0.0143 0.5148 1.6524
z[3] -2.1395 -0.9969 -0.4242 0.1216 1.1852
z[4] -3.6146 -2.4272 -1.7949 -1.1602 -0.0812
z[5] 0.3435 1.5433 2.1675 2.8132 4.1123
z[6] -1.6238 -0.5354 0.0150 0.5751 1.6811
z[7] -0.8444 0.1984 0.7448 1.3241 2.4888
z[8] -4.5015 -3.1984 -2.5031 -1.8468 -0.6189
z[9] -2.2544 -1.1394 -0.5636 -0.0166 1.0860Using 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)[ Info: Tuning eps ⏳
[ Info: Tuning L ⏳
[ Info: Tuning sigma ⏳
Tuning: 0%|▏ | ETA: 0:06:05
ϵ: 1.1895469487199226
L: 3.1622776601683795
dE/d: -0.034692248626156184
Tuning: 1%|▍ | ETA: 0:04:14
ϵ: 1.3608013131938246
L: 6.654182401789463
dE/d: 0.010563357763462377
Tuning: 100%|███████████████████████████████████████████| Time: 0:00:02
ϵ: 2.252302063022756
L: 954.2150049761135
dE/d: -0.014340325985904557Chains MCMC chain (10000×11×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 5.09 seconds
Compute duration = 5.09 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 ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
θ 0.9568 0.7540 0.0169 2055.8121 1882.3516 1.0044 ⋯
z[1] 0.6684 0.8394 0.0172 2394.1480 3325.6951 1.0003 ⋯
z[2] -0.0190 0.7472 0.0122 3766.2744 4949.7258 1.0372 ⋯
z[3] -0.4532 0.8389 0.0165 2613.2581 3633.3088 1.0023 ⋯
z[4] -1.8132 0.9175 0.0155 3541.1333 4387.8983 1.0001 ⋯
z[5] 2.2142 1.0178 0.0200 2580.9428 3953.9982 1.0011 ⋯
z[6] 0.0100 0.8429 0.0129 4259.2696 5828.5831 0.9999 ⋯
z[7] 0.7738 0.8862 0.0183 2353.1778 3240.5408 1.0005 ⋯
z[8] -2.5711 1.0300 0.0201 2637.2177 3401.8526 1.0011 ⋯
z[9] -0.5933 0.8634 0.0145 3535.1486 4623.7053 1.0071 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -0.6144 0.4886 0.9883 1.4784 2.3559
z[1] -1.0024 0.1322 0.6524 1.1905 2.4004
z[2] -1.5438 -0.4794 -0.0164 0.4465 1.5021
z[3] -2.1523 -0.9989 -0.4411 0.0962 1.1942
z[4] -3.6666 -2.4148 -1.7947 -1.1904 -0.0598
z[5] 0.2409 1.5166 2.2035 2.8971 4.2161
z[6] -1.7359 -0.4890 0.0088 0.5034 1.7474
z[7] -0.8953 0.1732 0.7474 1.3645 2.5745
z[8] -4.6578 -3.2451 -2.5495 -1.8884 -0.5722
z[9] -2.3220 -1.1421 -0.5799 -0.0422 1.1384The 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
endThere 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:
Footnotes
Xu et al., AdvancedHMC.jl: A robust, modular and efficient implementation of advanced HMC algorithms, 2019↩︎
Zhang et al., Pathfinder: Parallel quasi-Newton variational inference, 2021↩︎
Robnik et al, Microcanonical Hamiltonian Monte Carlo, 2022↩︎
Robnik and Seljak, Langevine Hamiltonian Monte Carlo, 2023↩︎