# 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 -0.1832 0.7262 0.1013 50.4107 82.6611 1.0015 ⋯
z[1] 0.4362 0.6635 0.0819 62.9988 110.2627 1.0524 ⋯
z[2] 0.1563 0.6261 0.0691 83.3282 144.1523 1.0080 ⋯
z[3] 1.4439 0.8958 0.1309 47.8547 42.0160 1.0073 ⋯
z[4] 0.0244 0.7051 0.0768 80.4296 128.9979 1.0048 ⋯
z[5] -0.4065 0.7177 0.0898 65.1409 170.8391 1.0123 ⋯
z[6] -0.3280 0.6908 0.0827 65.0407 125.0655 1.0540 ⋯
z[7] -0.1040 0.7313 0.0884 69.7145 84.9138 1.0062 ⋯
z[8] 0.7738 0.5638 0.0662 68.9657 126.8716 1.0087 ⋯
z[9] -0.2450 0.6389 0.0740 72.6826 188.9812 1.0169 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -1.6953 -0.6483 -0.2681 0.3156 1.2798
z[1] -0.8733 -0.0164 0.4011 0.9036 1.6399
z[2] -0.9529 -0.2889 0.0588 0.4976 1.4132
z[3] 0.1062 0.8056 1.2617 2.1108 3.6173
z[4] -1.1098 -0.4276 -0.0185 0.5125 1.5865
z[5] -1.6493 -0.9210 -0.4721 0.1161 0.8521
z[6] -1.8516 -0.7287 -0.2885 0.0660 1.1104
z[7] -1.6838 -0.6206 -0.0394 0.3321 1.2631
z[8] -0.2681 0.4358 0.8167 1.1285 1.9536
z[9] -1.5707 -0.6545 -0.2368 0.2486 0.8987Going 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 = 1.2 > 1. Corresponding importance sampling estimates are likely to be unstable and are unlikely to converge with additional samples.
└ @ PSIS ~/.julia/packages/PSIS/fU76x/src/core.jl:362
[ Info: Found initial step size 3.2Chains MCMC chain (10000×23×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 6.08 seconds
Compute duration = 6.08 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 -0.8773 1.0734 0.0266 1540.4691 1273.1077 1.0006 ⋯
z[1] 0.2944 0.6102 0.0076 6944.2030 5327.6766 1.0006 ⋯
z[2] 0.1690 0.5818 0.0057 10907.4261 6650.7295 1.0003 ⋯
z[3] 0.9490 0.8135 0.0143 3322.8517 6323.4639 1.0003 ⋯
z[4] 0.0623 0.5701 0.0054 11384.0730 6129.6453 0.9999 ⋯
z[5] -0.1475 0.5892 0.0060 10457.6815 5206.1937 1.0009 ⋯
z[6] -0.2819 0.5953 0.0064 9495.3772 5887.4577 1.0000 ⋯
z[7] -0.1698 0.5851 0.0059 10187.9188 6336.8450 1.0000 ⋯
z[8] 0.5510 0.6755 0.0096 5531.6237 6360.2701 0.9999 ⋯
z[9] -0.2708 0.5943 0.0063 9478.8867 5920.2450 0.9999 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -2.8169 -1.7036 -0.8341 -0.0707 1.0832
z[1] -0.8131 -0.0948 0.2305 0.6284 1.6728
z[2] -0.9515 -0.1851 0.1354 0.4999 1.4452
z[3] -0.2384 0.3349 0.7953 1.4443 2.8340
z[4] -1.1046 -0.2706 0.0533 0.3804 1.2879
z[5] -1.4410 -0.4825 -0.1100 0.2087 0.9868
z[6] -1.6157 -0.6194 -0.2290 0.1022 0.7842
z[7] -1.4700 -0.4946 -0.1272 0.1909 0.9335
z[8] -0.5369 0.0843 0.4392 0.9235 2.1173
z[9] -1.6063 -0.5973 -0.2232 0.1162 0.7999Using 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: 0:05:56
ϵ: 0.5031087911132046
L: 3.1622776601683795
dE/d: -0.001473597870779031
Tuning: 1%|▍ | ETA: 0:04:01
ϵ: 0.6091743899079449
L: 2.885533838713511
dE/d: -0.018721259480051343
Tuning: 100%|███████████████████████████████████████████| Time: 0:00:02
ϵ: 0.8596767093247397
L: 379.13607708283774
dE/d: 0.006815566060573985Chains MCMC chain (10000×11×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
Wall duration = 4.94 seconds
Compute duration = 4.94 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 -0.9143 1.1474 0.0374 894.4458 958.3049 1.0003 ⋯
z[1] 0.2846 0.5495 0.0145 1540.0582 1482.6766 1.0048 ⋯
z[2] 0.1777 0.5559 0.0153 1384.2012 1557.9999 1.0063 ⋯
z[3] 0.9137 0.8241 0.0232 1421.5047 1677.5444 1.0006 ⋯
z[4] 0.0415 0.5841 0.0169 1282.3919 1343.2095 1.0091 ⋯
z[5] -0.1159 0.6134 0.0185 1138.9152 1300.4095 0.9999 ⋯
z[6] -0.2811 0.5699 0.0154 1452.3150 1432.9169 1.0001 ⋯
z[7] -0.1883 0.5374 0.0148 1382.2526 1410.3457 1.0065 ⋯
z[8] 0.5294 0.6984 0.0185 1599.5889 1628.8640 1.0022 ⋯
z[9] -0.2628 0.5508 0.0144 1574.5016 1491.6321 1.0001 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
θ -2.8917 -1.8148 -0.9223 -0.0600 1.2682
z[1] -0.6482 -0.0744 0.2290 0.6006 1.4872
z[2] -0.8714 -0.1610 0.1297 0.4859 1.4089
z[3] -0.2823 0.2862 0.7596 1.4143 2.8570
z[4] -1.2064 -0.2790 0.0364 0.3659 1.2684
z[5] -1.3844 -0.4772 -0.1010 0.2706 1.0652
z[6] -1.5907 -0.6171 -0.2099 0.1006 0.7185
z[7] -1.4167 -0.4824 -0.1430 0.1312 0.8210
z[8] -0.5894 0.0530 0.4123 0.9061 2.1824
z[9] -1.5338 -0.5634 -0.2080 0.0917 0.6893The 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↩︎