EqualitySampler.jl

Author vandenman
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October 2020

EqualitySampler.jl

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EqualitySampler.jl is a Julia library for considering all possible equality constraints across parameters and sampling fromt the posterior distribution over equality constraints.

Installation

EqualitySampler is a registered package, so it can be installed with the usual incantations:

julia> ]add EqualitySampler

or alternatively

julia> import Pkg; Pkg.add("EqualitySampler")

Functionality

EqualitySampler defines four distributions over partitions of a set:

  • UniformPartitionDistribution
  • BetaBinomialPartitionDistribution
  • CustomInclusionPartitionDistribution
  • RandomProcessPartitionDistribution

Each of these is a subtype of the abstract type AbstractPartitionDistribution, which is a subtype of Distributions.DiscreteMultivariateDistribution.

Thus, each of these types can be called with e.g., rand and logpdf.

While a partition is usually defined without duplicates, the methods here do assume duplicates are present. For example, given 3 parameters (θ₁, θ₂, θ₃) there exist 5 unique partitions:

partition constraints representation
{{θ₁, θ₂, θ₃}} θ₁ == θ₂ == θ₃ [1, 1, 1]
{{θ₁, θ₂}, {θ₃}} θ₁ == θ₂ != θ₃ [1, 1, 2]
{{θ₁, θ₃}, {θ₂}} θ₁ == θ₃ != θ₂ [1, 2, 1]
{{θ₁}, {θ₂, θ₃}} θ₁ != θ₂ == θ₃ [1, 2, 2]
{{θ₁}, {θ₂}, {θ₃}} θ₁ != θ₂ != θ₃ [1, 2, 3]

However, we also consider [2, 2, 2] and [3, 3, 3] to be valid and identical to [1, 1, 1]. The main reason for this is that in a Gibbs sampling scheme, a transition from [1, 2, 2] to [1, 1, 1] by updating only the first element would be a natural but impossible without duplicated partitions. The default logpdf accounts for duplicated partitions, use logpdf_model_distinct to evaluate the logpdf without duplicated partitions.

Built-in tests

The package contains two functions to explore equality constraints in specific models. Both use Turing.jl under the hood and return a Chains object with posterior samples from MCMCChains.jl.

Independent Binomials

proportion_test can be used to explore equality constraints across a series of independent Binomials.

using EqualitySampler, EqualitySampler.Simulations, Distributions, Statistics
import Random, AbstractMCMC, MCMCChains

# simulate some data
Random.seed!(42) # on julia 1.7.3
n_groups = 5 # no. binomials.
true_partition     = rand(UniformPartitionDistribution(n_groups))
temp_probabilities = rand(n_groups)
true_probabilities = average_equality_constraints(temp_probabilities, true_partition)
# total no. trials
observations = rand(100:200, n_groups)
# no. successes
successes = rand(product_distribution(Binomial.(observations, true_probabilities)))

obs_proportions = successes ./ observations
[true_probabilities obs_proportions]
# 5×2 Matrix{Float64}:
#  0.30743   0.301205
#  0.640909  0.640244
#  0.640909  0.701493
#  0.30743   0.275591
#  0.30743   0.296053

# specify no mcmc iterations, no chains, parallelization. burnin and thinning can also be specified.
mcmc_settings = MCMCSettings(;iterations = 15_000, chains = 4, parallel = AbstractMCMC.MCMCThreads)

# nothing indicates no equality sampling is done and instead the full model is sampled from
chn_full = proportion_test(successes, observations, nothing; mcmc_settings = mcmc_settings)
# use a BetaBinomial(1, k) over the partitions
partition_prior = BetaBinomialPartitionDistribution(n_groups, 1, n_groups)
chn_eqs  = proportion_test(successes, observations, partition_prior; mcmc_settings = mcmc_settings)

# extract posterior mean of full model and averaged across equality constraints
estimated_probabilities_full = mean(chn_full).nt.mean
estimated_probabilities_eqs = mean(MCMCChains.group(chn_eqs, :p_constrained)).nt.mean
[true_probabilities obs_proportions estimated_probabilities_full estimated_probabilities_eqs]
# 5×4 Matrix{Float64}:
#  0.30743   0.301205  0.303421  0.296359
#  0.640909  0.640244  0.638429  0.662943
#  0.640909  0.701493  0.698563  0.666477
#  0.30743   0.275591  0.278896  0.295154
#  0.30743   0.296053  0.298635  0.296189

# posterior probabilities of equalities among the probabilities
compute_post_prob_eq(chn_eqs)
# 5×5 Matrix{Float64}:
#  0.0       0.0      0.0  0.0       0.0
#  0.0       0.0      0.0  0.0       0.0
#  0.0       0.94185  0.0  0.0       0.0
#  0.930683  0.0      0.0  0.0       0.0
#  0.937     0.0      0.0  0.931667  0.0
# true matrix
[i > j && p == q for (i, p) in enumerate(true_partition), (j, q) in enumerate(true_partition)]
# 5×5 Matrix{Bool}:
#  0  0  0  0  0
#  0  0  0  0  0
#  0  1  0  0  0
#  1  0  0  0  0
#  1  0  0  1  0

# list the visited models (use compute_model_probs to obtain their posterior probability)
compute_model_counts(chn_eqs, false)
# OrderedCollections.OrderedDict{String, Int64} with 10 entries:
# "12211" => 51488
# "12215" => 1512
# "12241" => 1808
# "12244" => 1562
# "12245" => 141
# "12311" => 2596
# "12315" => 245
# "12341" => 328
# "12344" => 254
# "12345" => 66

# convert true partition to normalized form and print as string
join(EqualitySampler.reduce_model(true_partition))
# "12211"
# and it so happens to be that the most visited model is also the true model

Post hoc tests in One-Way ANOVA

anova_test can be used to explore equality constraints across the levels of a single categorical predictor. The setup uses a grand mean $\mu$ and offsets $\theta_i$ for every level of the categorical predictor. To identify the model, the constraint $\sum_i\theta_i = 1$ is imposed.

using EqualitySampler, EqualitySampler.Simulations, Distributions, Statistics
import Random, AbstractMCMC, MCMCChains

# Simulate some data
Random.seed!(42)
n_groups = 5
n_obs_per_group = 1000
true_partition = rand(UniformPartitionDistribution(n_groups))
temp_θ = randn(n_groups)
temp_θ .-= mean(temp_θ) # center temp_θ to avoid identification constraints
true_θ = average_equality_constraints(temp_θ, true_partition)

g = repeat(1:n_groups; inner = n_obs_per_group)
μ, σ = 0.0, 1.0

# Important: this is the same parametrization as used by the model!
Dy = MvNormal.+ σ .* true_θ[g], σ)
y = rand(Dy)

# observed cell offsets
obs_offset = ([mean(y[g .== i]) for i in unique(g)] .- mean(y)) / std(y)
[true_θ obs_offset]
# 5×2 Matrix{Float64}:
#   0.191185   0.24118
#  -0.286777  -0.290348
#  -0.286777  -0.243936
#   0.191185   0.142092
#   0.191185   0.151012

# specify no mcmc iterations, no chains, parallelization. burnin and thinning can also be specified.
mcmc_settings = MCMCSettings(;iterations = 15_000, chains = 4, parallel = AbstractMCMC.MCMCThreads)

# nothing indicates no equality sampling is done and instead the full model is sampled from
chn_full = anova_test(y, g, nothing; mcmc_settings = mcmc_settings)
# use a BetaBinomial(1, k) over the partitions
partition_prior = BetaBinomialPartitionDistribution(n_groups, 1, n_groups)
chn_eqs  = anova_test(y, g, partition_prior; mcmc_settings = mcmc_settings)

estimated_θ_full = Statistics.mean(MCMCChains.group(chn_full, :θ_cs)).nt.mean
estimated_θ_eqs  = Statistics.mean(MCMCChains.group(chn_eqs , :θ_cs)).nt.mean
[true_θ obs_offset estimated_θ_full estimated_θ_eqs]
# 5×4 Matrix{Float64}:
#   0.191185   0.24118    0.245577   0.194745
#  -0.286777  -0.290348  -0.296143  -0.252687
#  -0.286777  -0.243936  -0.248339  -0.25913
#   0.191185   0.142092   0.145534   0.165273
#   0.191185   0.151012   0.153371   0.1518

# posterior probabilities of equalities among the probabilities
compute_post_prob_eq(chn_eqs)
# 5×5 Matrix{Float64}:
#  0.0         0.0        0.0        0.0     0.0
#  0.00858333  0.0        0.0        0.0     0.0
#  0.0         0.874517   0.0        0.0     0.0
#  0.772967    0.0256833  0.0428167  0.0     0.0
#  0.73465     0.10215    0.0279333  0.8664  0.0
# true matrix
[i > j && p == q for (i, p) in enumerate(true_partition), (j, q) in enumerate(true_partition)]
# 5×5 Matrix{Bool}:
#  0  0  0  0  0
#  0  0  0  0  0
#  0  1  0  0  0
#  1  0  0  0  0
#  1  0  0  1  0

# list the visited models (use compute_model_probs to obtain their posterior probability)
compute_model_counts(chn_eqs, false)
# OrderedCollections.OrderedDict{String, Int64} with 21 entries:
#   "11311" => 421
#   "11341" => 94
#   "12211" => 41409
#   "12212" => 346
#   "12215" => 1161
#   "12221" => 3
#   "12222" => 949
#   "12241" => 427
#   "12242" => 381
#   "12244" => 7699
#   "12245" => 96
#   "12311" => 723
#   "12312" => 2261
#   "12315" => 57
#   "12322" => 168
#   "12331" => 963
#   "12335" => 654
#   "12341" => 39
#   "12342" => 1509
#   "12344" => 615
#   "12345" => 25

# note that there is more uncertainty in the results here, probably because this model is more compelex than the previous.

# convert true partition to normalized form and print as string
join(EqualitySampler.reduce_model(true_partition))
# "12211"
# and it so happens to be that the most visited model is also the true model

Supplementary Analyses

The simulations and analyses for the manuscript 'Flexible Bayesian Multiple Comparison Adjustment Using Dirichlet Process and Beta-Binomial Model Priors' are in the folder "simulations". Note that this folder is a Julia project, so in order to rerun the simulations it is necessary to first activate and instantiate the project.