From 9d1d3f2069a822edc41e2b35321e12296c8bea1b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Kiant=C3=A9=20Fernandez?= <61021880+kiante-fernandez@users.noreply.github.com> Date: Tue, 15 Apr 2025 11:44:33 -0700 Subject: [PATCH 1/4] add amorized inference example This is the first of two examples I want to include for using simulation-based inference with models in SSM.jl. Here, I am using the package NeuralEstimators.jl. Feedback is welcome at this point. https://github.com/msainsburydale/NeuralEstimators Next, I am working on building likelihood approximation networks in Flux based on methods from Fengler, and I will have it up and running to sample with Turing. That will be the next example. For now, I tried to use the LCA, as similar examples have shown online (see https://bayesflow.org/stable-legacy/_examples/LCA_Model_Posterior_Estimation.html). I was able to get this running local but I think we have to env coflict we have to resolve to get NeuralEstimators working on the docs --- docs/Project.toml | 1 + docs/make.jl | 3 +- docs/src/assets/lca_amorized_recovery.png | Bin 0 -> 195463 bytes docs/src/neuralestimators_amorized.md | 261 ++++++++++++++++++++++ 4 files changed, 264 insertions(+), 1 deletion(-) create mode 100644 docs/src/assets/lca_amorized_recovery.png create mode 100644 docs/src/neuralestimators_amorized.md diff --git a/docs/Project.toml b/docs/Project.toml index 66788195..4cf6b198 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -4,6 +4,7 @@ DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b" +Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" KernelDensity = "5ab0869b-81aa-558d-bb23-cbf5423bbe9b" ParetoSmooth = "a68b5a21-f429-434e-8bfa-46b447300aac" Pigeons = "0eb8d820-af6a-4919-95ae-11206f830c31" diff --git a/docs/make.jl b/docs/make.jl index 828c1256..3226b064 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -53,7 +53,8 @@ makedocs( "Mode Estimation" => "mode_estimation.md", "Simple Bayesian Model" => "turing_simple.md", "Advanced Model Specification" => "turing_advanced.md", - "Hierarchical Models" => "turing_hierarchical.md" + "Hierarchical Models" => "turing_hierarchical.md", + "Amorized Inference" => "neuralestimators_amorized.md", ], "Model Comparison" => [ "Bayes Factors" => "bayes_factor.md", diff --git a/docs/src/assets/lca_amorized_recovery.png b/docs/src/assets/lca_amorized_recovery.png new file mode 100644 index 0000000000000000000000000000000000000000..adb8ab72abc7a35520f92fe591f732b4ed7e2be2 GIT binary patch literal 195463 zcmafbWmuM7(CvdDB?uy2A|;JTNFyK}(jnd5-QC^Y-Q6wH-JQ}6f;61X`+euS&cAc3 zDDlL-_w1Q9YppRzN>T_J5f>2xfgpn5oHlx6jZNRV1u)>Jt;=2nNL52#J&#PyyyJ;c-CjVyJ|BwJetmCbK0~!DI5*+ 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z2l}u$(!!c`@Obwr_{z$(!;TT13v+kMo9G}=6_Vg!F(-q+b`=nW8BxUKNAl2yH!7{# zB2?kb0B>3yf_Pi^r0YsrD66n1vJgZxsg#A48Dnk+4t91!BO@+E4he}HNbnEt*(nN< z#LkWmYrn#QN|(dF%XNA%WHLgPle3LL-%OrFU;BZN~KjCh(*b#B@LayLY-Vi1< zpVm^lSdIjwptb2T|Mg$o5s;KJGBW`cqHkZ!gH~uib{4Y_Ghl?g4A~8$vnjUa-6F3Eq)aUnR2Yy~$~WJ;D)=Uqk!? zdSe_iH!*?G+gn*tG5w{Uh>+jXvc3Ma_Wmf04gnefK!wjj!GckUqH7;TkaF=X$nU3F z7)|vKbd{m}cv<>C$TK~(i~Y!dKgug}{eS%QTS3_{ifKpRokRj&XU}Mz&QrSV|34p5 B#Hj!P literal 0 HcmV?d00001 diff --git a/docs/src/neuralestimators_amorized.md b/docs/src/neuralestimators_amorized.md new file mode 100644 index 00000000..705511a7 --- /dev/null +++ b/docs/src/neuralestimators_amorized.md @@ -0,0 +1,261 @@ +# Neural Parameter Estimation + +Neural parameter estimation provides a likelihood-free approach to parameter recovery, especially useful for models with computationally intractable likelihoods. This method is based on training neural networks to learn the mapping from data to parameters. Once trained, these networks can perform inference rapidly across multiple datasets, making them particularly valuable for models like the Leaky Competing Accumulator (LCA). + +Below, we demonstrate how to estimate parameters of the LCA model using the [NeuralEstimators.jl](https://github.com/msainsburydale/NeuralEstimators) package. + +## Example + +We'll estimate parameters of the LCA model, which is particularly challenging due to its complex dynamics where parameters like leak rate (λ) and lateral inhibition (β) can be difficult to recover. + +## Load Packages + +```julia +using NeuralEstimators +using SequentialSamplingModels +using Flux +using Distributions +using Random + +using Plots + +Random.seed!(123) +``` + +## Define Parameter Bounds + +```julia +# Define parameter bounds for the LCA model +const ν_min, ν_max = 0.1, 4.0 # Drift rates +const α_min, α_max = 0.5, 3.5 # Threshold +const β_min, β_max = 0.0, 0.5 # Lateral inhibition +const λ_min, λ_max = 0.0, 0.5 # Leak rate +const τ_min, τ_max = 0.1, 0.5 # Non-decision time +``` + +## Define Parameter Sampling + +Unlike traditional Bayesian approaches, simulation based inference methods require us to define a prior sampling function to generate training data. We will use this function to sample a range of parameters for training: + +```julia +# Function to sample parameters from priors +function sample(K::Integer) + ν1 = rand(Gamma(2, 1/1.2), K) # Drift rate 1 + ν2 = rand(Gamma(2, 1/1.2), K) # Drift rate 2 + α = rand(Gamma(10, 1/6), K) # Threshold + β = rand(Beta(1, 5), K) # Lateral inhibition + λ = rand(Beta(1, 5), K) # Leak rate + τ = rand(Gamma(1.5, 1/5.0), K) # Non-decision time + + # Stack parameters into a matrix (d×K) + θ = vcat(ν1', ν2', α', β', λ', τ') + + return θ +end +``` + +## Define Data Simulator + +Neural estimators learn the mapping from data to parameters through simulation. Here we define a function to simulate LCA model data. To do so we will use the LCA function from SequentialSamplingModels. + +```julia +# Function to simulate data from the LCA model +function simulate(θ, n_trials_per_param) + # Simulate data for each parameter vector + simulated_data = map(eachcol(θ)) do param + # Extract parameters for this model + ν1, ν2, α, β, λ, τ = param + ν = [ν1, ν2] # Two-choice LCA + σ = 1.0 # Fixed diffusion noise + + # Create LCA model + model = LCA(; ν, α, β, λ, τ, σ) + + # Generate choices and reaction times + choices, rts = rand(model, n_trials_per_param) + + # Return as a transpose matrix where each column is a trial + return Float32.([choices rts]') + end + + return simulated_data +end +``` + +## Define Neural Network Architecture + +For LCA parameter recovery, we use a DeepSet architecture which respects the permutation invariance of trial data. For more details on the method see NeuralEstimators.jl documentation. To construct the network architecture we will use the Flux.jl package. + +```julia +# Create neural network architecture for parameter recovery +function create_neural_estimator() + # Input dimension: 2 (choice and RT for each trial) + n = 2 + # Output dimension: 6 parameters + d = 6 # ν[1], ν[2], α, β, λ, τ + # Width of hidden layers + w = 128 + + # Inner network - processes each trial independently + ψ = Chain( + Dense(n, w, relu), + Dense(w, w, relu), + Dense(w, w, relu) + ) + + # Final layer with parameter constraints + final_layer = Parallel( + vcat, + Dense(w, 1, x -> ν_min + (ν_max - ν_min) * σ(x)), # ν1 + Dense(w, 1, x -> ν_min + (ν_max - ν_min) * σ(x)), # ν2 + Dense(w, 1, x -> α_min + (α_max - α_min) * σ(x)), # α + Dense(w, 1, x -> β_min + (β_max - β_min) * σ(x)), # β + Dense(w, 1, x -> λ_min + (λ_max - λ_min) * σ(x)), # λ + Dense(w, 1, x -> τ_min + (τ_max - τ_min) * σ(x)) # τ + ) + + # Outer network - maps aggregated features to parameters + ϕ = Chain( + Dense(w, w, relu), + Dense(w, w, relu), + final_layer + ) + + # Combine into a DeepSet + network = DeepSet(ψ, ϕ) + + # Initialize neural Bayes estimator + estimator = PointEstimator(network) + + return estimator +end +``` + +## Training the Neural Estimator + +Neural estimators, like all deep learning methods, require a training phase where they learn the mapping from data to parameters. Here we will train the estimator, simulating data as we go where the sampler provides new parameter vectors from the prior, and a simulator can be provided to continuously simulate new data conditional on the parameters. + +```julia +# Create the neural estimator +estimator = create_neural_estimator() + +# Train with on-the-fly simulation +trained_estimator = train( + estimator, + sample, # Parameter sampler function + simulate, # Data simulator function + m = 100, # Number of trials per parameter vector + K = 1000, # Number of training parameter vectors + K_val = 200, # Number of validation parameter vectors + loss = Flux.mae, # Mean absolute error loss + epochs = 5, # Number of training epochs + epochs_per_Z_refresh = 1, # Refresh data every epoch + epochs_per_θ_refresh = 5, # Refresh parameters every 5 epochs + batchsize = 64, # Batch size for training + verbose = true +) +``` + +## Assessing Estimator Performance + +We can assess the performance of our trained estimator on held-out test data: + +```julia +# Generate test data +n_test = 100 +θ_test = sample(n_test) +Z_test = simulate(θ_test, 100) + +# Assess the estimator +parameter_names = ["ν1", "ν2", "α", "β", "λ", "τ"] +assessment = assess( + trained_estimator, + θ_test, + Z_test; + parameter_names = parameter_names +) + +# Calculate performance metrics +bias_results = bias(assessment) +rmse_results = rmse(assessment) +println("Bias: ", bias_results) +println("RMSE: ", rmse_results) +``` + +## Visualizing Parameter Recovery + +A key advantage of neural estimation is the ability to quickly conduct inference after training. For example, we can visualize the recovery of parameters: + +```julia +# Extract data from assessment +df = assessment.df + +# Create recovery plots for each parameter +params = unique(df.parameter) +p_plots = [] + +for param in params + param_data = filter(row -> row.parameter == param, df) + p = scatter( + param_data.truth, + param_data.estimate, + xlabel="Ground Truth", + ylabel="Estimated", + title=param, + legend=false + ) + plot!(p, [minimum(param_data.truth), maximum(param_data.truth)], + [minimum(param_data.truth), maximum(param_data.truth)], + line=:dash, color=:black) + push!(p_plots, p) +end + +# Combine plots +p_combined = plot(p_plots..., layout=(3,2), size=(800, 600)) +display(p_combined) +``` +![](assets/lca_amorized_recovery.png) + +## Using the Trained Estimator + +Once trained, the estimator can instantly recover parameters from new data via a forward pass: + +```julia +# Generate "observed" data +ν = [2.5, 2.0] +α = 1.5 +β = 0.2 +λ = 0.1 +τ = 0.3 +σ = 1.0 + +# Create model and generate data +true_model = LCA(; ν, α, β, λ, τ, σ) +observed_choices, observed_rts = rand(true_model, 100) + +# Format the data +observed_data = Float32.([observed_choices observed_rts]') + +# Recover parameters +recovered_params = NeuralEstimators.estimate(trained_estimator, [observed_data]) + +# Compare true and recovered parameters +println("True parameters: ", [ν[1], ν[2], α, β, λ, τ]) +println("Recovered parameters: ", recovered_params) +``` + +## Notes on Performance + +Neural estimators are particularly effective for models with computationally intractable likelihoods like the LCA model. However, certain parameters (particularly β and λ) can be difficult to recover, even with advanced neural network architectures. This is a property of the LCA model rather than a limitation of the estimation technique. + +Additional details can be found in the [NeuralEstimators.jl documentation](https://github.com/msainsburydale/NeuralEstimators). + +# References + +Miletić, S., Turner, B. M., Forstmann, B. U., & van Maanen, L. (2017). Parameter recovery for the leaky competing accumulator model. Journal of Mathematical Psychology, 76, 25-50. + +Sainsbury-Dale, Matthew, Andrew Zammit-Mangion, and Raphaël Huser. "Likelihood-free parameter estimation with neural Bayes estimators." The American Statistician 78.1 (2024): 1-14. + +Zammit-Mangion, Andrew, Matthew Sainsbury-Dale, and Raphaël Huser. "Neural methods for amortized inference." Annual Review of Statistics and Its Application 12 (2024). + +Radev, S. T., Schmitt, M., Schumacher, L., Elsemüller, L., Pratz, V., Schälte, Y., ... & Bürkner, P. C. (2023). BayesFlow: Amortized Bayesian workflows with neural networks. arXiv preprint arXiv:2306.16015. From 5858fdd48e6571b3a291db7a98c0cf3412384f6b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Kiant=C3=A9=20Fernandez?= <61021880+kiante-fernandez@users.noreply.github.com> Date: Tue, 15 Apr 2025 12:46:33 -0700 Subject: [PATCH 2/4] Remove Flux dependency from Project.toml and update neural estimators documentation for clarity and additional references --- docs/Project.toml | 1 - docs/src/neuralestimators_amorized.md | 49 ++++++++++++++------------- 2 files changed, 25 insertions(+), 25 deletions(-) diff --git a/docs/Project.toml b/docs/Project.toml index 4cf6b198..66788195 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -4,7 +4,6 @@ DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b" -Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" KernelDensity = "5ab0869b-81aa-558d-bb23-cbf5423bbe9b" ParetoSmooth = "a68b5a21-f429-434e-8bfa-46b447300aac" Pigeons = "0eb8d820-af6a-4919-95ae-11206f830c31" diff --git a/docs/src/neuralestimators_amorized.md b/docs/src/neuralestimators_amorized.md index 705511a7..23a33552 100644 --- a/docs/src/neuralestimators_amorized.md +++ b/docs/src/neuralestimators_amorized.md @@ -1,16 +1,16 @@ # Neural Parameter Estimation -Neural parameter estimation provides a likelihood-free approach to parameter recovery, especially useful for models with computationally intractable likelihoods. This method is based on training neural networks to learn the mapping from data to parameters. Once trained, these networks can perform inference rapidly across multiple datasets, making them particularly valuable for models like the Leaky Competing Accumulator (LCA). +Neural parameter estimation provides a likelihood-free approach to parameter recovery, especially useful for models with computationally intractable likelihoods. This method is based on training neural networks to learn the mapping from data to parameters. See the review paper by Zammit-Mangion et al. (2025) for more details. Once trained, these networks can perform inference rapidly across multiple datasets, making them particularly valuable for models like the Leaky Competing Accumulator (LCA; Usher & McClelland, 2001). Below, we demonstrate how to estimate parameters of the LCA model using the [NeuralEstimators.jl](https://github.com/msainsburydale/NeuralEstimators) package. ## Example -We'll estimate parameters of the LCA model, which is particularly challenging due to its complex dynamics where parameters like leak rate (λ) and lateral inhibition (β) can be difficult to recover. +We'll estimate parameters of the LCA model, which is particularly challenging due to its complex dynamics, where parameters like leak rate (λ) and lateral inhibition (β) can be difficult to recover. This example draws from a more in-depth case that highlights many of the steps one ought to consider when utilizing amortized inference for cognitive modeling; see [Principled Amortized Bayesian Workflow for Cognitive Modeling](https://bayesflow.org/stable-legacy/_examples/Amortized_Bayesian_Workflow_for_Cognitive_Modeling.html). ## Load Packages -```julia +```@example using NeuralEstimators using SequentialSamplingModels using Flux @@ -24,7 +24,7 @@ Random.seed!(123) ## Define Parameter Bounds -```julia +```@example # Define parameter bounds for the LCA model const ν_min, ν_max = 0.1, 4.0 # Drift rates const α_min, α_max = 0.5, 3.5 # Threshold @@ -37,7 +37,7 @@ const τ_min, τ_max = 0.1, 0.5 # Non-decision time Unlike traditional Bayesian approaches, simulation based inference methods require us to define a prior sampling function to generate training data. We will use this function to sample a range of parameters for training: -```julia +```@example # Function to sample parameters from priors function sample(K::Integer) ν1 = rand(Gamma(2, 1/1.2), K) # Drift rate 1 @@ -56,20 +56,19 @@ end ## Define Data Simulator -Neural estimators learn the mapping from data to parameters through simulation. Here we define a function to simulate LCA model data. To do so we will use the LCA function from SequentialSamplingModels. +Neural estimators learn the mapping from data to parameters through simulation. Here we define a function to simulate LCA model data. To do so we will use the [LCA](https://itsdfish.github.io/SequentialSamplingModels.jl/dev/lca/). -```julia +```@example # Function to simulate data from the LCA model function simulate(θ, n_trials_per_param) # Simulate data for each parameter vector simulated_data = map(eachcol(θ)) do param # Extract parameters for this model ν1, ν2, α, β, λ, τ = param - ν = [ν1, ν2] # Two-choice LCA - σ = 1.0 # Fixed diffusion noise + ν = [ν1, ν2] # Two-choice LCA - # Create LCA model - model = LCA(; ν, α, β, λ, τ, σ) + # Create LCA model with SSM + model = LCA(; ν, α, β, λ, τ) # Generate choices and reaction times choices, rts = rand(model, n_trials_per_param) @@ -86,7 +85,7 @@ end For LCA parameter recovery, we use a DeepSet architecture which respects the permutation invariance of trial data. For more details on the method see NeuralEstimators.jl documentation. To construct the network architecture we will use the Flux.jl package. -```julia +```@example # Create neural network architecture for parameter recovery function create_neural_estimator() # Input dimension: 2 (choice and RT for each trial) @@ -133,22 +132,22 @@ end ## Training the Neural Estimator -Neural estimators, like all deep learning methods, require a training phase where they learn the mapping from data to parameters. Here we will train the estimator, simulating data as we go where the sampler provides new parameter vectors from the prior, and a simulator can be provided to continuously simulate new data conditional on the parameters. +Neural estimators, like all deep learning methods, require a training phase where they learn the mapping from data to parameters. Here we will train the estimator, simulating data as we go where the sampler provides new parameter vectors from the prior, and a simulator can be provided to continuously simulate new data conditional on the parameters. For more details on the training see the API for arguments [here](https://msainsburydale.github.io/NeuralEstimators.jl/dev/API/core/#Training). -```julia +```@example # Create the neural estimator estimator = create_neural_estimator() -# Train with on-the-fly simulation +# Train network trained_estimator = train( estimator, sample, # Parameter sampler function simulate, # Data simulator function m = 100, # Number of trials per parameter vector - K = 1000, # Number of training parameter vectors - K_val = 200, # Number of validation parameter vectors + K = 10000, # Number of training parameter vectors + K_val = 2000, # Number of validation parameter vectors loss = Flux.mae, # Mean absolute error loss - epochs = 5, # Number of training epochs + epochs = 50, # Number of training epochs epochs_per_Z_refresh = 1, # Refresh data every epoch epochs_per_θ_refresh = 5, # Refresh parameters every 5 epochs batchsize = 64, # Batch size for training @@ -160,7 +159,7 @@ trained_estimator = train( We can assess the performance of our trained estimator on held-out test data: -```julia +```@example # Generate test data n_test = 100 θ_test = sample(n_test) @@ -186,7 +185,7 @@ println("RMSE: ", rmse_results) A key advantage of neural estimation is the ability to quickly conduct inference after training. For example, we can visualize the recovery of parameters: -```julia +```@example # Extract data from assessment df = assessment.df @@ -220,7 +219,7 @@ display(p_combined) Once trained, the estimator can instantly recover parameters from new data via a forward pass: -```julia +```@example # Generate "observed" data ν = [2.5, 2.0] α = 1.5 @@ -230,7 +229,7 @@ Once trained, the estimator can instantly recover parameters from new data via a σ = 1.0 # Create model and generate data -true_model = LCA(; ν, α, β, λ, τ, σ) +true_model = LCA(; ν, α, β, λ, τ) observed_choices, observed_rts = rand(true_model, 100) # Format the data @@ -256,6 +255,8 @@ Miletić, S., Turner, B. M., Forstmann, B. U., & van Maanen, L. (2017). Paramete Sainsbury-Dale, Matthew, Andrew Zammit-Mangion, and Raphaël Huser. "Likelihood-free parameter estimation with neural Bayes estimators." The American Statistician 78.1 (2024): 1-14. -Zammit-Mangion, Andrew, Matthew Sainsbury-Dale, and Raphaël Huser. "Neural methods for amortized inference." Annual Review of Statistics and Its Application 12 (2024). - Radev, S. T., Schmitt, M., Schumacher, L., Elsemüller, L., Pratz, V., Schälte, Y., ... & Bürkner, P. C. (2023). BayesFlow: Amortized Bayesian workflows with neural networks. arXiv preprint arXiv:2306.16015. + +Usher, M., & McClelland, J. L. (2001). The time course of perceptual choice: The leaky, competing accumulator model. Psychological Review, 108 3, 550–592. https://doi.org/10.1037/0033-295X.108.3.550 + +Zammit-Mangion, Andrew, Matthew Sainsbury-Dale, and Raphaël Huser. "Neural methods for amortized inference." Annual Review of Statistics and Its Application 12 (2024). \ No newline at end of file From 0635ea3f18645267f373697c8d23f3416176aab8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Kiant=C3=A9=20Fernandez?= <61021880+kiante-fernandez@users.noreply.github.com> Date: Tue, 15 Apr 2025 17:34:57 -0700 Subject: [PATCH 3/4] Update documentation for amortized inference example --- docs/src/neuralestimators_amorized.md | 286 ++++++++++++++++++++++---- 1 file changed, 244 insertions(+), 42 deletions(-) diff --git a/docs/src/neuralestimators_amorized.md b/docs/src/neuralestimators_amorized.md index 23a33552..3c68aae7 100644 --- a/docs/src/neuralestimators_amorized.md +++ b/docs/src/neuralestimators_amorized.md @@ -6,11 +6,11 @@ Below, we demonstrate how to estimate parameters of the LCA model using the [Neu ## Example -We'll estimate parameters of the LCA model, which is particularly challenging due to its complex dynamics, where parameters like leak rate (λ) and lateral inhibition (β) can be difficult to recover. This example draws from a more in-depth case that highlights many of the steps one ought to consider when utilizing amortized inference for cognitive modeling; see [Principled Amortized Bayesian Workflow for Cognitive Modeling](https://bayesflow.org/stable-legacy/_examples/Amortized_Bayesian_Workflow_for_Cognitive_Modeling.html). +We'll estimate parameters of the LCA model, which is particularly challenging due to its complex dynamics, where parameters like leak rate (λ) and lateral inhibition (β) can be difficult to recover. This example draws from a more in-depth case that highlights many of the steps one ought to consider when utilizing amortized inference for cognitive modeling; see [Principled Amortized Bayesian Workflow for Cognitive Modeling](https://bayesflow.org/stable-legacy/_examples/LCA_Model_Posterior_Estimation.html). ## Load Packages -```@example +```julia using NeuralEstimators using SequentialSamplingModels using Flux @@ -22,30 +22,19 @@ using Plots Random.seed!(123) ``` -## Define Parameter Bounds - -```@example -# Define parameter bounds for the LCA model -const ν_min, ν_max = 0.1, 4.0 # Drift rates -const α_min, α_max = 0.5, 3.5 # Threshold -const β_min, β_max = 0.0, 0.5 # Lateral inhibition -const λ_min, λ_max = 0.0, 0.5 # Leak rate -const τ_min, τ_max = 0.1, 0.5 # Non-decision time -``` - ## Define Parameter Sampling -Unlike traditional Bayesian approaches, simulation based inference methods require us to define a prior sampling function to generate training data. We will use this function to sample a range of parameters for training: +Unlike traditional Bayesian inference methods, simulation-based inference approaches require us to define a prior sampling function specifically to generate synthetic training data. While traditional methods like MCMC also sample from the prior, those samples are used directly during inference rather than to create a separate training dataset. In SBI, we use the prior to sample a wide range of parameters and simulate corresponding data, which we then use to train a model (e.g., a neural network) to approximate the posterior. We will use the following function to sample a range of parameters for training: -```@example +```julia # Function to sample parameters from priors function sample(K::Integer) - ν1 = rand(Gamma(2, 1/1.2), K) # Drift rate 1 - ν2 = rand(Gamma(2, 1/1.2), K) # Drift rate 2 - α = rand(Gamma(10, 1/6), K) # Threshold - β = rand(Beta(1, 5), K) # Lateral inhibition - λ = rand(Beta(1, 5), K) # Leak rate - τ = rand(Gamma(1.5, 1/5.0), K) # Non-decision time + ν1 = rand(Gamma(2, 1/1.2f0), K) # Drift rate 1 + ν2 = rand(Gamma(2, 1/1.2f0), K) # Drift rate 2 + α = rand(Gamma(10, 1/6f0), K) # Threshold + β = rand(Beta(1, 5f0), K) # Lateral inhibition + λ = rand(Beta(1, 5f0), K) # Leak rate + τ = rand(Gamma(1.5, 1/5.0f0), K) # Non-decision time # Stack parameters into a matrix (d×K) θ = vcat(ν1', ν2', α', β', λ', τ') @@ -58,7 +47,7 @@ end Neural estimators learn the mapping from data to parameters through simulation. Here we define a function to simulate LCA model data. To do so we will use the [LCA](https://itsdfish.github.io/SequentialSamplingModels.jl/dev/lca/). -```@example +```julia # Function to simulate data from the LCA model function simulate(θ, n_trials_per_param) # Simulate data for each parameter vector @@ -74,7 +63,7 @@ function simulate(θ, n_trials_per_param) choices, rts = rand(model, n_trials_per_param) # Return as a transpose matrix where each column is a trial - return Float32.([choices rts]') + return [choices rts]' end return simulated_data @@ -83,11 +72,24 @@ end ## Define Neural Network Architecture -For LCA parameter recovery, we use a DeepSet architecture which respects the permutation invariance of trial data. For more details on the method see NeuralEstimators.jl documentation. To construct the network architecture we will use the Flux.jl package. +For LCA parameter recovery, we use a DeepSet architecture which respects the permutation invariance of trial data. For more details on the method [see NeuralEstimators.jl documentation](https://msainsburydale.github.io/NeuralEstimators.jl/dev/API/architectures/#NeuralEstimators.DeepSet). To construct the network architecture we will use the Flux.jl package. -```@example +```julia # Create neural network architecture for parameter recovery -function create_neural_estimator() +function create_neural_estimator(; + ν_bounds = (0.1, 4.0), + α_bounds = (0.5, 3.5), + β_bounds = (0.0, 0.5), + λ_bounds = (0.0, 0.5), + τ_bounds = (0.1, 0.5) +) + # Unpack defined parameter Bounds + ν_min, ν_max = ν_bounds # Drift rates + α_min, α_max = α_bounds # Threshold + β_min, β_max = β_bounds # Lateral inhibition + λ_min, λ_max = λ_bounds # Leak rate + τ_min, τ_max = τ_bounds # Non-decision time + # Input dimension: 2 (choice and RT for each trial) n = 2 # Output dimension: 6 parameters @@ -132,25 +134,25 @@ end ## Training the Neural Estimator -Neural estimators, like all deep learning methods, require a training phase where they learn the mapping from data to parameters. Here we will train the estimator, simulating data as we go where the sampler provides new parameter vectors from the prior, and a simulator can be provided to continuously simulate new data conditional on the parameters. For more details on the training see the API for arguments [here](https://msainsburydale.github.io/NeuralEstimators.jl/dev/API/core/#Training). +Neural estimators, like all deep learning methods, require a training phase during which they learn the mapping from data to parameters. Here, we train the estimator by simulating data on the fly: the sampler provides new parameter vectors from the prior, and the simulator generates corresponding data conditional on those parameters. Since we use online training and the network never sees the same simulated dataset twice, overfitting is less likely. For more details on training, see the API for arguments [here](https://msainsburydale.github.io/NeuralEstimators.jl/dev/API/core/#Training). -```@example +```julia # Create the neural estimator estimator = create_neural_estimator() # Train network trained_estimator = train( estimator, - sample, # Parameter sampler function - simulate, # Data simulator function - m = 100, # Number of trials per parameter vector - K = 10000, # Number of training parameter vectors - K_val = 2000, # Number of validation parameter vectors - loss = Flux.mae, # Mean absolute error loss - epochs = 50, # Number of training epochs - epochs_per_Z_refresh = 1, # Refresh data every epoch - epochs_per_θ_refresh = 5, # Refresh parameters every 5 epochs - batchsize = 64, # Batch size for training + sample, # Parameter sampler function + simulate, # Data simulator function + m = 100, # Number of trials per parameter vector + K = 10000, # Number of training parameter vectors + K_val = 2000, # Number of validation parameter vectors + loss = Flux.mae, # Mean absolute error loss + epochs = 60, # Number of training epochs + epochs_per_Z_refresh = 1, # Refresh data every epoch + epochs_per_θ_refresh = 5, # Refresh parameters every 5 epochs + batchsize = 16, # Batch size for training verbose = true ) ``` @@ -159,7 +161,7 @@ trained_estimator = train( We can assess the performance of our trained estimator on held-out test data: -```@example +```julia # Generate test data n_test = 100 θ_test = sample(n_test) @@ -183,9 +185,9 @@ println("RMSE: ", rmse_results) ## Visualizing Parameter Recovery -A key advantage of neural estimation is the ability to quickly conduct inference after training. For example, we can visualize the recovery of parameters: +A key advantage of neural estimation is the ability to quickly conduct inference after training. For example, we can visualize the recovery of parameters. While NeuralEstimators provides built-in visualization capabilities through the [AlgebraOfGraphics.jl](https://github.com/MakieOrg/AlgebraOfGraphics.jl), we will demonstrate custom plotting below: -```@example +```julia # Extract data from assessment df = assessment.df @@ -219,7 +221,7 @@ display(p_combined) Once trained, the estimator can instantly recover parameters from new data via a forward pass: -```@example +```julia # Generate "observed" data ν = [2.5, 2.0] α = 1.5 @@ -249,6 +251,206 @@ Neural estimators are particularly effective for models with computationally int Additional details can be found in the [NeuralEstimators.jl documentation](https://github.com/msainsburydale/NeuralEstimators). +# Complete Code +```@raw html +
+Show Details +``` +```julia +using NeuralEstimators +using SequentialSamplingModels +using Flux +using Distributions +using Random +using Plots + +Random.seed!(123) + +# Function to sample parameters from priors +function sample(K::Integer) + ν1 = rand(Gamma(2, 1/1.2f0), K) # Drift rate 1 + ν2 = rand(Gamma(2, 1/1.2f0), K) # Drift rate 2 + α = rand(Gamma(10, 1/6f0), K) # Threshold + β = rand(Beta(1, 5f0), K) # Lateral inhibition + λ = rand(Beta(1, 5f0), K) # Leak rate + τ = rand(Gamma(1.5, 1/5.0f0), K) # Non-decision time + + # Stack parameters into a matrix (d×K) + θ = vcat(ν1', ν2', α', β', λ', τ') + + return θ +end + +# Function to simulate data from the LCA model +function simulate(θ, n_trials_per_param) + # Simulate data for each parameter vector + simulated_data = map(eachcol(θ)) do param + # Extract parameters for this model + ν1, ν2, α, β, λ, τ = param + ν = [ν1, ν2] # Two-choice LCA + + # Create LCA model with SSM + model = LCA(; ν, α, β, λ, τ) + + # Generate choices and reaction times + choices, rts = rand(model, n_trials_per_param) + + # Return as a transpose matrix where each column is a trial + return [choices rts]' + + end + + return simulated_data +end + +# Create neural network architecture for parameter recovery +function create_neural_estimator(; + ν_bounds = (0.1, 4.0), + α_bounds = (0.5, 3.5), + β_bounds = (0.0, 0.5), + λ_bounds = (0.0, 0.5), + τ_bounds = (0.1, 0.5) +) + # Unpack defined parameter Bounds + ν_min, ν_max = ν_bounds # Drift rates + α_min, α_max = α_bounds # Threshold + β_min, β_max = β_bounds # Lateral inhibition + λ_min, λ_max = λ_bounds # Leak rate + τ_min, τ_max = τ_bounds # Non-decision time + + # Input dimension: 2 (choice and RT for each trial) + n = 2 + # Output dimension: 6 parameters + d = 6 # ν[1], ν[2], α, β, λ, τ + # Width of hidden layers + w = 128 + + # Inner network - processes each trial independently + ψ = Chain( + Dense(n, w, relu), + Dense(w, w, relu), + Dense(w, w, relu) + ) + + # Final layer with parameter constraints + final_layer = Parallel( + vcat, + Dense(w, 1, x -> ν_min + (ν_max - ν_min) * σ(x)), # ν1 + Dense(w, 1, x -> ν_min + (ν_max - ν_min) * σ(x)), # ν2 + Dense(w, 1, x -> α_min + (α_max - α_min) * σ(x)), # α + Dense(w, 1, x -> β_min + (β_max - β_min) * σ(x)), # β + Dense(w, 1, x -> λ_min + (λ_max - λ_min) * σ(x)), # λ + Dense(w, 1, x -> τ_min + (τ_max - τ_min) * σ(x)) # τ + ) + + # Outer network - maps aggregated features to parameters + ϕ = Chain( + Dense(w, w, relu), + Dense(w, w, relu), + final_layer + ) + + # Combine into a DeepSet + network = DeepSet(ψ, ϕ) + + # Initialize neural Bayes estimator + estimator = PointEstimator(network) + + return estimator +end + +# Create the neural estimator +estimator = create_neural_estimator() + +# Train network +trained_estimator = train( + estimator, + sample, # Parameter sampler function + simulate, # Data simulator function + m = 100, # Number of trials per parameter vector + K = 10000, # Number of training parameter vectors + K_val = 2000, # Number of validation parameter vectors + loss = Flux.mae, # Mean absolute error loss + epochs = 60, # Number of training epochs + epochs_per_Z_refresh = 1, # Refresh data every epoch + epochs_per_θ_refresh = 5, # Refresh parameters every 5 epochs + batchsize = 16, # Batch size for training + verbose = true +) + +# Generate test data +n_test = 100 +θ_test = sample(n_test) +Z_test = simulate(θ_test, 100) + +# Assess the estimator +parameter_names = ["ν1", "ν2", "α", "β", "λ", "τ"] +assessment = assess( + trained_estimator, + θ_test, + Z_test; + parameter_names = parameter_names +) + +# Calculate performance metrics +bias_results = bias(assessment) +rmse_results = rmse(assessment) +println("Bias: ", bias_results) +println("RMSE: ", rmse_results) + +# Extract data from assessment +df = assessment.df + +# Create recovery plots for each parameter +params = unique(df.parameter) +p_plots = [] + +for param in params + param_data = filter(row -> row.parameter == param, df) + p = scatter( + param_data.truth, + param_data.estimate, + xlabel="Ground Truth", + ylabel="Estimated", + title=param, + legend=false + ) + plot!(p, [minimum(param_data.truth), maximum(param_data.truth)], + [minimum(param_data.truth), maximum(param_data.truth)], + line=:dash, color=:black) + push!(p_plots, p) +end + +# Combine plots +p_combined = plot(p_plots..., layout=(3,2), size=(800, 600)) +display(p_combined) + +# Generate "observed" data +ν = [2.5, 2.0] +α = 1.5 +β = 0.2 +λ = 0.1 +τ = 0.3 +σ = 1.0 + +# Create model and generate data +true_model = LCA(; ν, α, β, λ, τ) +observed_choices, observed_rts = rand(true_model, 100) + +# Format the data +observed_data = Float32.([observed_choices observed_rts]') + +# Recover parameters +recovered_params = NeuralEstimators.estimate(trained_estimator, [observed_data]) + +# Compare true and recovered parameters +println("True parameters: ", [ν[1], ν[2], α, β, λ, τ]) +println("Recovered parameters: ", recovered_params) +``` +```@raw html +
+``` + # References Miletić, S., Turner, B. M., Forstmann, B. U., & van Maanen, L. (2017). Parameter recovery for the leaky competing accumulator model. Journal of Mathematical Psychology, 76, 25-50. 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