### Configure Working Memory Model Estimation (MATLAB)

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

The Configuration_BMW function sets up parameters for model estimation and assessment. It requires a structure defining the model, data source, fitting options (method, algorithm, display, RFXBMS), criteria for assessment, and optional parameter constraints (start values, bounds). It returns a structured configuration object for model fitting.

```matlab
% Define model and data
MA.Model.Model = 'Variable Precision';
MA.Model.Variants = {'ContinuousK', 'ResponseNoise'};
MA.Data = 'path/to/data.mat';  % Or pass cell array directly

% Set fit options
MA.FitOptions.Method = 'MAP';  % 'MAP' or 'MLE'
MA.FitOptions.Algorithm = 'DE-MCMC';  % Default algorithm
MA.FitOptions.Display = 'iter';  % 'iter', 'detail', 'off'
MA.FitOptions.RFXBMS = 1;  % Enable random effects BMS

% Define criteria for model assessment
MA.Criteria = {'LLH', 'AIC', 'BIC', 'DIC2', 'WAIC2', 'LME_GHM'};

% Define parameter constraints (optional - auto-configured if omitted)
MA.Constraints.start = [50, 20, 2, 100];  % Initial values
MA.Constraints.lb = [0, 0, 0, 0];  % Lower bounds
MA.Constraints.ub = [700, 200, 10, 700];  % Upper bounds

% Run configuration
MA_Config = Configuration_BMW(MA);
```

--------------------------------

### MATLAB: Complete Working Memory Model Fitting and Comparison Workflow

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

This MATLAB script demonstrates a full workflow for fitting and comparing Bayesian working memory models. It requires the BMW Toolbox and experimental data in a specific format. The script defines a model space, fits each model using DE-MCMC, and performs model comparison using criteria like BIC and RFX-BMS.

```matlab
%% Setup
close all; clear variables;
addpath('path/to/BMW_Toolbox');
BMW('AddPath');

%% Prepare Data
% Data structure requirements:
% - Data.sample: Target stimulus values (e.g., orientations in degrees)
% - Data.response: Subject responses
% - Data.SS: Set size for each trial
% - Data.error_range: Range of possible errors

% Load experimental data (cell array, one per subject)
load('experiment_data.mat');  % Contains Data_All cell array

%% Define Model Space
ModelSpace = {
    'Item Limit',
    'Standard Mixture',
    'Slots-plus-Averaging',
    'Variable Precision',
    'Variable Precision with Capacity'
};

%% Fit All Models
MA_All = cell(1, length(ModelSpace));
for i = 1:length(ModelSpace)
    fprintf('\n=== Fitting Model: %s ===\n', ModelSpace{i});

    % Configure
    MA.Model.Model = ModelSpace{i};
    MA.Model.Variants = {'ContinuousK'};  % Optional variants
    MA.Data = Data_All;

    % Fit options
    MA.FitOptions.Method = 'MAP';
    MA.FitOptions.Algorithm = 'DE-MCMC';
    MA.FitOptions.Display = 'iter';
    MA.FitOptions.RFXBMS = 1;

    % Criteria
    MA.Criteria = {'LLH', 'AIC', 'BIC', 'DIC2', 'WAIC2', 'LME_GHM'};

    % Run
    MA = Configuration_BMW(MA);
    MA_All{i} = ModelFit_BMW(MA);
end

%% Compare Models
MC = ModelComparison_BMW(MA_All);

%% Display Results
fprintf('\n=== Model Comparison Results ===\n');
fprintf('\nBIC-based Group Posterior Probabilities:\n');
for i = 1:length(ModelSpace)
    fprintf('%s: %.3f\n', ModelSpace{i}, MC{1}.BIC.Group_PP(i));
end

fprintf('\nRFX-BMS Exceedance Probabilities:\n');
for i = 1:length(ModelSpace)
    fprintf('%s: %.3f\n', ModelSpace{i}, MC{1}.BIC.EP(i));
end

%% Save Results
save('ModelComparison_Results.mat', 'MA_All', 'MC');

%% Cleanup
BMW('ClearPath');

```

--------------------------------

### Launch and Manage BMW Toolbox Paths (MATLAB)

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

The main BMW function serves as an entry point for launching the graphical user interface (GUI), managing MATLAB paths for the toolbox, checking available optimization methods, and opening the user manual. It takes string arguments to specify the desired action.

```matlab
% Launch the BMW GUI
BMW();

% Add BMW paths to MATLAB path
BMW('AddPath');

% Remove BMW paths
BMW('ClearPath');

% Check available optimization algorithms
BMW('CheckMethods');

% Open the manual
BMW('Manual');
```

--------------------------------

### Core Model Fitting Engine for Working Memory Models (MATLAB)

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

BMW_Fit is the low-level engine for fitting working memory models using various optimization algorithms. It requires data, configuration settings, the model name, parameter constraints, and fit options. It returns the fitted parameters and quality metrics, including posterior samples if requested. Supported algorithms include DE-MCMC, MH-MCMC, fmincon, BADS, GA, and SA.

```matlab
% Prepare data structure
Data.sample = [45, 90, 135, 180];  % Sample orientations (degrees)
Data.response = [47, 88, 140, 175];  % Response orientations
Data.SS = [2, 2, 4, 4];  % Set sizes
Data.error_range = -90:1:90;  % Error range

% Configure model
Config.Output = 'LP';  % Log posterior
Config.Variants = {'ResponseNoise'};

% Set constraints
Constraints.start = [50, 20, 2, 100];  % [kappa1_bar, tau, power, kappa_r]
Constraints.lb = [0, 0, 0, 0];
Constraints.ub = [700, 200, 10, 700];

% Set fit options
FitOptions.Algorithm = 'DE-MCMC';  % Options: 'DE-MCMC', 'MH-MCMC', 'fmincon: sqp', 'BADS', 'GA', 'SA'
FitOptions.Display = 'iter';

% Fit model
[Param, Quality] = BMW_Fit(Data, Config, 'Variable_Precision', Constraints, FitOptions);

% Results
disp(['Fitted parameters: ', num2str(Param)]);
disp(['Log posterior: ', num2str(Quality.Output)]);
mcmc_result = Quality.MCMCResult;  % Contains posterior samples
```

--------------------------------

### Fit and Assess Working Memory Models (MATLAB)

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

ModelFit_BMW fits a specified working memory model to data for multiple subjects and calculates information criteria for assessment. It takes a pre-configured structure from Configuration_BMW and returns results including fitted parameters, subject-wise criteria values, and MCMC posterior samples. Dependencies include the configuration structure and the underlying BMW fitting functions.

```matlab
% Setup model configuration
MA.Model.Model = 'Standard Mixture';
MA.Data = {SubjectData1, SubjectData2, SubjectData3};  % Cell array of subject data
MA.FitOptions.Method = 'MAP';
MA.FitOptions.Algorithm = 'DE-MCMC';
MA.Criteria = {'AIC', 'BIC', 'DIC2', 'WAIC2'};

% Configure and fit
MA_Config = Configuration_BMW(MA);
MA_Results = ModelFit_BMW(MA_Config);

% Access results
fitted_params = MA_Results.Param;  % Nsubj x Nparam matrix
aic_values = MA_Results.AIC;  % AIC for each subject
bic_values = MA_Results.BIC;  % BIC for each subject
mcmc_samples = MA_Results.MCMC;  % MCMC posterior samples
```

--------------------------------

### Calculate Information Criteria with BMW_GetIC_MCMC in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Calculates various information criteria (DIC, WAIC, Log Model Evidence) from MCMC posterior samples. It supports multiple estimation methods and requires the MCMC raw sampling output, model, data, and a configuration structure specifying the IC method.

```matlab
% After MCMC sampling
[RawSampling, ~] = BMW_MCMC(model, Data, config);

% Configure IC method
Method.IC = 'LME_BridgeSampling';  % Options: 'DIC1', 'DIC2', 'DIC*', 'WAIC1', 'WAIC2', 'LME_HarmonicMean', 'LME_BridgeSampling'
Method.Verbosity = 1;  % 0=off, 1=progress, 2=detailed
Method.Transform = 'Probit';

% Calculate DIC1
Method.IC = 'DIC1';
DIC1 = BMW_GetIC_MCMC(RawSampling, model, Data, Method);

% Calculate DIC2 (variance-based penalty)
Method.IC = 'DIC2';
DIC2 = BMW_GetIC_MCMC(RawSampling, model, Data, Method);

% Calculate WAIC1 (pointwise deviance penalty)
Method.IC = 'WAIC1';
WAIC1 = BMW_GetIC_MCMC(RawSampling, model, Data, Method);

% Calculate WAIC2 (variance-based penalty)
Method.IC = 'WAIC2';
WAIC2 = BMW_GetIC_MCMC(RawSampling, model, Data, Method);

% Calculate Log Model Evidence using Harmonic Mean
Method.IC = 'LME_HarmonicMean';
LME_GHM = BMW_GetIC_MCMC(RawSampling, model, Data, Method);

% Calculate Log Model Evidence using Bridge Sampling (recommended)
Method.IC = 'LME_BridgeSampling';
LME_BS = BMW_GetIC_MCMC(RawSampling, model, Data, Method);
```

--------------------------------

### Perform MCMC Sampling with BMW_MCMC in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Configures and runs Monte Carlo Markov Chain (MCMC) sampling for Bayesian inference using either Differential Evolution (DE) or Metropolis-Hastings (MH) algorithms. It requires model configuration, data, and sampling parameters, outputting raw samples and summary statistics.

```matlab
% Configure model
model.Model = 'Variable_Precision';
model.Constraints.start = repmat([50, 20, 2], 4, 1);  % 4 chains
model.Constraints.lb = [0, 0, 0];
model.Constraints.ub = [700, 200, 10];
model.Output = 'LP';

% Configure MCMC
config.Algorithm = 'DE';  % 'DE' or 'MH'
config.Nsample = 10000;  % Number of samples after burn-in
config.Verbosity = 'iter';  % 'off', 'iter', 'notify', 'final'
config.Transform = 'Probit';  % Parameter transformation
config.Convergence.Diagnostic = 'GR';  % Gelman-Rubin diagnostic
config.Convergence.Nbatchburnin = 200;  % Burn-in samples per batch
config.Convergence.Nmaxbatchburnin = 50;  % Max burn-in batches
config.Convergence.Tol = 0.2;  % Convergence tolerance

% Run MCMC
[RawSampling, Summary] = BMW_MCMC(model, Data, config);

% Access results
posterior_samples = RawSampling.Samples;  % N x Nparam matrix
log_posterior = RawSampling.logPosterior;  % Log posterior values
log_likelihood = RawSampling.logLikelihood;  % Log likelihood values
best_params = Summary.FitParam;  % Best fitting parameters
max_posterior = Summary.MAXposterior;  % Maximum posterior value
```

--------------------------------

### Variable Precision Model Implementation in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Implements the Variable Precision model where memory precision varies trial-by-trial following a gamma distribution. Precision declines as a power function of set size. Supports various model variants like 'ResponseNoise'.

```matlab
% Model parameters: [kappa1_bar, tau, power, kappa_r]
% kappa1_bar: Mean precision at set size 1
% tau: Resource allocation variability
% power: Set size effect exponent
% kappa_r: Response noise (if ResponseNoise variant)
param = [100, 20, 2, 50];

% Prepare data
Data.sample = [45, 90, 135, 180, 225, 270];  % Sample orientations
Data.response = [43, 92, 130, 182, 220, 275];  % Responses
Data.SS = [2, 2, 4, 4, 6, 6];  % Set sizes
Data.error_range = [-90, 90];  % Continuous mode

% Configure model
Input.Output = 'LLH';  % 'LLH', 'LP', 'Prior', 'LPPD', 'All'
Input.Variants = {'ResponseNoise'};

% Calculate log likelihood
LLH = Variable_Precision(param, Data, Input);

% Get probability density function
Data.error_range = -90:1:90;  % Discrete mode for PDF
Input.Output = 'LPPD';
Input.PDF = 1;
PDF_result = Variable_Precision(param, Data, Input);

% Available variants:
% 'ResponseNoise' - Add response noise parameter
% 'Bias' - Add systematic response bias
% 'BiasF' - Add stimulus-dependent bias fluctuation
% 'PrecF' - Add stimulus-dependent precision fluctuation
% 'Swap' - Add swap errors (misbinding)
% 'ContinuousK' - Continuous capacity parameter
% 'Category (Within-Item)' - Categorical encoding within items
% 'Category (Between-Item)' - Categorical encoding between items
```

--------------------------------

### Compare Models with ModelComparison_BMW in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Compares multiple fitted Bayesian models using criteria like BIC, DIC, WAIC, and Log Model Evidence. It computes Bayes factors, model weights, and supports random-effects Bayesian model selection. The function takes a cell array of fitted models and optionally a family grouping.

```matlab
% Fit multiple models (see SampleScript1 pattern)
ModelSpace = {'Item Limit', 'Standard Mixture', 'Slots-plus-Averaging', ...
              'Variable Precision', 'Variable Precision with Capacity'};

MA_All = cell(1, length(ModelSpace));
for i = 1:length(ModelSpace)
    MA.Model.Model = ModelSpace{i};
    MA.Data = ExperimentalData;  % Cell array of subject data
    MA.FitOptions.Method = 'MAP';
    MA.Criteria = {'BIC', 'DIC2', 'WAIC2', 'LME_GHM'};

    MA = Configuration_BMW(MA);
    MA_All{i} = ModelFit_BMW(MA);
end

% Compare models
MC = ModelComparison_BMW(MA_All);

% Access comparison results (for each criterion)
best_model_bic = MC{1}.BIC.BestModel;  % Best model per subject
bic_weights = MC{1}.BIC.BIC_weight;  % Model weights per subject
group_posterior = MC{1}.BIC.Group_PP;  % Group-level posterior probability
bayes_factors = MC{1}.BIC.BIC_GBF;  % Group Bayes factors

% Random-effects BMS results (if enabled)
model_frequency = MC{1}.BIC.ModelFreq;  % Expected model frequency
exceedance_prob = MC{1}.BIC.EP;  % Exceedance probability

% Compare model families
family = [1, 1, 1, 2, 2];  % Group models into families
MC_Family = ModelComparison_BMW(MA_All, family);
family_freq = MC_Family{1}.BIC.FamilyFreq;
family_ep = MC_Family{1}.BIC.EP_Family;
```

--------------------------------

### Bayesian Model Selection with BMW_BMS in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Performs hierarchical random-effects Bayesian model selection across subjects using variational Bayes approximation. It estimates expected model frequencies and exceedance probabilities from a Log Model Evidence matrix.

```matlab
% Prepare Log Model Evidence matrix (Nmodel x Nsubject)
LME = zeros(5, 20);  % 5 models, 20 subjects
for model = 1:5
    LME(model, :) = MA_All{model}.LME_GHM';  % Or use BIC: -MA_All{model}.BIC'/2
end

% Configure BMS
Config.Start = 1e-6 * ones(5, 1);  % Dirichlet prior parameters
Config.MaxIter = 1e6;  % Maximum iterations
Config.Stop = 1e-6;  % Convergence criterion
Config.Verbosity = 1;  % Display progress
Config.Rec = 1;  % Record iteration history

% Run BMS
BMS_Results = BMW_BMS(LME, Config);

% Access results
dirichlet_params = BMS_Results.a;  % Posterior Dirichlet parameters
model_frequency = BMS_Results.r;  % Expected model frequency
exceedance_prob = BMS_Results.EP;  % Exceedance probability
iteration_history = BMS_Results.RecIter;  % Parameter history (if Rec=1)

% Display results
disp('Model Frequencies:');
disp(model_frequency);
disp('Exceedance Probabilities:');
disp(exceedance_prob);
```

--------------------------------

### Circular Statistics Calculation with CircSummary_BMW in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Computes essential circular statistics for angular data, commonly used in working memory experiments. Functions include calculating circular standard deviation, mean, and variance, handling data periodicity.

```matlab
% Calculate circular standard deviation
error_data = [-10, 5, -3, 8, -15, 20, 2, -7];  % Response errors in degrees
period = 180;  % Data period (180 for orientations, 360 for colors)
circular_sd = CircSummary_BMW('CircSD', error_data, period);

% Calculate other circular statistics
circular_mean = CircSummary_BMW('CircMean', error_data, period);
circular_var = CircSummary_BMW('CircVar', error_data, period);

disp(['Circular SD: ', num2str(circular_sd), ' degrees']);
```

--------------------------------

### Circular Distance Calculation with CircDist_BMW in MATLAB

Source: https://context7.com/nian-jingqing/bayesian_modeling_of_working_memory/llms.txt

Computes circular distances between angles, correctly handling the wrap-around nature of periodic data. It can calculate both direct circular differences and absolute circular distances.

```matlab
% Calculate circular difference (errors)
samples = [45, 90, 135, 180];
responses = [50, 85, 140, 175];
period = 180;  % For orientation data

errors = CircDist_BMW('Diff', responses, samples, period);
% Returns: [5, -5, 5, -5]

% Calculate absolute circular distance
abs_errors = CircDist_BMW('Abs', responses, samples, period);
```

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