-module(mlx_causal). -export([ % Causal Discovery pc_algorithm/2, ges_algorithm/2, fast_causal_inference/2, causal_discovery_with_latents/2, constraint_based_discovery/3, score_based_discovery/3, hybrid_causal_discovery/3, % Causal Estimation instrumental_variables/3, regression_discontinuity/3, difference_in_differences/4, synthetic_control/4, causal_forests/4, double_ml/4, targeted_ml/4, % Causal Deep Learning causal_vae/3, causal_gan/4, causal_transformer/3, neural_causal_model/4, deep_structural_model/4, causal_representation_learning/3, counterfactual_generator/3, % Interventional Inference do_calculus/3, backdoor_adjustment/3, frontdoor_adjustment/3, mediation_analysis/4, moderation_analysis/4, causal_effect_estimation/4, attribution_analysis/3, % Counterfactual Reasoning counterfactual_inference/4, closest_world_counterfactuals/3, structural_counterfactuals/4, probabilistic_counterfactuals/3, contrastive_explanation/4, causal_explanation/3, % Causal Reinforcement Learning causal_policy_learning/4, confounded_bandits/4, causal_meta_learning/4, offline_causal_rl/4, invariant_risk_minimization/3, domain_adaptation_causal/4, % Time Series Causal Analysis granger_causality/3, var_causal_analysis/3, causal_time_series_forecasting/4, dynamic_causal_modeling/4, temporal_causal_discovery/3, regime_switching_causal/4, % Advanced Causal Methods quantum_causal_models/3, probabilistic_causal_programming/3, causal_graph_neural_networks/4, federated_causal_learning/4, causal_fairness_analysis/4, robust_causal_inference/4, % Evaluation and Validation causal_model_validation/3, sensitivity_analysis/4, causal_benchmark_suite/2, intervention_simulation/4, causal_discovery_metrics/3, treatment_effect_validation/4 ]). %% Revolutionary Causal Discovery Algorithms pc_algorithm(Data, Alpha) -> % Peter-Clark algorithm for causal structure learning % Uses conditional independence tests to discover causal graphs Variables = get_variables(Data), Graph = initialize_complete_graph(Variables), % Skeleton discovery phase SkeletonGraph = discover_skeleton(Data, Graph, Alpha), % Orientation phase using v-structures OrientedGraph = orient_edges(Data, SkeletonGraph, Alpha), % Final orientation rules FinalGraph = apply_orientation_rules(OrientedGraph), #{graph => FinalGraph, statistics => compute_discovery_stats(Data, FinalGraph), confidence => compute_edge_confidence(Data, FinalGraph, Alpha)}. ges_algorithm(Data, Penalties) -> % Greedy Equivalence Search for causal discovery % Score-based approach using BIC or other scoring functions InitialGraph = empty_graph(get_variables(Data)), % Forward phase - add edges ForwardGraph = ges_forward_phase(Data, InitialGraph, Penalties), % Backward phase - remove edges BackwardGraph = ges_backward_phase(Data, ForwardGraph, Penalties), % Turn phase - orient edges FinalGraph = ges_turn_phase(Data, BackwardGraph, Penalties), #{graph => FinalGraph, score => compute_graph_score(Data, FinalGraph, Penalties), equivalence_class => compute_equivalence_class(FinalGraph)}. fast_causal_inference(Data, Options) -> % GPU-accelerated causal discovery using advanced algorithms #{algorithm := Algorithm, parallel := Parallel} = Options, case Algorithm of pc_parallel -> parallel_pc_algorithm(Data, Options, Parallel); ges_cuda -> cuda_ges_algorithm(Data, Options); neural_causal -> neural_causal_discovery(Data, Options); quantum_causal -> quantum_causal_algorithm(Data, Options) end. %% Advanced Causal Estimation Methods instrumental_variables(Treatment, Outcome, Instrument) -> % Two-stage least squares with weak instrument tests FirstStage = regress(Treatment, Instrument), WeakInstrumentTest = weak_instrument_test(FirstStage), PredictedTreatment = predict(FirstStage, Instrument), SecondStage = regress(Outcome, PredictedTreatment), CausalEffect = extract_coefficient(SecondStage), StandardError = compute_robust_se(SecondStage, FirstStage), #{causal_effect => CausalEffect, standard_error => StandardError, weak_instrument_test => WeakInstrumentTest, f_statistic => compute_f_statistic(FirstStage)}. double_ml(Treatment, Outcome, Confounders, MLModels) -> % Double Machine Learning for causal inference % Uses cross-fitting to avoid overfitting bias #{treatment_model := TModel, outcome_model := OModel} = MLModels, % Cross-fitting procedure Folds = create_cross_fitting_folds(length(Treatment), 5), Results = lists:map(fun(Fold) -> {TrainIdx, TestIdx} = Fold, % Fit models on training data TreatmentModel = fit_ml_model(TModel, Confounders, Treatment, TrainIdx), OutcomeModel = fit_ml_model(OModel, Confounders, Outcome, TrainIdx), % Predict on test data TreatmentResiduals = compute_residuals(TreatmentModel, Treatment, TestIdx), OutcomeResiduals = compute_residuals(OutcomeModel, Outcome, TestIdx), % Estimate causal effect estimate_ate(TreatmentResiduals, OutcomeResiduals) end, Folds), aggregate_double_ml_results(Results). %% Causal Deep Learning causal_vae(Data, TreatmentVar, LatentDim) -> % Causal Variational Autoencoder for representation learning EncoderArch = #{ layers => [ #{type => dense, units => 512, activation => relu}, #{type => dense, units => 256, activation => relu}, #{type => dense, units => LatentDim * 2, activation => linear} % mean and logvar ] }, DecoderArch = #{ layers => [ #{type => dense, units => 256, activation => relu}, #{type => dense, units => 512, activation => relu}, #{type => dense, units => mlx:size(Data, 2), activation => sigmoid} ] }, % Causal regularization terms CausalLoss = fun(Z, T, Y) -> ReconstructionLoss = reconstruction_loss(Data, decode(Z)), KLDivergence = kl_divergence_loss(Z), CausalRegularization = causal_regularization_term(Z, T, Y), ReconstructionLoss + KLDivergence + CausalRegularization end, train_causal_vae(Data, TreatmentVar, EncoderArch, DecoderArch, CausalLoss). neural_causal_model(Features, Treatment, Outcome, Architecture) -> % Deep neural network for causal effect estimation % Incorporates causal assumptions into network structure RepresentationNetwork = #{ layers => [ #{type => dense, units => 512, activation => relu, dropout => 0.2}, #{type => dense, units => 256, activation => relu, dropout => 0.2}, #{type => dense, units => 128, activation => relu} ] }, TreatmentNetwork = #{ layers => [ #{type => dense, units => 64, activation => relu}, #{type => dense, units => 1, activation => sigmoid} ] }, OutcomeNetwork = #{ layers => [ #{type => dense, units => 64, activation => relu}, #{type => dense, units => 1, activation => linear} ] }, % Training with adversarial balance train_neural_causal_model(Features, Treatment, Outcome, RepresentationNetwork, TreatmentNetwork, OutcomeNetwork). %% Counterfactual Reasoning counterfactual_inference(StructuralModel, Intervention, Evidence, Query) -> % Three-step counterfactual inference procedure % 1. Abduction: infer unobserved variables % 2. Action: modify model according to intervention % 3. Prediction: compute query under modified model % Step 1: Abduction UnobservedVars = abduction_step(StructuralModel, Evidence), % Step 2: Action (intervention) ModifiedModel = apply_intervention(StructuralModel, Intervention), % Step 3: Prediction CounterfactualOutcome = prediction_step(ModifiedModel, UnobservedVars, Query), #{counterfactual_outcome => CounterfactualOutcome, probability => compute_counterfactual_probability(CounterfactualOutcome), explanation => generate_counterfactual_explanation(StructuralModel, Intervention, CounterfactualOutcome)}. closest_world_counterfactuals(Model, FactualWorld, CounterfactualQuery) -> % Find closest possible world where counterfactual holds % Uses similarity metrics and constraint satisfaction PossibleWorlds = generate_possible_worlds(Model), SimilarityScores = lists:map(fun(World) -> Similarity = compute_world_similarity(FactualWorld, World), Satisfies = satisfies_query(World, CounterfactualQuery), {World, Similarity, Satisfies} end, PossibleWorlds), ValidWorlds = lists:filter(fun({_, _, Satisfies}) -> Satisfies end, SimilarityScores), case ValidWorlds of [] -> {error, no_valid_counterfactual}; _ -> {ClosestWorld, _, _} = lists:max(fun({_, S1, _}, {_, S2, _}) -> S1 >= S2 end, ValidWorlds), {ok, ClosestWorld} end. %% Causal Reinforcement Learning causal_policy_learning(States, Actions, Rewards, CausalGraph) -> % Policy learning that respects causal constraints % Incorporates causal knowledge into RL algorithms PolicyNetwork = #{ layers => [ #{type => dense, units => 256, activation => relu}, #{type => dense, units => 128, activation => relu}, #{type => dense, units => length(Actions), activation => softmax} ] }, ValueNetwork = #{ layers => [ #{type => dense, units => 256, activation => relu}, #{type => dense, units => 128, activation => relu}, #{type => dense, units => 1, activation => linear} ] }, % Causal-aware training CausalConstraints = extract_causal_constraints(CausalGraph), train_causal_policy(States, Actions, Rewards, PolicyNetwork, ValueNetwork, CausalConstraints). invariant_risk_minimization(Domains, Features, Labels) -> % Learn invariant predictors across domains % Discovers stable causal relationships RepresentationFunction = #{ layers => [ #{type => dense, units => 512, activation => relu}, #{type => dense, units => 256, activation => relu}, #{type => dense, units => 128, activation => relu} ] }, ClassifierFunction = #{ layers => [ #{type => dense, units => 64, activation => relu}, #{type => dense, units => length(unique(Labels)), activation => softmax} ] }, % IRM loss: ERM + invariance penalty IRMLoss = fun(Phi, W, Domain) -> Representations = apply_network(RepresentationFunction, Features), Predictions = apply_network(ClassifierFunction, Representations), ERMLoss = cross_entropy_loss(Predictions, Labels), InvariancePenalty = compute_invariance_penalty(Phi, W, Domain), ERMLoss + InvariancePenalty end, train_irm_model(Domains, Features, Labels, RepresentationFunction, ClassifierFunction, IRMLoss). %% Advanced Causal Methods quantum_causal_models(QuantumStates, CausalStructure, Measurements) -> % Quantum causal models for quantum systems % Incorporates quantum superposition and entanglement QuantumGraph = quantum_causal_graph(CausalStructure), % Quantum do-calculus QuantumInterventions = lists:map(fun(Intervention) -> apply_quantum_intervention(QuantumStates, Intervention, QuantumGraph) end, Measurements), % Quantum causal effects QuantumEffects = compute_quantum_causal_effects(QuantumInterventions), #{quantum_graph => QuantumGraph, quantum_effects => QuantumEffects, entanglement_structure => analyze_causal_entanglement(QuantumStates), quantum_confounding => detect_quantum_confounding(QuantumStates, CausalStructure)}. causal_graph_neural_networks(GraphData, NodeFeatures, CausalAdjacency, TargetNodes) -> % Graph Neural Networks with causal constraints % Learns representations that respect causal structure CausalGCNLayer = fun(H, A) -> % Message passing with causal masking CausalMask = create_causal_mask(A, CausalAdjacency), MaskedAdjacency = element_wise_multiply(A, CausalMask), % Graph convolution with causal constraints Messages = matrix_multiply(MaskedAdjacency, H), apply_activation(relu, Messages) end, % Multi-layer causal GNN Layers = [ CausalGCNLayer, CausalGCNLayer, fun(H, _) -> apply_dense_layer(H, 64, relu) end, fun(H, _) -> apply_dense_layer(H, length(TargetNodes), softmax) end ], train_causal_gnn(GraphData, NodeFeatures, CausalAdjacency, TargetNodes, Layers). %% Evaluation and Validation causal_model_validation(Model, TestData, GroundTruthGraph) -> % Comprehensive validation of causal models PredictedGraph = extract_causal_graph(Model), StructuralMetrics = #{ precision => compute_precision(PredictedGraph, GroundTruthGraph), recall => compute_recall(PredictedGraph, GroundTruthGraph), f1_score => compute_f1_score(PredictedGraph, GroundTruthGraph), shd => structural_hamming_distance(PredictedGraph, GroundTruthGraph) }, CausalEffectMetrics = validate_causal_effects(Model, TestData, GroundTruthGraph), InterventionalMetrics = validate_interventional_predictions(Model, TestData), #{structural_metrics => StructuralMetrics, causal_effect_metrics => CausalEffectMetrics, interventional_metrics => InterventionalMetrics, overall_score => compute_overall_validation_score(StructuralMetrics, CausalEffectMetrics, InterventionalMetrics)}. sensitivity_analysis(CausalModel, Parameters, Perturbations, Metrics) -> % Analyze sensitivity of causal conclusions to assumptions BaselineResults = evaluate_model(CausalModel, Parameters, Metrics), SensitivityResults = lists:map(fun(Perturbation) -> PerturbedParameters = apply_perturbation(Parameters, Perturbation), PerturbedResults = evaluate_model(CausalModel, PerturbedParameters, Metrics), Sensitivity = compute_sensitivity_measure(BaselineResults, PerturbedResults), #{perturbation => Perturbation, results => PerturbedResults, sensitivity => Sensitivity} end, Perturbations), #{baseline_results => BaselineResults, sensitivity_results => SensitivityResults, robustness_score => compute_robustness_score(SensitivityResults)}. %% Helper Functions get_variables(Data) -> case size(Data) of {Rows, Cols} -> lists:seq(1, Cols); _ -> error(invalid_data_format) end. initialize_complete_graph(Variables) -> % Create complete undirected graph lists:foldl(fun(V1, Acc1) -> lists:foldl(fun(V2, Acc2) -> case V1 =/= V2 of true -> maps:put({V1, V2}, undirected, Acc2); false -> Acc2 end end, Acc1, Variables) end, #{}, Variables). discover_skeleton(Data, Graph, Alpha) -> % Remove edges based on conditional independence tests Edges = maps:keys(Graph), lists:foldl(fun(Edge, CurrentGraph) -> {V1, V2} = Edge, IsIndependent = conditional_independence_test(Data, V1, V2, [], Alpha), case IsIndependent of true -> maps:remove(Edge, CurrentGraph); false -> CurrentGraph end end, Graph, Edges). conditional_independence_test(Data, V1, V2, ConditioningSet, Alpha) -> % Perform statistical test for conditional independence % Using partial correlation or mutual information TestStatistic = compute_test_statistic(Data, V1, V2, ConditioningSet), PValue = compute_p_value(TestStatistic), PValue > Alpha. compute_test_statistic(Data, V1, V2, ConditioningSet) -> % Placeholder for actual statistical test % Would implement Fisher's z-transform for partial correlation % or conditional mutual information test 0.5. compute_p_value(TestStatistic) -> % Placeholder for p-value computation % Would use appropriate statistical distribution 0.1. orient_edges(Data, Graph, Alpha) -> % Apply orientation rules to discover edge directions Graph. apply_orientation_rules(Graph) -> % Apply Meek rules for edge orientation Graph. compute_discovery_stats(Data, Graph) -> #{edges => length(maps:keys(Graph)), density => compute_graph_density(Graph)}. compute_edge_confidence(Data, Graph, Alpha) -> % Compute confidence scores for discovered edges #{}. empty_graph(Variables) -> #{}. ges_forward_phase(Data, Graph, Penalties) -> Graph. ges_backward_phase(Data, Graph, Penalties) -> Graph. ges_turn_phase(Data, Graph, Penalties) -> Graph. compute_graph_score(Data, Graph, Penalties) -> 0.0. compute_equivalence_class(Graph) -> [Graph]. parallel_pc_algorithm(Data, Options, Parallel) -> #{}. cuda_ges_algorithm(Data, Options) -> #{}. neural_causal_discovery(Data, Options) -> #{}. quantum_causal_algorithm(Data, Options) -> #{}. regress(Y, X) -> #{}. weak_instrument_test(Model) -> #{}. predict(Model, X) -> []. extract_coefficient(Model) -> 0.0. compute_robust_se(SecondStage, FirstStage) -> 0.0. compute_f_statistic(Model) -> 0.0. create_cross_fitting_folds(N, K) -> []. fit_ml_model(Model, X, Y, Indices) -> #{}. compute_residuals(Model, Y, Indices) -> []. estimate_ate(TreatmentResiduals, OutcomeResiduals) -> 0.0. aggregate_double_ml_results(Results) -> #{}. reconstruction_loss(Original, Reconstructed) -> 0.0. kl_divergence_loss(Z) -> 0.0. causal_regularization_term(Z, T, Y) -> 0.0. decode(Z) -> []. train_causal_vae(Data, Treatment, Encoder, Decoder, Loss) -> #{}. train_neural_causal_model(Features, Treatment, Outcome, RepNet, TreatNet, OutNet) -> #{}. abduction_step(Model, Evidence) -> #{}. apply_intervention(Model, Intervention) -> Model. prediction_step(Model, Unobserved, Query) -> #{}. compute_counterfactual_probability(Outcome) -> 0.5. generate_counterfactual_explanation(Model, Intervention, Outcome) -> "Explanation". generate_possible_worlds(Model) -> []. compute_world_similarity(World1, World2) -> 0.0. satisfies_query(World, Query) -> true. extract_causal_constraints(Graph) -> []. train_causal_policy(States, Actions, Rewards, PolicyNet, ValueNet, Constraints) -> #{}. unique(List) -> sets:to_list(sets:from_list(List)). apply_network(Network, Input) -> []. cross_entropy_loss(Predictions, Labels) -> 0.0. compute_invariance_penalty(Phi, W, Domain) -> 0.0. train_irm_model(Domains, Features, Labels, RepNet, ClassNet, Loss) -> #{}. quantum_causal_graph(Structure) -> #{}. apply_quantum_intervention(States, Intervention, Graph) -> #{}. compute_quantum_causal_effects(Interventions) -> []. analyze_causal_entanglement(States) -> #{}. detect_quantum_confounding(States, Structure) -> false. create_causal_mask(A, CausalAdjacency) -> A. element_wise_multiply(A, B) -> A. matrix_multiply(A, B) -> B. apply_activation(relu, X) -> X. apply_dense_layer(H, Units, Activation) -> H. train_causal_gnn(GraphData, NodeFeatures, CausalAdjacency, TargetNodes, Layers) -> #{}. extract_causal_graph(Model) -> #{}. compute_precision(Pred, Truth) -> 0.0. compute_recall(Pred, Truth) -> 0.0. compute_f1_score(Pred, Truth) -> 0.0. structural_hamming_distance(Pred, Truth) -> 0. validate_causal_effects(Model, Data, Truth) -> #{}. validate_interventional_predictions(Model, Data) -> #{}. compute_overall_validation_score(Structural, CausalEffect, Interventional) -> 0.0. evaluate_model(Model, Parameters, Metrics) -> #{}. apply_perturbation(Parameters, Perturbation) -> Parameters. compute_sensitivity_measure(Baseline, Perturbed) -> 0.0. compute_robustness_score(Results) -> 0.0. compute_graph_density(Graph) -> 0.0. %% Missing exported function implementations (stubs) attribution_analysis(_Data, _Treatment, _Options) -> #{effect => 0.0, confidence => 0.95}. backdoor_adjustment(_Graph, _Treatment, _Outcome) -> #{adjustment_set => [], effect => 0.0}. confounded_bandits(_Arms, _Confounders, _Rewards, _Policy) -> #{optimal_arm => 1, reward => 0.0}. contrastive_explanation(_Model, _Factual, _Counterfactual, _Options) -> #{explanation => "No contrast found"}. difference_in_differences(_PreTreatment, _PostTreatment, _Control, _Options) -> #{ate => 0.0, se => 0.1, pvalue => 0.5}. do_calculus(_Graph, _Query, _Evidence) -> #{identifiable => false, expression => undefined}. domain_adaptation_causal(_SourceData, _TargetData, _Model, _Options) -> #{adapted_model => undefined, performance => 0.0}. frontdoor_adjustment(_Graph, _Treatment, _Outcome) -> #{mediator_set => [], effect => 0.0}. granger_causality(_TimeSeries1, _TimeSeries2, _Options) -> #{causal => false, pvalue => 0.5}. hybrid_causal_discovery(_Data, _Constraints, _Options) -> #{graph => [], score => 0.0}. intervention_simulation(_Model, _Interventions, _Targets, _Options) -> #{effects => [], confidence => 0.95}. moderation_analysis(_Data, _Treatment, _Moderator, _Outcome) -> #{interaction_effect => 0.0, significance => 0.05}. offline_causal_rl(_States, _Actions, _Rewards, _Policy) -> #{policy => undefined, value => 0.0}. regime_switching_causal(_Data, _Regimes, _Transitions, _Options) -> #{regimes => [], transitions => []}. score_based_discovery(_Data, _Score, _Options) -> #{graph => [], score => 0.0}. structural_counterfactuals(_Model, _Evidence, _Intervention, _Query) -> #{counterfactual => undefined, probability => 0.5}. synthetic_control(_Treatment, _Control, _Outcome, _Options) -> #{effect => 0.0, weights => []}. targeted_ml(_Data, _Treatment, _Outcome, _Models) -> #{ate => 0.0, se => 0.1}. treatment_effect_validation(_Model, _Data, _TrueEffects, _Options) -> #{accuracy => 0.0, bias => 0.0}. causal_benchmark_suite(_Datasets, _Methods) -> #{results => [], rankings => []}. causal_discovery_metrics(_Predicted, _True, _Options) -> #{precision => 0.0, recall => 0.0, f1 => 0.0}. causal_discovery_with_latents(_Data, _Options) -> #{graph => [], latents => []}. causal_effect_estimation(_Data, _Treatment, _Outcome, _Confounders) -> #{ate => 0.0, se => 0.1}. causal_explanation(_Model, _Instance, _Options) -> #{explanation => "No explanation available"}. causal_fairness_analysis(_Data, _Protected, _Treatment, _Outcome) -> #{fairness_metrics => [], bias => 0.0}. causal_forests(_Data, _Treatment, _Outcome, _Features) -> #{model => undefined, effects => []}. causal_gan(_Data, _Treatment, _Generator, _Discriminator) -> #{generator => undefined, discriminator => undefined}. causal_meta_learning(_Tasks, _Models, _Adaptation, _Options) -> #{meta_model => undefined, performance => 0.0}. causal_representation_learning(_Data, _Architecture, _Options) -> #{representations => [], model => undefined}. causal_time_series_forecasting(_TimeSeries, _Treatment, _Horizon, _Options) -> #{forecasts => [], effects => []}. causal_transformer(_Data, _Architecture, _Options) -> #{model => undefined, attention => []}. constraint_based_discovery(_Data, _Constraints, _Options) -> #{graph => [], constraints_satisfied => true}. counterfactual_generator(_Model, _Evidence, _Options) -> #{counterfactuals => [], likelihood => []}. deep_structural_model(_Data, _Structure, _Architecture, _Options) -> #{model => undefined, structure => []}. dynamic_causal_modeling(_TimeSeries, _Structure, _Parameters, _Options) -> #{model => undefined, dynamics => []}. federated_causal_learning(_Clients, _Data, _Models, _Options) -> #{global_model => undefined, effects => []}. mediation_analysis(_Data, _Treatment, _Mediator, _Outcome) -> #{direct_effect => 0.0, indirect_effect => 0.0}. probabilistic_causal_programming(_Program, _Evidence, _Query) -> #{posterior => [], probability => 0.5}. probabilistic_counterfactuals(_Model, _Evidence, _Query) -> #{counterfactuals => [], probabilities => []}. regression_discontinuity(_Data, _Cutoff, _Options) -> #{effect => 0.0, bandwidth => 1.0}. robust_causal_inference(_Data, _Models, _Robustness, _Options) -> #{robust_effect => 0.0, uncertainty => 0.1}. temporal_causal_discovery(_TimeSeries, _Lags, _Options) -> #{temporal_graph => [], lags => []}. var_causal_analysis(_TimeSeries, _Order, _Options) -> #{var_model => undefined, causality => []}.