deep-inference

Deep Learning for Individual Heterogeneity with Valid Inference

deep-inference enriches structural economic models with deep learning while maintaining valid statistical inference. It implements the Farrell, Liang, and Misra (2021, 2025) framework.

Valid Inference

95% confidence intervals that actually cover 95% of the time

13 Model Families

Linear, Logit, Poisson, Gamma, NegBin, Weibull, Gumbel, Tobit, Gaussian, Probit, Beta, ZIP, Multinomial Logit

Economic Targets

AME, Elasticity, WTP, Dose-Response, Profit, Consumer Welfare, Conditional Variance, and custom targets with autodiff

Regime Detection

Auto-selects optimal Lambda strategy for RCTs vs observational data

PyTorch Backend

Automatic differentiation for exact gradients and Hessians

Quick Start

import numpy as np
import torch
from deep_inference import structural_dml

# Heterogeneous logistic demand (binary outcomes)
np.random.seed(42)
torch.manual_seed(42)
n = 2000
X = np.random.randn(n, 5)
T = np.random.randn(n)

# Heterogeneous treatment effect: β(X) = 0.5 + 0.3*X₁
alpha = 0.2 * X[:, 0]
beta = 0.5 + 0.3 * X[:, 1]
prob = 1 / (1 + np.exp(-(alpha + beta * T)))
Y = np.random.binomial(1, prob).astype(float)

# Run influence function inference
result = structural_dml(
    Y=Y, T=T, X=X,
    family='logit',
    hidden_dims=[64, 32],
    epochs=100,
    n_folds=50
)

print(result.summary())

Output (Date/Time will vary):

==============================================================================
                            Structural DML Results
==============================================================================
Family:           Logit                Target:           E[beta]
No. Observations: 2000                 No. Folds:        50
Date:             Fri, 16 Jan 2026     Time:             13:54:12
==============================================================================
                  coef     std err         z     P>|z|      [0.025    0.975]
------------------------------------------------------------------------------
     E[beta]    0.4704      0.0496     9.492  0.000      0.3733    0.5675
==============================================================================
Diagnostics:
  Min Lambda eigenvalue:    0.134946
  Mean condition number:    1.17
  Correction ratio:         45.2493
  Pct regularized:          0.0%
------------------------------------------------------------------------------

Predictions & Visualization

# Predict treatment effects for new observations
X_new = np.random.randn(5, 5)
beta_new = result.predict_beta(X_new)
print(f"Predicted β(X) for new data: {beta_new}")

# Predict probabilities at treatment level T=1
proba = result.predict_proba(X_new, t_value=1.0)
print(f"P(Y=1|X,T=1): {proba}")

# Visualize heterogeneity distributions
result.plot_distributions()
result.plot_heterogeneity(feature_idx=1)  # β(X) vs X₁

Distribution of estimated parameters:

Treatment effect heterogeneity (β vs X₁):

Economic Targets

The inference() API supports built-in economic targets:

from deep_inference import inference

# Built-in: average marginal effect
result = inference(Y, T, X, model='logit', target='ame', t_tilde=0.0)
print(result.summary())

The same target can be defined from scratch with a custom loss and custom target — autodiff handles all derivatives:

import torch

# Custom loss (logit negative log-likelihood)
def my_loss(y, t, theta):
    p = torch.sigmoid(theta[0] + theta[1] * t)
    return -y * torch.log(p + 1e-7) - (1 - y) * torch.log(1 - p + 1e-7)

# Custom target (average marginal effect at t_tilde)
def my_ame(x, theta, t_tilde):
    p = torch.sigmoid(theta[0] + theta[1] * t_tilde)
    return p * (1 - p) * theta[1]

result = inference(Y, T, X, loss=my_loss, target_fn=my_ame, theta_dim=2, t_tilde=0.0)

Built-in output (Date/Time will vary):

==============================================================================
                         Structural Inference Results
==============================================================================
Family:           Logit                Target:           ame
No. Observations: 2000                 No. Folds:        50
Date:             Sat, 07 Feb 2026     Time:             23:37:29
==============================================================================
                  coef     std err         z     P>|z|      [0.025    0.975]
------------------------------------------------------------------------------
         ame    0.1181      0.0127     9.290  0.000      0.0932    0.1431
==============================================================================
Diagnostics:
------------------------------------------------------------------------------

Custom output:

==============================================================================
                         Structural Inference Results
==============================================================================
No. Observations: 2000                 Target:           E[beta]
Date:             Sat, 07 Feb 2026     No. Folds:        50
                                       Time:             23:41:09
==============================================================================
                  coef     std err         z     P>|z|      [0.025    0.975]
------------------------------------------------------------------------------
     E[beta]    0.1251      0.0128     9.763  0.000      0.1000    0.1502
==============================================================================
Diagnostics:
------------------------------------------------------------------------------

Both approaches produce matching estimates (0.118 vs 0.125, within MC noise). Output generated by docs/generate_quickstart_plots.py.

More built-in targets:

# Price elasticity at P=2.0
result = inference(Y, T, X, model='logit', target='elasticity', t_tilde=2.0)

# Consumer welfare
result = inference(Y, T, X, model='logit', target='welfare', t_tilde=2.0)

# Dose-response: average predicted outcome at treatment level
result = inference(Y, T, X, model='logit', target='dose_response', t_tilde=1.0)

# Expected profit/revenue at price level
result = inference(Y, T, X, model='logit', target='profit', t_tilde=2.0)

# Conditional variance (risk heterogeneity)
result = inference(Y, T, X, model='logit', target='conditional_variance', t_tilde=0.0)

# Randomized experiment (compute Lambda instead of estimating)
from deep_inference.lambda_.compute import Normal
result = inference(Y, T, X, model='logit', target='beta',
                   is_randomized=True, treatment_dist=Normal(0, 1))

Why deep-inference?

The Problem: Neural networks are great at prediction but naive inference produces invalid confidence intervals with coverage far below 95%.

The Solution: Influence function-based debiasing corrects for regularization bias, providing valid confidence intervals for economic targets like average treatment effects.

Method	Coverage	SE Ratio
Naive	8%	0.27
Influence	95%	1.08

Documentation

Contents

• Getting Started
  • Installation
  • Quickstart
  • Usage Guide
  • Overview
  • Why Influence Functions?
  • Supported Models
• Flowchart
  • Interactive Decision Flowchart
  • Decision 1: Model Family
  • Decision 2: Target Functional
  • Decision 3: Regime & Lambda Strategy
  • Common Paths
  • Quick Reference
  • See Also
• Tutorials
  • Gallery
  • Linear Model Tutorial
  • Logit Model Tutorial
  • Poisson Model Tutorial
  • Tobit Model Tutorial
  • Negative Binomial Model Tutorial
  • Gamma Model Tutorial
  • Gumbel Model Tutorial
  • Weibull Model Tutorial
  • Multinomial Logit (Conditional Logit) Tutorial
  • Multimodal Tutorial: Text & Image Embeddings
  • Personalized Pricing Tutorial
  • Continuous Treatment Tutorial
  • Overview
  • Model Selection Guide
  • Common Workflow
• Theory
  • Deep Learning for Individual Heterogeneity
  • Overview
  • Key References
  • The Core Insight
• Algorithm
  • Overview
  • Algorithm Flow
  • Lambda Estimation (Critical)
  • Three-Way Splitting
  • Regularization
  • Variance Estimation
  • Two APIs
  • Implementation Files
  • Pseudocode (Actual Implementation)
  • References
• Validation
  • Eval 01: Parameter Recovery
  • Eval 02: Autodiff vs Calculus
  • Eval 03: Lambda Estimation
  • Eval 04: Target Jacobian
  • Eval 05: Influence Function Assembly
  • Eval 06: Frequentist Coverage
  • Eval 07: End-to-End Workflow
  • Eval 09: Multinomial Logit
  • Verification Against FLM2
  • Eval Suite Overview
  • Quick Summary
  • Running Evals
  • References
• API Reference
  • Inference API
  • Families Module
  • Targets Module
  • Lambda Strategies
  • Models Module
  • Metrics Module
  • Quick Reference
  • Module Overview

References

Core Framework

• Farrell, Liang, Misra (2021): “Deep Neural Networks for Estimation and Inference” Econometrica (pdf)

• Farrell, Liang, Misra (2025): “Deep Learning for Individual Heterogeneity” Working Paper (pdf)

Applications

• Dubé, Misra (2023): “Personalized Pricing and Consumer Welfare” Journal of Political Economy (pdf)

• Hetzenecker, Osterhaus (2024): “Deep Learning for Heterogeneous Parameters in Discrete Choice Models” arXiv 2408.09560 (pdf)

• Colangelo, Lee (2026): “Double Debiased Machine Learning Nonparametric Inference with Continuous Treatments” JBES (pdf)

• Momin (2025): “Heterogeneous Treatment Effects and Counterfactual Policy Targeting Using Deep Neural Networks” SSRN 5149650

• Chen, Liu, Ma, Zhang (2024): “Causal Inference of General Treatment Effects using Neural Networks” Journal of Econometrics (pdf)

• Ye, Zhang, Zhang, Zhang, Zhang (2025): “Deep-Learning-Based Causal Inference for Large-Scale Combinatorial Experiments” Management Science (pdf)

Automatic Debiasing / Riesz Representation

• Chernozhukov, Newey, Quintas-Martinez, Syrgkanis (2022): “RieszNet and ForestRiesz: Automatic Debiased Machine Learning with Neural Nets” ICML (pdf)

• Chernozhukov, Newey, Singh (2022): “Automatic Debiased Machine Learning of Causal and Structural Effects” Econometrica (pdf)

• Chernozhukov, Newey, Quintas-Martinez, Syrgkanis (2021): “Automatic Debiased Machine Learning via Neural Nets for Generalized Linear Regression” Working Paper (pdf)

• Hines, Hines (2025): “Automatic Debiasing of Neural Networks via Moment-Constrained Learning” CLeaR (pdf)

DNN Architecture + Influence Functions

• Shi, Blei, Veitch (2019): “Adapting Neural Networks for the Estimation of Treatment Effects” NeurIPS (pdf)

• Li, McCoy et al. (2025): “Targeted Deep Architectures for Estimation and Inference” arXiv 2507.12435 (pdf)

• Shirakawa et al. (2024): “Deep Longitudinal Targeted Minimum Loss-based Estimation” ICML (pdf)

• Liu et al. (2024): “DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimation” ICML (pdf)

• Cai, Fonseca, Hou, Namkoong (2025): “C-Learner: Constrained Learning for Causal Inference and Semiparametric Statistics” arXiv 2405.09493 (pdf)

Theory

• Yan, Chen, Yao (2025): “Overparameterized Neural Networks in Semiparametric Inference” arXiv 2504.19089 (pdf)

• Metzger (2022): “Adversarial Estimators” arXiv 2204.10495 (pdf)

• Foster, Syrgkanis (2023): “Orthogonal Statistical Learning” Annals of Statistics (pdf)

Frontier

• Melnychuk, Feuerriegel (2026): “GDR-Learners: Generalized Doubly Robust Learners for Causal Inference” ICLR (pdf)

• Nguyen (2025): “Neural Network Estimation and Simulation for Dynamic Discrete Choice Models” Georgetown JMP (pdf)

License

MIT License