deep-inference 0.1.0

Production-grade econometrics with neural networks

deep-inference

Deep Learning for Individual Heterogeneity

deep-inference is a Python package for enriching structural economic models with deep learning. It implements the framework developed by Farrell, Liang, and Misra (2021, 2025) to recover rich, non-linear parameter heterogeneity ($\theta(X)$) while maintaining the interpretability and validity of structural economics.

Standard deep learning minimizes prediction error, which leads to biased parameter estimates ("The Inference Trap"). This package implements Influence Function-based Debiasing (a form of Double Machine Learning) to provide valid confidence intervals and p-values for economic targets. 1

Why deep-inference?

Economic structure and Machine Learning are complements, not substitutes.

• Deep Learning provides the capacity to learn complex, high-dimensional heterogeneity.
• Structural Models provide the constraints necessary for causal interpretation and counterfactuals.

deep-inference enforces the structural loss (e.g., Tobit likelihood) on the output of a neural network, ensuring that the estimated parameters $\alpha(X), \beta(X)$ respect the economic theory.

Installation

pip install deep-inference

Quickstart

Estimate a linear demand model where Price Elasticity $\beta$ varies non-linearly with customer characteristics $X$.

import torch
from deep_inference import get_dgp, get_family, influence

# 1. Generate synthetic data (Linear Demand with Heterogeneity)
dgp = get_dgp("linear", seed=42)
data = dgp.generate(n=2000)

# 2. Define the Structural Family
# (Defines the Loss, Score, and Hessian automatically)
family = get_family("linear")

# 3. Run Inference
# The package trains the structural network, computes the influence function,
# and aggregates results via 10-fold cross-fitting.
result = influence(
    X=data.X, T=data.T, Y=data.Y,
    family=family,
    config={
        "hidden_dims": [128, 64, 32],
        "epochs": 50,
        "lr": 0.01
    }
)

print(f"Avg Elasticity (Truth): {data.mu_true:.4f}")
print(f"Avg Elasticity (Est):   {result.mu_hat:.4f}")
print(f"Standard Error:         {result.se:.4f}")
print(f"95% CI:                 [{result.ci_lower:.4f}, {result.ci_upper:.4f}]")

Supported Structural Families

The package abstracts the math of influence functions. You simply select the family that matches your outcome variable.

Family	Model Structure	Use Case
Linear	$Y = \alpha(X) + \beta(X)T + \varepsilon$	Wages, Test Scores, consumption
Logit	$P(Y=1) = \sigma(\alpha(X) + \beta(X)T)$	Binary Choice, Market Entry
Tobit	$Y = \max(0, \alpha(X) + \beta(X)T + \varepsilon)$	Labor Supply, Censored Demand
Poisson	$Y \sim \text{Pois}(\exp(\alpha(X) + \beta(X)T))$	Patent Counts, Doctor Visits
Gamma	$Y \sim \text{Gamma}(k(X), \theta(X))$	Healthcare Costs, Insurance Claims
Weibull	$Y \sim \text{Weibull}(\dots)$	Duration Analysis, Unemployment Spells

Note: For complex models like Tobit, deep-inference automatically handles the joint estimation of structural variance $\sigma(X)$ required for consistent inference.

Methodological Details

1. The Enriched Model

We replace fixed parameters $\theta$ with neural networks $\theta(X)$:

\hat{\theta}(\cdot) = \arg \min_{\theta \in \mathcal{F}_{DNN}} \sum \ell(y_i, t_i, \theta(x_i))

2. The Influence Function Correction

Naive averaging of $\hat{\theta}(X)$ yields biased inference. We construct a Neyman-Orthogonal score $\psi$ using the Influence Function:

\psi(z) = H(\hat{\theta}) + \nabla_\theta H \cdot \Lambda(x)^{-1} \cdot \nabla_\theta \ell(z, \hat{\theta})

Where $\Lambda(x) = \mathbb{E}[\nabla^2 \ell \mid X=x]$ is the conditional Hessian.

• Automatic Differentiation: deep-inference uses PyTorch Autograd to compute exact Jacobians and Hessians for any model family.
• Stability: Includes Tikhonov regularization for inverting Hessians in non-linear models (e.g., Logit/Tobit).

Validation (Monte Carlo Results)

M=30 simulations, N=10,000 observations. Target: 95% coverage, SE ratio ≈ 1.0.

Linear Model

Config	K	Network	Coverage	SE Ratio	RMSE	Bias²/MSE
Best	50	[64,32]	93.3%	1.03	0.032	22%
Deep	20	[128,64,32]	93.3%	1.02	0.033	21%
E=100	20	[64,32]	90.0%	0.90	0.036	18%
Separate	20	[128,64,32]×2	80.0%	0.82	0.036	14%
Naive	—	—	10%	0.09	0.083	1%

Finding: K=50 folds achieves best coverage and lowest RMSE. Separate networks don't help.

Logit Model

Target: E[β(X)] = average log-odds ratio

Method	Coverage	SE Ratio	RMSE	Hessian min λ
Influence	90.0%	0.93	0.054	0.051
Naive	3.3%	0.03	0.108	—

Summary

Model	Target	Naive Cov	IF Cov	RMSE Improvement
Linear	E[β(X)]	10%	93%	2.6×
Logit	E[β(X)]	3%	90%	2.0×
Poisson	E[β(X)]	TBD	TBD	—
Gamma	E[β(X)]	TBD	TBD	—

Citation

@article{farrell2021deep,
  title={Deep Neural Networks for Estimation and Inference},
  author={Farrell, Max H. and Liang, Tengyuan and Misra, Sanjog},
  journal={Econometrica},
  volume={89},
  number={1},
  pages={181--213},
  year={2021}
}

@article{farrell2025heterogeneity,
  title={Deep Learning for Individual Heterogeneity},
  author={Farrell, Max H. and Liang, Tengyuan and Misra, Sanjog},
  journal={Working Paper},
  year={2025}
}

License

MIT