Primary Debts and Sources
Non Parametric Causal Inference with PyMC
Propensity Scores, Debiased ML and Causal Mediation
Nathaniel Forde
Data Science @ Personio
and Open Source Contributor @ PyMC
3/1/24
Preliminaries
Profile
- I’m a data scientist at Personio
- Bayesian statistician,
- Reformed philosopher and logician.
- Website: https://nathanielf.github.io/
Disclaimer
None of Personio’s data was used in this presentation
Code or it didn’t Happen
The worked examples used here can be found here
The Pitch
Agnostic statistical methods in causal inference are well motivated but limited in scope.
We’ll show how Bayesian non-parametrics are natural framwork in with which to couch and extend these methods
Agenda
Propaganda
Full Luxury Bayesianism
Posterior Predictive Imputation of Treatment Effects
Non Parametric Bayes Formula
Full Luxury Bayesianism
\[ p(\color{blue}{\theta} |D) \propto p(D | \color{blue}{\theta})p(D) \]
where \(\color{blue}{\theta}\) is an explict model parameter becomes
\[ p(\color{blue}{G} |D) \propto p(D | \color{blue}{G})p(D) \]
where \(\color{blue}{G}\) is a general stochastic process
Three Acts
- Propensity Scores and Non-Parametric Causal Inference
- Confounding and Debiasing
- Causal Structure and Mediation
Act One
Propensity Scores and Non-Parametric Causal Inference
- Strong Ignorability and Propensity Scores
- BART models and Non-Parametric Estimation
- Balance and Inverse Propensity Weighting
- Robust and Doubly Robust methods
Strong Ignorability and Propensity Scores
Definitions
- Potential Outcomes
- \(Y(0)\) and \(Y(1)\) under different treatment regimes \(T \in \{ 0, 1\}\)
- Strong Ignorability
- Outcomes are independent of the treatment assignment given a covariate profile \(X\): \(Y(0), Y(1) \perp\!\!\!\perp T | X\)
- Propensity Scores
- An estimate of the probability for a particular treatment status conditional on the covariate profile \(X\): \(0 \leq p_{t}(X) \leq 1\)
BART Models and Non-Parametric Estimation
- BART
- Bayesian Additive Regression Trees
- “[B]lack-box method based on the sum of many trees where priors are used to regularize inference”
- Non-Parametric Estimation
- Outcomes and Propensity Scores can be estimated using non-parametric methods
- Causal estimands can be estimated using posterior predictive imputation under different treatment regimes
- Benefit of minimalist structural assumptions.
Inverse Propensity Score Weighting
- Adjustment by representative Weighting
- Using the propensity scores as a summary metric for group membership, we down-weight and upweight the prevalence of high and low propensity score in each group to induce strong ignorability like conditions.
Robust and Doubly Robust Methods
Differing Weighting Schemes
- Raw
- \(\sum\frac{1}{N}\Big[ Y(1) \cdot \frac{1}{p_{T}(X)} - (Y(0)\cdot\frac{1}{1-p_{T}(X)}) \Big]\)
- Doubly Robust
- \[ \hat{Y(1)} = \frac{1}{n} \sum_{0}^{N} \Bigg[ \frac{T(Y - m_{1}(X))}{p_{T}(X)} + m_{1}(X) \Bigg] \\ \hat{Y(0)} = \frac{1}{n} \sum_{0}^{N} \Bigg[ \frac{(1-T)(Y - m_{0}(X))}{(1-p_{T}(X))} + m_{0}(X) \Bigg] \]
Go to Code
Act Two
Confounding and Debiasing
- Propensity Scores Miscalibrated
- BART models and Overfitting
- Debiased Machine Learning
- CATE estimation
Miscalibrated Propensity Scores
What is the Estimand?
“Each theoretical estimand is linked to an empirical estimand involving only observable quantities (e.g. a difference in means in a population) by assumptions about the relationship between the data we observe and the data we do not.” - Lundberg et al in What is your Estimand
- Q1. What are we aiming at when we estimate propensity scores for highly granular covariate profiles?
- Q2. What happens when the sample data has no treatment data cases for a particular covariate profile?
Overfitting
BART models can achieve perfect Allocation
Overfitting
Debiasing Machine Learning
K-fold Propensity Estimation
CATE Estimation
Conditional Average Treatment Effect
Go to Code
Act Three
Causal Structure and Mediation
- Parametric Mediation
- Non-Parametric Mediation
- Escalating Structural Assumptions and Bayesian Inference
Parametric Mediation
Traditional Model Based Mediation
Causal Mediation Analysis
Non-Parametric Estimation
- NDE: \(E[Y(t, M(t^{*})) - Y(t^{*}, M(t^{*}))]\)
- Which is to say we’re interested in the differences in the imputed outcomes under different treatments, mediated by values for M under a specific treatment regime.
- NIE: \(E[(Y(t, M(t))) - Y(t, M(t^{*}))]\)
- Which amounts to the imputed differences in the outcome Y due to differences in the treatment regimes which generated the mediation values M.
- TE: NDE + NIE
Go To Code
Conclusions
Structural Beliefs and Bayesian Inference
- Propensity score adjustment reflects the belief in the need for adjustment
- It reflects a belief in the adequacy of the propensity score model for achieving balance
- Regression based imputation of treatment effects reflects the belief that covariate controls eliminates selection effects
- Doubly Robust methods reflect the belief that either the propensity or outcome model is mispecified.
- But that one ought to be adequately specified.
- Debiased ML methods reflect the belief that mis-specification can be corrected by cross-validation and non-parametric estimation of residuals in FWL theorem
- Mediation Analysis reflects the belief that the causal influence must be interepreted with particular causal structure to avoid confounding.
- There are no truly Agnostic statistics:In each case sound inference proceeds as we take steps to adjust our model or the conditions of its assessment as informed by our best beliefs regarding the problem to hand.
Non Parametric Causal Inference
Non Parametric Causal Inference with PyMC Propensity Scores, Debiased ML and Causal Mediation Nathaniel Forde Data Science @ Personio and Open Source Contributor @ PyMC 3/1/24