Dual machine learning for causal inference (2023)

Thoughts and theory,BCN CAUSES SOMETHING

This post is the result of a joint effort withAleixa Ruiza de Villi,Jesus seeks, and the wholeCausality SOMETHING BCNclub. Note: We assume that the reader is familiar with the basic concepts of causal inference.

If you dabble in the waters of causal inference, you may have heard of the concept of dual machine learning. And if you haven't heard of it yet, I personally bet it will be possible in the near future. Or better yet, you can use it without even knowing you're using it. Like any great technology, Double Machine Learning for causal inference can become quite ubiquitous. But let us cool the heat of this writer and return to our work. In this post I try to explain, briefly but quite comprehensively, what Dual Machine Learning is and how it works. To this end, we will cover the topic from its theoretical background to a typical example of its use in causal inference.

Entrance

So what is dual machine learning? This idea was introduced and developed by Chernozukov et al. in a series of articles (Ten,TenITen) and generally aims to provide:

1. A general framework for estimating causal effects using machine learning techniques
2. Confidence intervals for these estimates
3. An estimator that is "root consistent," that is, an estimator that has good data convergence and performance properties.

And where did this whole idea come from? On the one hand, to state the obvious, it comes from the idea of ​​causal inference using machine learning. But if we look a little deeper, two non-obvious key ideas emerge:

1. From a statistical point of view, Machine Learning is a set of non-parametric or semi-parametric estimation methods,
2. there is a very large body of theoretical work on non-parametric and semi-parametric estimation methods (with bounds, returns, etc.)

Double Machine Learning combines both, drawing inspiration and useful results from the latter to draw causal inferences from the former.

composition

Let's start. We start by defining the DAG of the data generation process we will work on, as shown in the figure below:

Furthermore, we define the following partially linear model that sets the relationships between variables from the DAG:

where Y is the outcome variable, D is the binary treatment, Z is the covariate vector, and U and V are disturbances. Equation 1.1 is the main equation, and θ0 is the parameter of interest we would like to estimate, which is the derivative of ATE with respect to D. Equation 1.2 tracks confounding, that is, the relationship of treatment to covariates. This relationship is modeled using the function m0(Z) and the dependence of the outcome on the covariates using the function g0(Z). We will see later that Double Machine Learning also works for fully nonlinear models, but we will start assuming this partially linear model to make the method easier to explain and interpret. It should also be noted that we assume all the usual traceability conditions in causal inference, i.e. no hidden confounders, positivity and consistency.

Then, recalling and complementing the definition of Dual Machine Learning given in the introduction, the goal of the method is to obtain the root of the n-consistent estimator and the confidence intervals of the interesting (low dimensional) parameter θ0, in the presence of the nuisance parameter (potentially large dimension) η₀=(g0, m0). In this context, perturbation means that we do not directly care about the correctness of our estimate of η₀, as long as we have a good (root n-consistent) estimator of θ₀.

But why would we care to use machine learning to accomplish this task at all? Mainly for three reasons. First and most obvious, because of the power machine learning methods have in modeling functions and/or expectations. Second, because machine learning models predict better than traditional statistical methods (ie linear regression and OLS) for multivariate data, which is becoming the norm in our world. And last and most important, because compared to traditional statistical methods, machine learning methods do not impose such strong assumptions on the functional forms of m₀(Z) and especially g0(Z), but learn these forms from data. This is a good precaution against model misspecification, which is a problem that in our example would lead to a biased estimator even in the absence of unmeasured confounders.

Naive estimator

So why not use machine learning to estimate θ₀ directly? For example, we could use an alternating iterative approach, estimating g0 using random forests, then estimating θ0 using OLS, repeating until convergence (note that in this case, using OLS for θ₀ does not mean that the model is not well specified because the model is linear in θ0 ). . Well, life is not that simple. The figure below shows the θ0-θ distribution for this approximation compared to the normal distribution of mean 0. As can be seen from the difference between the two distributions, the estimator is biased. Note that g₀ is set to a smooth function with few parameters that should in principle be well approximated by random forests.

The key observation for understanding this phenomenon is that g₀(Z)≠[Y|Z]. Thus, it is generally not possible to obtain a good estimate of g0(Z) by "regressing" Y on Z, and this in turn leads to the inability to obtain a good estimate of θ0. Nevertheless, it is perfectly possible to make very good predictions about Y given Z and D. This is why we usually say that machine learning is good for prediction but bad for causal inference. Bias has two sources: adjustment and overadjustment. Double Machine Learning aims to correct both regularization error by orthogonalization and overfitting error by cross-fitting. The following sections explain how these two deviation correction strategies work. A detailed explanation of the sources and patterns of bias can be found in the authors' original work orthis presentation.

Orthogonality and the Neyman orthogonality condition

To show how orthogonalization works, we first state and briefly explain the Frisch-Waugh-Lovell theorem. This theorem states that, given the linear model Y=β0+β1D+β2Z+U, the following two approaches to estimating β1 produce the same result:

1. Linear regression of Y on D and Z using OLS.
2. A three-step process: 1) Regress D on Z. 2) Regress Y on Z. 3) regress the residuals of 2 on the residuals of 1 to obtain β1 (all regressions using OLS).

Similarly, returning to our semi-linear example, we can proceed as follows:

1. Predict D from Z using machine learning.
2. Predict Y from Ω using machine learning.
3. Linear regression of the residuals of 2 on the residuals of 1 to obtain an estimate of θ₀.

This procedure ensures that the model from step 3 is "orthogonal", resulting in an unbiased square root estimator consistent with n. See the figure below for the θ0-θ distribution for this approximation.

How can we formalize and generalize this process? For this, we first need to introduce the concept of scaling functions and moment conditions (for an introduction to the Generalized Method of Moments, see the articlethis page). Specifically, the scoring function used in this case is ψ=(D−m₀(Z))×(Y−g₀(Z)−(D−m₀(Z))θ), where the multiplicative terms are the error terms of the partially linear model, although other alternatives are available. Now we require this resultant function to be zero, ψ=0, which is our moment condition. This is a mathematical expression that we want our regressor and our error to be orthogonal, which is like saying we want them to be uncorrelated. Operationally, this means that once we calculate g₀ and m₀, we can get θ₀ from the momentum equation of state.

We are finally ready to express and defineNeyman rectangles Stan, How

∂η𝔼[ψ(W;θ₀,η₀)][η−η₀] = 0

where W is our chunk of data. This equation is interpreted as follows: The left-hand side is the directional derivative of the gate of our outcome function with respect to our perturbation parameter η in the neighborhood of η₀. We say we want this derivative to vanish. And since the derivative is the instantaneous rate of change, we say that our outcome function (and thus our estimate of θ₀) should be robust to "small" perturbations of the disturbance parameter n.

In summary, imposing the Neyman orthogonality condition on our outcome function (and thus on our θ0 and η0 estimators) renders the θ0 estimator free from one of two sources of systematic error—regularization error.

Splitting and Crossing Samples

Now it is time to get rid of the second source of bias, namely the overfitting error (again, for a detailed explanation of the shape of these errors, seethis presentation). A possible strategy for this would be the so-calledsample splitgetting closer. The functionality is as follows:

1. We randomly divide our data into two subsets.
2. We fit the machine learning models for D and Y to the first subset.
3. In the second subset, we calculate θ₀ using the models obtained in step 2.

The downside of this strategy is that it reduces performance and statistical power. But it can be solved withjunction, which is the strategy used in the Double Machine Learning method. It looks like this:

1. We randomly divide our data into two subsets.
2. We fit the machine learning models for D and Y to the first subset.
3. In the second subset, we estimate θ0,1 using the models obtained in step 2.
4. We align the machine learning models with the second subset.
5. In the first subset, we calculate θ0,2 using the models obtained in step 4.
6. The final estimate of θ0 is taken as the average of θ0,1 and θ0,2.

Note that we can gain strength by repeating the process for K folds with K greater than 2.

Algorithm

We have finally reached the end of this post and it is time to gather all the previously explained ingredients and put them into an algorithm.

Let us define a fully interactive model, more general than a partially linear model. This model does not assume that D (machining) is additively separated:

Our causal parameter of interest is ATE, 𝔼[g₀(1;Z) - g₀(0;Z)] and our outcome function is

provided by Robins and Rotnizsky (1995) and is Neyman orthogonal (and doubly robust), satisfying ∂𝔼ψ(W; θ₀, η)= 0.

In such a setting, Double Machine Learning providesunbiased n-connected root estimatorATE and herconfidence intervals(with alpha confidence level) by following these steps:

(Video) Philipp Bach and Sven Klaassen: Tutorial on DoubleML for double machine learning in Python and R

And so we have come to the end of our post. Note that there are packages available for Python and R that implement this and other related algorithms, which can be found athttps://github.com/DoubleML.