Machine learning for conditional density estimation

This post is part of our Q&A series.

A question from graduate students in our Spring 2021 offering of the new course “Targeted Learning in Practice” at UC Berkeley:

Question:

Hi Mark,

I was curious in general about approaching problems that involve machine learning-based estimation of densities rather than scalar quantities (i.e., regression), particularly for continuous variables. As a grounding example, for continuous treatments in the TMLE framework one needs to estimate $P(A \mid W)$, where $A$ is a continuous random variable. My (perhaps naive) impression is that it is difficult to get reasonable estimates of such a probability with limited samples. Even when $A$ is discrete, I could imagine that these probabilities are difficult to estimate with high-dimensional $W$ and limited samples. Any insight into how different or not-so different your approach to density estimation is compared to regression would be really interesting!

Best,

H.N.

Answer:

Hi H.N.,

Thank you for your important question.

Yes, we are very much involved in conditional density estimation. We have various approaches. In a 2010 paper on longitudinal TMLE, I proposed discretization of the continuous variables to parametrize the conditional density in terms of its hazard $p(A=a \mid A \geq a,W)$, so that it becomes a repeated measures pooled logistic regression problem, and thereby we can utilize the whole world of machine learning algorithms and parametric models, with super learning. We used this approach to construct a TMLE (Stitelman et al.) for multiple time point intervention-specific mean outcomes based on general longitudinal data structures. In a 2011 paper by Iván Díaz and myself, we proposed estimation of the conditional density with a similar approach also involving selecting of the bin width with cross-validation. Oleg Sofrygin also implemented this logistic regression approach as part of his condensier R package on causal inference, which was useful for network causal inference, in which we have to estimate a conditional density of multiple summary measures of the treatment of the friends and the person itself. In more recent work, we use the highly adaptive lasso (HAL). For example, in work with Helene Rytgaard on TMLE for continuous-time survival analysis, we estimate intensities and conditional hazards, and thus conditional densities, with HAL using the Poisson family. Nima Hejazi is currently writing an article where we compare different HAL-based estimators of the conditional density $g(A \mid W)$ (using his haldensify R package) and its impact on efficient estimators of the effect of shift interventions, including tuning of the $L_1$ norm of these HAL estimators for the purpose of the estimator of the target parameter. So, the overall message is that conditional density estimation is not much harder than going after the conditional mean. Formally, the parameter space of conditional density is one dimension larger than that from the conditional mean, and optimal rates of convergence are determined by the entropy/covering number of the function class. When using HAL, the rate this implies (at most) an extra $\log(n)$ factor contirbution. On the other hand, when using inverse weighting by conditional density in TMLE or IPTW for causal inference target estimands, there is more concern for positivity violations so that this requires care. This also motivates to start developing TMLE for conditional independence models that assume that $L(t)$ only depends on the past through recent time points (Markov type), which implies that the efficient influence curve will involve much less inverse weighting, adding stability to these estimators, while such conditional independence assumptions could still be reasonable given the right definition of $L(t)$.

Best Wishes,

Mark

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