Double robustness

Suppose we are interested in estimating the average treatment effect (ATE), defined in potential-outcome notation as $latex \tau = E(E(Y^1 \mid X) - E(Y^0 \mid X))$ where the outer expectation is over $latex X&s=1$. Assuming strong ignorability, so that $latex E(Y \mid D = 1, X) = E(Y^1 \mid X)&s=1$ and $latex E(Y \mid D…

Monte Carlo and MCMC slides

I believe that one of these was never officially posted. Here are the two slide decks covering the core computational material. Monte Carlo slides MCMC slides

direct regression adjustment vs IPW

In class tonight we examined and R script which generates some "observational" data with confounding and compares inverse probability weighting to direct regression adjustment for the purpose of estimating the average treatment effect. More concretely, we want to compare two estimators that are based on the following representations of the average treatment effect: $latex \tau…

Problem set

Here is a problem set, to be completed in one sitting like an exam, to help you judge your understanding of some key topics we've covered. We will go over the answers in class next week.

Protecting against underflow when using Bayes rule

If we have data that we assume has arisen via an i.i.d. parametric model and a discrete set of $latex K&s=1$ values that parameter can take, then Bayes rule has the form $latex \pi(\theta \mid y_{1:n}) = c^{-1} \pi(\theta) \prod_{i = 1}^n f(y_i \mid \theta),&s=1$ where $latex c = \sum_{k = 1}^K \pi(\theta_k) \prod_{i =…

Papers to read

Rosenbaum and Rubin. Please read this for class next Monday. Also, here are several papers about vitamin D, which we discussed in class this evening. A call to public health authorities Men's Journal article. (Read the comment section.) Obesity and vitamin D Fracture risk and vitamin D Meta-analysis of vitamin D's effect on mortality

New Causal Midterm

I have substantially revised the midterm and changed the due date. Below are the relevant links. Please reach out to me as early as possible with questions. I will be in the office until 4pm today to take questions. The new format is more straightforward I think (not easier, per se). Causal inference midterm Midterm…

Bayesian linear regression notes

Here is a great explanation of the Stein phenomenon. Here is a truly fantastic set of notes on Bayesian linear regression. Here is a short R script implementing a Gibbs sampler for the Bayesian linear regression model under the independent Normal-Gamma prior from section 1.4 of the above notes. Here is a picture of the Mogollon…

Hierarchical models

I'll have more to say here later, but for now here are the slides the R script the data

Midterms!

Due before midnight of Sunday March 18th. Bayesian midterm Causal midterm