1. Discussion of Profs. Robins’ and M  ller’s Papers S.A. Murphy ENAR 2003

2. Insert Slides commenting on Robins’ Paper

3. Both are decision problems in which multiple decisions must be made: “Multi-stage Decisions” Connection Between Papers

4. Collect information: L 1 (Robins) or y 1 (M  ller) Choose a treatment (or decision) based on this info  a 1 (or d 1 ) Collect more information: L 2 (or y 2 ) Choose an treatment based on info to date  a 2 (or d 2 ) And so on….. until time K (or T) GOAL: Choose a1, a2,…. to maximize E[u(a 1,…a K,L 1,….L K+1 ;  )]

5. Information Used to Achieve Goal Robins: Random Sample of Observations: L 1, A 1, L 2, A 2, …., A K, L K+1 (A’s are randomized treatments)

6. Information Used to Achieve Goal M  ller: Known multivariate distribution of L 1, L 2, …., L K+1 indexed by decisions, a 1, a 2, …., a K and  + prior for .

7. A Commonality The information at time t: L 1, L 2, …., L t is high dimensional; both authors must use summary statistics, a.k.a. “Feature Extraction” Open problem: Best methods for Feature Extraction in multi-stage decision problems

8. Delayed Effects of Decisions Treatment decision influences L t and other unobserved individual characteristics: Robins’ setting: known L t and other unobserved characteristics may interact with next treatment effect. M  ller’s setting: Optimal decision depends on latent . L t is used to update the distribution of .

9. Computational Development Robins’ and Murphy’s methods: undeveloped with unknown computational issues. M  ller’s, Bayesians, Reinforcement Learning methods: well developed with sophisticated understanding of computational issues.

10. Interesting Philosophical Note Statistical analysis should be conducted as if it is a confirmatory trial versus Statistical analysis should be conducted as if this trial is part of a sequence of trials with a confirmatory study to follow. Sequential Experimentation