Seven steps to formalizing medical decisions

Here are 7 steps for using probability to formalize medical decisions. For more on this topic, see papers on (dynamic) treatment regimes, reinforcement learning, or decision analysis.

1. Gather information. Suppose we have some patient covariates S that are continuous, for simplicity.

2. Decide how you might be able to act. Suppose we have a binary treatment, A, that will be taken according to S. Draw A from the conditional distribution \pi(A|S), which we call a “treatment policy”.

3. Understand how baseline covariates and treatments affect the patient. Suppose we have some future, post-treatment covariates, S', which depend on the initial covariates and the treatment. Draw S' from p(S'|A,S).

4. Think about what is truly desired. Define a reward, R(S,A,S') that you wish to maximize. For example, R might pick out one covariate that we want to be large (for simplicity, assume the function R is not random).

5. Relate the treatment policy, \pi, to the reward, R. Write the expectation E_{\pi} R = \int_s \int_{s'}\sum_a R(s,a,s') p(s'|a,s)\pi(a|s)p(s) ds'ds.

6. Find the treatment policy that gives the most reward, in expectation. Estimate \pi^*, where \pi^*=\arg\max_{\pi} E_{\pi} R.

7. Doing all of this in practice is difficult. However, the framework provides a way in which data can be used to make healthcare decisions, which will ultimately lead to better outcomes for patients.

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