The Bayesian recipe

Outline

Topics

• Introduce the Bayesian Recipe (synonym for the Bayes estimator)
• Illustrate it using this week’s running example

Rationale

The Bayesian Recipe/Bayes estimator is the guide for all “full Bayesian” statistical analyses. This week we apply it to an example that builds on last week’s review.

This week’s running example

• You are consulting for a satellite operator
• They are about to send a $100M satellite on a Delta 7925H rocket

• Data: as of Jan 2024, Delta 7925H rockets have been launched 3 times, with 0 failed launches
  • Note: Delta 7925H is not reusable, so each rocket is “copy- built” from the same blueprint
• Should you recommend buying a $2M insurance policy?

Convention: use 1 for a success, 0 for a failure.

The Bayesian recipe

The goal this week is to undersand the 3 steps in the Bayesian recipe:

1. Construct a probability model including
   • random variables for what we will measure/observe
   • random variables for the unknown quantities
     • those we are interested in (“parameters”, “predictions”)
     • others that just help us formulate the problem (“nuisance”, “random effects”).
2. Compute the posterior distribution (condition on the data)
3. Use the posterior distribution to (decision theory):
   • make prediction (point estimate)
   • estimate uncertainty (credible intervals)
   • make a decision

Plan

• Understanding Step 1 (“construct a probability model”):
  • we reviewed probability models last week,
  • in fact, the model we use for this week’s problem is the same as last week’s!
  • We will just need to add some Bayesian terminology: prior, likelihood.
• Understanding Step 2 (“condition on the data”):
  • we reviewed conditional probability last week,
  • in fact, the conditional probability calculation for this week’s problem is the same as last week’s!
  • We just need to add some Bayesian terminology: posterior distributions.
• Understanding Step 3: this is where most of the new material will be for this week.