#---------------------------------------------------------
# File:   MIT18_05S22_in-class16-script.txt
# Author: Jeremy Orloff
#
# MIT OpenCourseWare: https://ocw.mit.edu
# 18.05 Introduction to Probability and Statistics
# Spring 2022
# For information about citing these materials or our Terms of Use, visit:
# https://ocw.mit.edu/terms.
#
#---------------------------------------------------------
Class 16 Choosing priors. Probability Intervals

NOTE: The slides will show in class have more slides than the posted version. They include some of the tables in the solution to the first board question.

Jerry
  Slide 1

  Slide 2: Announcements/Agenda (2 minutes)

  Slide 3-8 Two parameter tables Malaria (6 minutes)
      QUICKLY: Point out this is a review from reading.
      Go through the mechanics of 2 param. tables
      Next board question covers the same ground

  Slide 9:  continuous 2 parameter example (2 minutes)
      Don't over explain
      In practice, if we use the computer we'll discretize

  Slides 10,11:  BOARD QUESTION Respiratory failure (Work 15 minutes, discussion 6 minutes)
       Make  flat and informed priors and likelihood table,
       The computation for this is long. We should get them to see what they have to do, but not finish the calculation
       Discussion goes with next slides which show tables

  *** These 3 slides are not in the posted slides. We'll have them in class to show if it seems better
  Slide 1: Solution priors (part of Board question discussion)
       Describe rationale for informed prior
  Slide 2: Likelihood
       Formula is on the slide. The exponents come from the data
  Slide 3: Posteriors
       P(theta_E > theta_C | Harvard data) = 0.94 for both posterios
       Point out that posteriors are similar --robustness to choice of prior is good.
       DISCUSSION

Jen
  Slide 12  Probability intervals (2 minutes)

  Slide 13  Prob. intervals in Bayesian updating (2 minutes)
      NOTE: We wanted them to see these. But we won't do much more with them.
      They are useful for helping to create subjective priors

  Slides 14 Normal prob. intervals (2 minutes)
      Note: uncentered intervals tend to be longer than centered ones for the same value.
      
  Slide 15 Beta intervals (1 minute)
      Just show, no need to spend any time -- similar to normal

  Slide 16 Concept Question --how to increase p (4 minutes)

  Slides 17-21  Reading question subjective prob. intervals (5 minutes)
      For airline: some people may have answered for average yearly deaths.
      No need to talk about any but the first
      Used data from 2022
      These are good examples of the difficulty in taking data.
      Notes --People interpreted questions differently
           e.g. question 1, it looks like some people gave an answer for average yearly # of deaths.
            --A good percentage didn't do them
            -- Some people clearly didn't take them seriously
            -- Beliefs vary greatly
      Given how many intervals don't cover the true value, we may want to learn to widen our intervals.

  Slides 22, 23 Citations for the numbers quoted in above slides (1 minute)
     Just point out these exist and that they give references for where we got the true values.

For ocw don't have these images See if can reproduce heat maps. Use xkcd for discussion

  If have time:
  Slide 24 Meteor intro

  Slide 25-27 Board questions --subjective priors and probability intervals
     We'll keep them standing as we change the slides. Discussion at end

  Slide 28: Answer to direction question
     Lesson: We're easily swayed by incorrect interpretations of pictures
     The second heat map shows the number of calls corresponds to population centers29

  Slide   XKCD commentary