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# File:   MIT18_05S22_in-class15-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.
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Class 15 Beta distributions, Conjugate priors
# Class 15 in spring 2022 is a combination of previous years' class14 and class15. We skipped general continuous-continuous updating and only do the normal-normal as a conjugate pair

Jen
  Slide 1: One key point of today --don't actually compute integrals (1 minute)

  Slide 2: Agenda/Announcements (1 minute)

  Slide 3: Advertisement: (2 minutes)
   What is this good for.
   
  Slide 4:  Beta distribution (5 minutes)
    Use board to write c\theta^{a-1}(1-\theta)^{b-1}
    Show applet for left-right and spikiness 

  Slide 5: Observation (2 minutes)
    We've made this point many times.
    It's worth pointing out that it is the key to avoiding computing integrals.

  Slide 6: Flat prior (2 minute)
    Point is Beta(1,1) is special
    
  Slide 7: PREAMBLE to board question (1 minute)

  Slide 8: BOARD QUESTION beta-binomial update (Work 10 minutes, discuss 5 minutes)
     Discussion: This is just what we did in the last class. What's new is we now have a name for the distribution and a formula for the normalizing factor.
     The point of (b) is that the presence or absence of the binomial coefficient doesn't change the posterior.

  Slide 9: Conjugate priors (2 minutes)
    This is just a reminder of the use of the term in the reading
    and what they just saw.
    Don't dwell on this. The next concept question covers it again.

  Slide 10: CONCEPT QUESTION Binomial/Beta updating (5 minutes)
    Beta prior; beta posterior
    This is just for attendance. Everyone should get it.

Jerry
  Slide 11: CONCEPT QUESTION Strong priors (5 minutes)
    Moral: If your prior is 0, no amount of data will change your mind. Always leave some probability for things you think are unlikely.
    
  Slides 12,13: normal/normal formulas (4 minutes)
    Do not go over the formulas. They will work with them on the next board question.
    Just note: this follows by a mess of algebra involving completing the square. Don't attempt to explain this.
    With the stronger groups we can explain it by waving our hands at the formula they have in part b of the next board question 

  Slide 14: BOARD QUESTION normal-normal (Work 10 minutes, discussion 5 minutes)
     They will find the updating easy.
     We should push them to put in the formula for the normal pdf in the table in part b
     Discussion: Hardest part is keeping the terms straight

  Slides 15, 16: Concept questions on normal-normal updating (6 minutes: 4,2)
     Key point for normal-normal updating: 
         mu_post is between u_prior and xbar
         sigma_post < sigma_prior (variance decreases)
     SKIP DISCUSSION of part (b) if no time. 
     (Can point them to posted solutions or office hours)
     Key to (b): xbar = 8, so mu_post between 6 and 8

  Slide 17: BOARD QUESTION: Normal-normal (Work 8 minutes, discussion 2 minute)
     This is normal-normal updating.
     The hard part is identifying the parts.
     With help they all should get it. No need to finish the calculations

Only if time:

Slide 18: Class discussion: Likelihood principle (6 minutes)
        (All the inferential information from the data is in the likelihood function)


Slide 19 Conjugate priors --table of pairs
        ***Careful explanation for TWO-- NOT ALL-- pairs. 

Slide 20 CLASS DISCUSSION question conjugate priors
         DISCUSSION -use board