Exercise 1: Specify a prior model

The first step in learning about \(\pi\) is to specify a prior model for \(\pi\) (i.e. prior to collecting any data). Suppose your friend specifies their understanding of \(\pi\) through the “Beta(2, 20)” model. Plot this Beta model and discuss what it tells you about your friend’s prior understanding. For example:

• What do they think is the most likely value of \(\pi\)?
  
• What range of \(\pi\) values do they think are plausible?

plot_beta(alpha = 2, beta = 20)

Notes:

• proportion between 0 and 1 (not \(-\infty\) to \(+\infty\))

• this model, beta-2-20, is right skewed

Q: what is your friend saying is the most likely value of pie?

A: about .12, aka about 12% of people believe in climate change.

• spike of model is best estimate

• looking at range the prior model drops off above .25, so you friend believes under 25% of people believe in climate change.

Exercise 2: Check out some data

The second step in learning about \(\pi\), the proportion of U.S. adults that believe that climate change is real and caused by people, is to collect data. Your friend surveys 10 people and 6 believe that climate change is real and caused by people. The likelihood function of \(\pi\) plots the chance of getting this 6-out-of-10 survey result under different possible \(\pi\) values. Based on this plot:

• With what values of \(\pi\) are the 6-out-of-10 results most consistent?

  Approx. 60% of people belive in climate change. Shown by our graph spiking at that value.

• For what values of \(\pi\) would these 6-out-of-10 results be unlikely?

  Our data would not be very likely to happen for values below .25 and above .9.

plot_binomial_likelihood(y = 6, n = 10)

Notes:

• The next step after creating a model (beta-2-20) we collect data.

• This plot is showing us what the chance is that we got these survey results under different possible pie values.

Exercise 3: Build the posterior model

In a Bayesian analysis of \(\pi\), we build a posterior model of \(\pi\) by combining the prior model of \(\pi\) with the data (represented through the likelihood function). Plot all 3 components below. Summarize your observations:

• What’s your friend’s posterior understanding of \(\pi\)?

  My friend’s prior understanding of \(\pi\) is not as low as what it was before, but also not as high as what is suggested in the data.

• How does their posterior understanding compare to their prior and likelihood? Thus how does their posterior balance the prior and data?

plot_beta_binomial(alpha = 2, beta = 20, y = 6, n = 10)

Notes:

• Depends on a lot of factors

Exercise 4: Another friend

Consider another friend that saw the same 6-out-of-10 polling data but started with a Beta(1, 1) prior model for \(\pi\):

plot_beta(alpha = 1, beta = 1)

• Describe the new friend’s understanding of \(\pi\). Compared to the first friend, are they more or less sure about \(\pi\)?

• Do you think the new friend will have a different posterior model than the first friend? If so, how do you think it will compare?

• Test your intuition. Use plot_beta_binomial() to explore your new friend’s posterior model of \(\pi\).

plot_beta_binomial(alpha = 1, beta = 1, y = 6, n = 10)

Notes:

• this is a shoulder shrug, uncertain prior model. It could really be anything.

Exercise 5: More data

Your two friends come across more data. In the pulse_of_the_nation survey, 655 of 1000 people believed climate change was real and caused by people:

data("pulse_of_the_nation")
pulse_of_the_nation %>% 
  count(climate_change)

## # A tibble: 3 × 2
##   climate_change                    n
##   <fct>                         <int>
## 1 Not Real At All                 150
## 2 Real and Caused by People       655
## 3 Real but not Caused by People   195

• How do you think the additional data will impact your first friend’s posterior understanding of \(\pi\)? What about the second friend’s?

• Upon seeing the 1000-person survey results, do you think your two friends’ posterior understandings of \(\pi\) will disagree a lot or a little?

• Test your intuition. Use plot_beta_binomial() to explore both friends’ posterior models of \(\pi\).

# first friend 
plot_beta_binomial(alpha = 2, beta = 20, y = 655, n = 1000)

# second friend 
plot_beta_binomial(alpha = 1, beta = 1, y = 655, n = 1000)

–> See the link to the github repository above for the rest of this activity.