Bayesian Statistics and Everyday Life

Steve

Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.

Steve

Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.

Which is more likely?

1. Steve is a librarian.
2. Steve is a farmer.

Steve

Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.

Which is more likely?

1. Steve is a librarian.
2. Steve is a farmer.

Steve

• Let's think about the main question:
  • Farmer or librarian?
• There are 20-60x more farmers in the U.S. than librarians.
  • Say 20x.
• With no additional evidence, 1/21 chance of being a librarian.
  • This is called our prior hypothesis.

Steve

• We got new evidence in the statement:
  • "meek and tidy," "shy and withdrawn"
• ​Let's suppose 50% of librarians have those traits and 10% of farmers.
• How does this evidence update our hypothesis?
• Bayes box

Linda

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Linda

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more likely?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

Linda

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more likely?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

Steve, reworded

• The original wording wasn't exactly fair.
• How was Steve chosen?
  • Do I personally know him?
• Did I know his job when I wrote the description?
  • How did that influence the description?
• Why were farmers and librarians chosen for the jobs?
  • Was it to imply one or the other? Which?

Steve, reworded

Steve is picked uniformly at random from the American population. I then interview Steve and write the previous description of him. Then, I select uniformly at random two jobs that exist in the United States.  

Which is more likely?

1. Steve is a librarian.
2. Steve is a farmer.

Envelope Auction

• Earlier today, I rolled three dice.
• After rolling the dice, I placed three different denominations of U.S. notes (paper money) using a secret method.
• What do you want to bid for the envelope?

Envelope Auction

• Was the answer what you expected?
  • Did you account for being wrong?
  • Did you account for me having more information?
• Was the person who won "happy?"
  • Even without the sneaky answer, would they have been happy?
  • P(Guess<answer|highest guess)
• This is a combination of adverse selection and the winner's curse.

Adverse Selection

• Arises from people having imperfect information.
• People only choose to do something when their information gives them an advantage.

More examples

• Empty subway car/restaurant.
• Buying or selling something (used car, stock).

Are Humans Bayesian?

Are Humans Bayesian?

• Sometimes
• Maybe we should be more Bayesian...

Are Humans Bayesian?

• We are not perfect Bayesian networks
  • Cannot update beliefs every time
    • Exponential in number of variables
  • Can have "orphaned" beliefs

Are Humans Bayesian?

• Aumann's Agreement Theorem

Are Humans Bayesian?

• Confirmation bias in Bayesian terms
• Ex:
  • Rowan says she saw a deer in Williamsburg
  • Dan says he saw a polar bear in Williamsburg
  • My prior: >99% chance deer exist in Williamsburg, <1% chance polar bears are here
  • Neither update my prior very much
    • But they do slightly
• When could this fail?