1. MATH 643 Bayesian Statistics

2. 2 Discrete Case n There are 3 suspects in a murder case –Based on available information, the police think the following probabilities apply

3. 3 Discrete Case n New evidence comes to light –The shot came from 2000 feet –The police assess the following probabilities –These probabilities are called the likelihood –In this case, the likelihood of making that shot

4. 4 Discrete Case n How can we change our prior probabilities to account for the new evidence? –Bayes Theorem

5. 5 Discrete Case n What does this look like? –The likelihoods of making the shot have increased or decreased the prior probabilities.

6. 6 Continuous Case n You own a pretzel manufacturing company –An important consideration is market share –Y = # Customers out of N that buy your pretzel –We are uncertain about Y so we express this in terms of probabilities –Assume customers buy independently –Y ~ Binomial(N,p)

7. 7 Continuous Case n How can you estimate p? –Assume that we know the total daily pretzel market (N) –In one day suppose, y* people buy your brand –Before we said for fixed p, Y=y is this likely –Now, Y=y* is fixed and we wish to know p

8. 8 Continuous Case n What value of p would make observing Y=y* the most probable? –What is the maximum of L(p;y*) with respect to p? –This is the maximum likelihood estimate of p

9. 9 Continuous Case n MLEs are only a best guess –We can also say that we are 95% confident that p is somewhere in the interval

10. 10 Continuous Case n So really we are uncertain about the value of p –We are trying to express this uncertainty through confidence levels that act like, but are not really probabilities n What do we do when we are uncertain about something? We use probabilities!!

11. 11 Continuous Case n What would be a good distribution to express uncertainty about p? –p is a probability, lying between 0 and 1 –The beta distribution is very flexible for bounded variables like this Prior distribution Probability model

12. 12 Continuous Case n What does the prior distribution on p look like? –This is for n 0 = 4 and r 0 = 1

13. 13 Continuous Case n What does this mean we think our market looks like? Prior predictions

14. 14 Continuous Case n How can you estimate p? –Assume that we know the total daily pretzel market (N) –In one day suppose, y* people buy your brand –Update using Bayes Theorem

15. 15 Continuous Case n It turns out that the math works kind of nicely –The beta distribution is called the natural conjugate distribution for the binomial probability model –Remember that the p.d.f. for the beta and the p.m.f. for the binomial looked kind of similar Y* out of N buy our pretzel

16. 16 Continuous Case n What does this look like? –The prior distribution has been modified by the likelihood function

17. 17 Continuous Case n What does this mean we think our market looks like? Posterior predictions