Bayesian Statistics in Discrete and Continuous Cases - Murder Suspects & Pretzel Market

1. MATH 643 Bayesian Statistics

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

3. Discrete Case • 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. Discrete Case • How can we change our prior probabilities to account for the new evidence? • Bayes Theorem

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

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

10. Continuous Case • 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 • What do we do when we are uncertain about something? We use probabilities!!

11. Continuous Case • 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. Continuous Case • What does the prior distribution on p look like? • This is for n0 = 4 and r0 = 1

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

14. Continuous Case • 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. Continuous Case • 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. Continuous Case • What does this look like? • The prior distribution has been modified by the likelihood function

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