1. 1 Francisco José Vázquez Polo [www.personales.ulpgc.es/fjvpolo.dmc] Miguel Ángel Negrín Hernández [www.personales.ulpgc.es/mnegrin.dmc] {fjvpolo or mnegrin}@dmc.ulpgc.es Course on Bayesian Methods Basics (continued): Models for proportions and means

2. Binomial and Beta distributions Problem: Suppose that θ represents a percentage and we are interested in its estimation: Examples: -Probability of a single head occurs when we throw a coin. -probability of using public transport -Probability of paying for the entry to a natural park.

3. Binomial and Beta distributions Binomial distribution: X has a binomial distribution with parameters θ and n if its density function is: Moments:

4. Prior: Beta distribution 1.θ has a beta distribution with parameters α and β if its density function is: 2. Moments:

5. Prior: Beta distribution Advantages of the Beta distribution: - Its natural unit range from 0 to 1 - The beta distribution is a conjugate family for the binomial distribution - It is very flexible

6. Prior: Beta distribution

7. - Elicitation - Non-informative prior: Beta(1,1), Beta(0.5, 0.5)

8. Beta-Binomial Model 1.Model Given θ the observations X 1,…,X m are mutually independent with B(x|θ,1) density function: The joint density of X1,…,Xn given θ is:

9. The conjugate prior distribution for θ is the beta distribution Beta(α0, β0) with density: The posterior distribution of θ given X has density: Beta-Binomial Model

10. Updating parameters PriorPosterior

11. Posterior: Beta distribution Posterior moments:

12. Binomial and Beta distributions Example: We are studying the willingness to pay for a natural park in Gran Canaria (price of 5€). We have a sample of 20 individuals and 14 of them are willing to pay 5 euros for the entry. 1.Elicit the prior information 2.Obtain the posterior distribution (mean, mode, variance)

13. Poisson and Gamma distributions Problem: Suppose that λ represents a the mean of a discrete variable X. Model used in analyzing count data. Examples: -Number of visits to an specialist -Number of visitors to state parks -The number of people killed in road accidents

14. Poisson and Gamma distributions Poisson distribution: X has a Poisson distribution with parameters λ if its density function is: Moments:

15. Prior: Gamma distribution 1.λ has a gamma distribution with parameters α and β if its density function is: 2. Moments:

16. Prior: Gamma distribution Advantages of the Gamma distribution: - The gamma distribution is a conjugate family for the Poisson distribution - It is very flexible

17. Prior: Gamma distribution - Elicitation - Non-informative prior: Gamma(1,0), Gamma(0.5,0)

18. The conjugate prior distribution for λ is the gamma distribution Gamma(α 0, β 0 ) with density: The posterior distribution of θ given X has density: Poisson-Gamma Model

19. Updating parameters PriorPosterior

20. Posterior moments: Posterior: Gamma Distribution

21. Example: We are studying the number of visits to a natural park during the last two months. We have data of the weekly visits: {10, 8, 35, 15, 12, 6, 9, 17} 1.Elicit the prior information 2.Obtain the posterior distribution (mean, mode, variance) Posterior: Gamma Distribution

22. Other conjugated analysis

23. Good & Bad News Only simple models result in equations More complex models require numerical methods to compute posterior mean, posterior standard deviations, prediction, and so on. MCMC