1. Course on Bayesian Methods in Environmental Valuation
Basics (continued):
Models for proportions and means
Francisco José Vázquez Polo [
Miguel Ángel Negrín Hernández [
{fjvpolo or 1

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

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

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

10. Updating parameters
Prior Posterior

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.
Elicit the prior information
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
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. Poisson-Gamma Model
The conjugate prior distribution for λ is the gamma distribution Gamma(α0, β0) with density:
The posterior distribution of θ given X has density:

19. Updating parameters
Prior Posterior

20. Posterior: Gamma Distribution
Posterior moments:

21. Posterior: Gamma Distribution
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}
Elicit the prior information
Obtain the posterior distribution (mean, mode, variance)

22. Other conjugated analysis

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