1 / 23

Francisco José Vázquez Polo [personales.ulpgc.es/fjvpolo.dmc]

Course on Bayesian Methods. Basics (continued): Models for proportions and means. 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. 1. Binomial and Beta distributions

kobe
Download Presentation

Francisco José Vázquez Polo [personales.ulpgc.es/fjvpolo.dmc]

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Course on Bayesian Methods Basics (continued): Models for proportions and means 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 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 Only simple models result in equations More complex models require numerical methods to compute posterior mean, posterior standard deviations, prediction, and so on. MCMC

More Related