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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 - PowerPoint PPT Presentation

Transcript of Francisco Jos© Vzquez Polo [

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    Francisco Jos Vzquez Polo [www.personales.ulpgc.es/fjvpolo.dmc]Miguel ngel Negrn Hernndez [www.personales.ulpgc.es/mnegrin.dmc]

    {fjvpolo or mnegrin}@dmc.ulpgc.esCourse on Bayesian Methods Basics (continued):Models for proportions and means

  • Binomial and Beta distributions

    Problem:Suppose that 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.

  • Binomial and Beta distributions

    Binomial distribution:X has a binomial distribution with parameters and n if its density function is:

    Moments:

  • Prior: Beta distribution

    has a beta distribution with parameters and if its density function is:

    2. Moments:

  • 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

  • Prior: Beta distribution

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

  • Beta-Binomial Model

    1.ModelGiven the observations X1,,Xm are mutually independent with B(x|,1) density function:

    The joint density of X1,,Xn given is:

  • The conjugate prior distribution for is the beta distribution Beta(0, 0) with density:

    The posterior distribution of given X has density:

    Beta-Binomial Model

  • Updating parameters

    PriorPosterior

  • Posterior: Beta distributionPosterior moments:

  • 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 informationObtain the posterior distribution (mean, mode, variance)

  • Poisson and Gamma distributions

    Problem:Suppose that represents a the mean of a discrete variable X. Model used in analyzing count data.

    Examples:Number of visits to an specialistNumber of visitors to state parksThe number of people killed in road accidents

  • Poisson and Gamma distributions

    Poisson distribution:X has a Poisson distribution with parameters if its density function is:

    Moments:

  • Prior: Gamma distribution

    has a gamma distribution with parameters and if its density function is:

    2. Moments:

  • Prior: Gamma distribution

    Advantages of the Gamma distribution:

    - The gamma distribution is a conjugate family for the Poisson distribution - It is very flexible

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

  • The conjugate prior distribution for is the gamma distribution Gamma(0, 0) with density:

    The posterior distribution of given X has density:

    Poisson-Gamma Model

  • Updating parameters

    PriorPosterior

  • Posterior moments: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 informationObtain the posterior distribution (mean, mode, variance)Posterior: Gamma Distribution

  • Other conjugated analysis

  • 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