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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

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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

- 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

- 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