Course on Bayesian Methods in Environmental Valuation

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Presentation transcript:

Course on Bayesian Methods in Environmental Valuation 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 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.

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

Prior: Beta distribution θ has a beta distribution with parameters α and β 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.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:

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:

Updating parameters Prior Posterior

Posterior: Beta distribution Posterior 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 information Obtain the posterior distribution (mean, mode, variance)

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

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

Prior: Gamma distribution λ has a gamma distribution with parameters α and β 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)

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:

Updating parameters Prior Posterior

Posterior: Gamma Distribution 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 information Obtain the posterior distribution (mean, mode, variance)

Other conjugated analysis

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