Download presentation
Presentation is loading. Please wait.
Published byJeremy Maxwell Modified over 6 years ago
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
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.