1 Francisco José Vázquez Polo [ Miguel Ángel Negrín Hernández [ {fjvpolo or...
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Transcript of 1 Francisco José Vázquez Polo [ Miguel Ángel Negrín Hernández [ {fjvpolo or...
1
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
Course on Bayesian Methods
Basics (continued):
Models for proportions and means
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.
10
Binomial and Beta distributions
Binomial distribution:
X has a binomial distribution with parameters θ and n if its density function is:
Moments:
.0integrerand;10
;,...,1,0
1,|,|
n
nxfor
x
nnxBnx xnx
1,|,| nnXVandnXE
Prior: Beta distribution
1. θ has a beta distribution with parameters α and β if its density function is:
2. Moments:
0and;0;10for
1,
,| 11
Beta
12
Var
E
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
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
Beta(.25, .25)
Beta(3, 7)
Beta(1, 1)
Prior: Beta distribution
2
1
12
Mode
Var
E
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:
xxx 11|
ii xnx
nxx 1|,...,1
The conjugate prior distribution for θ is the beta distribution Beta(α0, β0) with density:
The posterior distribution of θ given X has density:
11
00
00 00 1,
in
in
nnn
xn
x
Betaxx
0
0
1 ,|,...,|
Beta-Binomial Model
Updating parameters
Prior Posterior
in
in
xn
x
00
00
2
1
1
00
0
002
00
00
00
0
n
xModa
nn
xnxVar
n
xE
i
ii
i
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.
1. Elicit the prior information
2. 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
0
Poisson and Gamma distributions
Poisson distribution:
X has a Poisson distribution with parameters λ if its density function is:
Moments:
.0
;,...,1,0!
||
nxforx
exPx
x
|| XVandXE
Prior: Gamma distribution
1. λ has a gamma distribution with parameters α and β if its density function is:
2. Moments:
0and;0;0for
,| 1
eGamma
1
; 2
Mode
VarE
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)
1
; 2
Mode
VarE
The conjugate prior distribution for λ is the gamma distribution Gamma(α0, β0) with density:
The posterior distribution of θ given X has density:
n
x
Gammaxx
n
in
nnn
0
0
1 ,|,...,|
Poisson-Gamma Model
00 1
0
0
e
Updating parameters
Prior Posterior
n
x
n
inn
00
0
n
xModa
n
xVar
n
xE
i
i
i
0
0
20
0
0
0
1
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}
1. Elicit the prior information
2. Obtain 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