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