1. Bayesian Linear Regression
In regression analysis, we look at the conditional distribution of the response variable at different levels of a predictor variable

2. Bayesian Linear Regression
Response variable
- Also called “dependent” or “outcome” variable
- What we want to explain or predict
- In simple linear regression, response variable is continuous
Predictor variables
Also called “independent” variables or “covariates”
In simple linear regression, predictor variable usually is also continuous
How we define which variable is response and which is predictor depends on our research question

3. Quick review of linear functions
Y is a response variable that is a linear function of the predictor variable
β0: intercept; the value of Y when X=0
β1: slope; how much Y changes when X increases by 1 unit

4. Intro to Bayesian simple linear regression
Likelihood
Prior distribution
The posterior distribution is not straighforward
We have to implement MCMC techniques with WinBUGS

5. Examples
Willingness to Pay for Environmental Improvements in a Large City. For example, we can study the social benefits of a set of environmental and urban improvements planned for the waterfront of the City of Valencia (Spain):
Response variable:
How much are you willing to pay for this policy?
Covariates: Sex, Age, Income
Data: 80 individuals

6. Examples

7. Discrete choice experiment
Random Utility Model
Probit Model
Logit Model

8. Objectives:
Revealed preference models use random utilities
Probit models assume that utilities are multivariate normal
Probit MCMC generates latent, random utilities
Logit models assume that the random utilities have extreme value distributions.
Logit MCMC uses the Hasting-Metropolis algorith

9. Random Utility Model
Utility for alternative m is:
where there are n subjects (sample),
M+1 alternatives in the choice set, and
Ji choice ocassiones for subject i

10. Subject picks alternative k if
for all m
The probability of selecting k is
Statistical Models:
{εi,j,m} are Normal  Probit Model
{εi,j,m} are Extreme Value  Logit Model

11. Logit model in WinBUGS
Likelihood:
Prior distributions:

12. Logit model in WinBUGS
Example: Discrete choice experiment
To study the value that car consumers place upon environmental concerns when purchasing a car
Response variable: Yes/No
Attributes: safety (Yes/No), carbon dioxide emissions, acceleration from 0 to 100 km/h(<10sec. And < 7.5 sec)2.5 sec), second hand, and annual cost (900€, 1400€, 2000€).
Sample size: 150

13. Logit model in WinBUGS

14. Probit model in WinBUGS
Likelihood:
Prior distributions:

15. Probit model in WinBUGS

16. Hierarchical Logit yij ~ Bernoulli(pij),
The hierarchical logistic regression model is a very easy extension of standard logit.
Likelihood
yij ~ Bernoulli(pij),
logit(pij) <- b1j + b2jx2ij + … bkjxkij
Priors
bjk ~ N(Bjk, Tk) for all j,k
Bjk <- k1 + k2 zj2 + … + km zjm
qr ~ N(0, .001) for all q,r
Tk ~ Gamma(.01, .01)