THOMAS BAYES TO THE RESCUEst5219: Bayesian hierarchical modelling

lecture 1.4

BAYES THEOREM: MATHS ALERT

(You know this already, right?)

BAYES THEOREM: APPLICATION

You are GP in country like SP Foreign worker comes for HIV test HIV test results come back +ve Does worker have HIV?

How to work out?Test sensitivity is 98%Test specificity is 96%

ie f(test +ve | HIV +ve) = 0.98

f(test +ve | HIV --ve) = 0.04

BAYES THEOREM: APPLICATION

Analogy to hypothesis testing Null hypothesis is not infected Test statistic is test result p-value is 4% Reject hypothesis of non-

infection, conclude infected

But we calculated:f(+ test | infected)

NOT f(infected | + test)

BAYES THEOREM: APPLICATION

How to work out?Test sensitivity is 98%Test specificity is 96%Infection rate is 1%

ie f(test +ve | HIV +ve) = 0.98

f(test +ve | HIV --ve) = 0.04f(HIV +ve) = 0.01

BAYES THEOREM: APPLICATION

BAYES THEOREM: APPLICATION

AIDS AND H0S

Frequentists happy to use Bayes’ formula here

But unhappy to use it to estimate parameters But...

If you think it is wrong to use the probability of a positive test given non-infection to decide if infected given a positive test why use the probability of (imaginary) data

given a null hypothesis to decide if a null hypothesis is true given

data?

THE BAYESIAN ID AND FREQUENTIST EGO

How do you normally estimate parameters?

Is theta hat the most likely parameter value?

THE BAYESIAN ID AND FREQUENTIST EGO

The parameter that maximises the likelihood function is not the most likely parameter value

How can we get the distribution of the parameters given the data?

Bayes’ formula tells us

posteriorlikelihood prior

(this is a constant)

UPDATING INFORMATION VIA BAYES

Can also work with

1. Start with information before the experiment: the prior

2. Add information from the experiment: the likelihood

3. Update to get final information: the posterior

• If more data come along later, the posterior becomes the prior for the next time

UPDATING INFORMATION VIA BAYES

1. Start with information before the experiment: the prior

2. Add information from the experiment: the likelihood

3. Update to get final information: the posterior

UPDATING INFORMATION VIA BAYES

1. Start with information before the experiment: the prior

2. Add information from the experiment: the likelihood

3. Update to get final information: the posterior

UPDATING INFORMATION VIA BAYES

1. Start with information before the experiment: the prior

2. Add information from the experiment: the likelihood

3. Update to get final information: the posterior

Mean:

SUMMARISING THE POSTERIOR

Median:

Mode:

SUMMARISING THE POSTERIOR

95% credible interval: chop off 2.5% from either side of posterior

SUMMARISING THE POSTERIOR

Bye bye

delta approximation

s!!!

SOUNDS TOO EASY! WHAT’S THE CATCH?!

Here are where the difficulties are:1. building the model2. obtaining the posterior3. model assessment

Same issues arise in frequentist statistics (1, 3); estimating MLEs and CIs difficult for non à la carte problems

Let’s see an example! Back to AIDS!