THOMAS BAYES TO THE RESCUE
st5219: 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) = 0.04
f(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

likelihood
(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
SUMMARISING THE POSTERIOR
Mean:
Median:
Mode:
SUMMARISING THE POSTERIOR

95% credible interval: chop off 2.5% from either
side of posterior
SUMMARISING THE POSTERIOR
Bye bye
delta
approxi
mations
!!!
SOUNDS TOO EASY! WHAT’S THE CATCH?!

Here are where the difficulties are:
1.
2.
3.


building the model
obtaining the posterior
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!