An introduction to Bayesian reasoning Learning from experience: H M Higgins Learning from experience:

Outline Key differences between the ‘Bayesian’ and ‘classical’ approaches to statistics Overview of how the Bayesian methods works. Discuss the controversy The different ‘Schools of Bayesians’ Designing clinical trials Take home messages 

Motivation Total number of ‘Bayesian papers’ published per year Number published

Key differences between ‘Bayesian’ and ‘classical’ statistics. 1. What do we mean by ‘probability’? ‘Probability’ is the mathematical way to describe uncertainty Two different types of uncertainty: ‘Can’t know’ ‘Don’t know’ = Aleatory = Epistemic

Can’t know for sure until its happened Don’t know, but we could find out. A quantity we have uncertainty about because it is intrinsically unpredictable. A quantity we have uncertainty about because currently we have imperfect knowledge. Name?

Can’t know Can’t know Don’t know ‘probability’ = Bayesian world: Classical statistics: ‘probability’ = Bayesian world: Can’t know Can’t know Don’t know A key idea in Bayesian philosophy is the importance of acknowledging (and embracing) all types of uncertainty

2. Who does the uncertainty belong to? Classical statistics: the uncertainty is attached to the event itself. Bayesian world: the uncertainty is attached to the person observing the event.

3. What type of event do we consider? Classical world: ‘the probability of an event’ is the relative frequency with which it occurs in a series of repeatable experiments Bayesian world: ‘the probability for the event’ is the observers ‘degree of belief’ about ANY unknown quantity, unique or repeatable

Bayesian world: Probability has a subjective interpretation; it is a reflection of personal uncertainty, or ‘degree of belief’ for the unknown parameter. A key idea in Bayesian philosophy is the importance of respecting subjectivity and being transparent about it

4. Parameters ‘Parameters’ are the things that we are interested in, but which are unknown. We generate data to provide us with information about the parameters. In any type of statistical analysis we have to specify a statistical model to link our data to the parameters. Almost invariably, parameters are things which we ‘don’t know, because we have imperfect knowledge’ about them.

Classical statistics: You cannot assign probability distributions to parameters. Because although we have uncertainty about them, it is the ‘don’t know’ type, and this is not recognised in our definition of ‘probability’. You can never talk about the probabilities of parameters.

The results of classical analysis appear to be making probability statements about parameters, even though they cannot.

Example Unknown parameter: ‘the average height of adults in the UK’ (population mean) Take a random sample. Measure each person. Calculate: sample mean height (=170cm) 95% confidence interval (165cm to 175cm). ‘there is a 95% chance that the population mean height is between 165cm and 175cm’ Wrong!

How do you interpret the results of a classical analysis? It’s very difficult! The results of a classical analysis are telling you about what will happen if you repeat the experiment over and over again. It is not directly telling you anything about this one particular experiment you’ve just done…

Interpreting a 95% confidence interval 175cm height

Bayesian world: we can assign probability distributions to parameters and you CAN talk directly about the probabilities of parameters. This has two important implications: Interpreting the results of a Bayesian analysis is easier and more natural Bayesian approach allows us to directly answer many more questions

5. How many sources of data do we consider? Classical statistics: ONE source of information (our data) is used to learn about unknown quantities. Bayesian world: TWO sources of information (our data AND ‘prior information’). ‘prior information’ is any relevant information, external to our current data

Key resources this talk is based on A primer on Bayesian Statistics in Health Economics and Outcomes Research (2003) O’Hagan and Lucy ‘Bayesian Statistics’ , M249 Practical modern statistics, The Open University. O’Hagan, A. (2009), ‘Bayesian principles’, in O’Hagan, T. (ed.), Bayesian Methods in Health Economics: A short course, The Biomedical & Life Sciences Collection, Henry Stewart Talks Ltd, London (online at http://hstalks.com/bio)

Thank-you for listening! Any questions? NB. If you want a sensible answer, you’d be better off asking Jed….