24-25 January 2007 An Overview of State-of-the-Art Data Modelling
The triplot
• A triplot gives a graphical representation of prior to posterior updating.
24-25 January 2007
An Overview of State-of-the-Art Data Modelling
Prior to posterior updating
Prior Data Posterior Bayes’s theorem is used to update our beliefs.
Bayesian methods, priors and Gaussian processes
John Paul Gosling Department of Probability and Statistics
Overview
• The Bayesian paradigm • Bayesian data modelling • Quantifying prior beliefs • Data modelling with Gaussian processes
• So, once we have our posterior, we have captured all our beliefs about the parameter of interest. • We can use this to do informal inference, i.e. intervals, summary statistics. • Formally, to make choices about the parameter, we must couple this with decision theory to calculate the optimal decision.
Bayes’s theorem for distributions
This Bayesian can probability be statistics, extended courses, to we continuous usewe Bayes’s • In early are taught theorem in distributions a particular : for events way: Bayes’s theorem :
24-25 January 2007
An Overview of State-of-the-Art Data Modelling
Bayesian methods
The beginning, the subjectivist philosophy, and an overview of Bayesian techniques.
P(data|parameters)
To a Bayesian, the parameters are uncertain, the observed data are not: P(parameters|data)
24-25 January 2007 An Overview of State-of-the-Art Data Modelling
Prior Likelihood Posterior
24-25 January 2007
An Overview of State-of-the-Art Data Modelling
Audience participation
Quantification of our prior beliefs • What proportion of people in this room are left handed? – call this parameter ψ • When I toss this coin, what’s the probability of me getting a tail? –tive probability
• Bayesian statistics involves a very different way of thinking about probability in comparison to classical inference. • The probability of a proposition is defined to a measure of a person’s degree of belief. • Wherever there is uncertainty, there is probability • This covers aleatory and epistemic uncertainty
24-25 January 2007 An Overview of State-of-the-Art Data Modelling
Sequential updating
Prior beliefs Posterior beliefs
Posterior beliefs
Today’s posterior is tomorrow’s prior
The posterior is proportional to the prior times the likelihood.
24-25 January 2007 An Overview of State-of-the-Art Data Modelling
Posterior distribution
24-25 January 2007 An Overview of State-of-the-Art Data Modelling
Differences with classical inference
To a frequentist, data are repeatable, parameters are not: