Course on Bayesian Methods in Environmental Valuation Basics (continued): Models for proportions and means Francisco José Vázquez Polo [www.personales.ulpgc.es/fjvpolo.dmc] Miguel Ángel Negrín Hernández [www.personales.ulpgc.es/mnegrin.dmc] {fjvpolo or mnegrin}@dmc.ulpgc.es 1

Introduction to Bayesian Analysis Course on Bayesian Methods in Environmental Valuation Contents Introduction to Bayesian Analysis 2. Bayesian inference. Conjugate priors 2.1 Analysis of proportions 2.2 Analysis of count data 3. Software: WinBUGS

Introduction to Bayesian Analysis Thomas Bayes (1702 - 1761)

Introduction to Bayesian Analysis He set out Bayes’s theory of probability in the paper “An essay towards solving a problem in the doctrine of chances” (Philosophical Transactions of the Royal Society of London, 1763). The paper was sent by Richard Price, a friend of Bayes’. This paper introduced the concept of inverse probability; set of hypothesis prior probabilities, likelihood of the data A

Introduction to Bayesian Analysis Bayes’ Theorem: The posterior probability of Hi given A is proportional to the product of the prior probability of Hi and the likelihood of A when Hi is true.

What does probability mean? The frequency definition of the probability of an event: The probability of an event is the proportion of the time it would occur in a long sequence of observations (i.e. as the number of trials tends to infinity). Example: when we say that the probability of getting head on a toss of a fair coin is 0.5, we mean that we would expect to get a head half the time if we flipped the coin a huge number of times under exactly the same conditions. Requires a sequence of repeatable experiments. No frequency interpretation possible for probabilities of many kinds of events:

Probability as degree of belief The subjective definition of probability is A probability of an event is a number between 0 and 1 that measures a particular person’s subjective opinion as to how likely that event is to occur (or to have occurred). Applies whenever the person in question has an opinion about the event If we count ignorance as an opinion. Different people may have different subjective probabilities regarding the same event. The same person’s subjective probability may change as more information comes in.

Properties of probabilities These properties apply to probability whichever definition is being used: Probabilities must not be negative. If A is any event, then P(A) ≥ 0 All possible outcomes together must have probability 1. If S is the sample space in a probability model then P(S) = 1

Example: Do you have a rare disease? Suppose your friend is diagnosed with a rare disease that has no obvious symptoms. You wish to determine how likely it is that you, too, have the disease. That is, you are uncertain about your true disease status. Your friend’s doctor has told him that the proportion of people in the general population who have the disease is 0.001. The disease is not contagious. A blood test exists to detect the disease, but it sometimes gives incorrect results (0.05)

Example (continuation) Two possible events: You have the disease You don’t have the disease Before taking any blood test, you think your chance of having the disease is similar to that of a randomly selected person in the population. So you assign the following prior probabilities to the two events: Prob (Have disease) = 0.001 Prob(Don’t have disease) = 0.999

Data You decide to take the blood test. The new information that you obtain to learn about the different models is called data. The different possible data results are called observations. The data in this example is the result of the blood test. The two possible observations are: - A “positive” blood test (+) suggests you have the disease. - A “negative” blood test (-) suggests you don’t have the disease.

Likelihood The probabilities of the two possible test results are different depending on whether you have the disease or not. These probabilities are called likelihoods – the probabilities of the different data outcomes conditional on each possible model. P(+ | have disease) = 0.95 P(+ | don’t have disease) = 0.05 P(- | have disease) = 0.05 P(- | don’t have disease) = 0.95

Using Bayes’ rule to update probabilities Bayes’ rule is the formula for updating your probabilities about the models given the data. Enables you to compute posterior probabilities given the observed data Bayes’ rule: P(event | data)  P(event) x P(data | event) Posterior  prior x likelihood

Bayes’ rule applied to the example You take the blood test and the result is positive (+). This is the data or observation. P(have disease | +) = 0.019 P(don’t have disease | +) = 0.981

What have you learned from the blood test? The probability of your having the disease has increased by a factor of 19. But the actual probability is still small. You decide to obtain more information by taking the blood test again.

Updating the probabilities again We assume that the blood tests are independent. The posterior probabilities after the first test will become your prior probabilities with respect to the second test. Suppose that the second test is also positive. The new posterior probabilities are: P(have disease | +,+) = 0.269 P(don’t have disease | +,+) = 0.731

P(don’t have disease | +,-) = 0.999 What if the second test had been negative? Suppose that the second test is negative. The new posterior probabilities are: P(have disease | +,-) = ? P(don’t have disease | +,-) = ? P(have disease | +,-) = 0.001 P(don’t have disease | +,-) = 0.999

Prior distribution Before we see any data, we have some idea about what values the parameters might take Experts, experience, previous studies, and so on. Example: e.g. there are very few people 3m tall It is our subjective uncertainty about the parameters before we see the data

Prior Terminology Uninformative prior Uniform, as wide as possible Sometimes called flat priors Problem: often difficult to define Informative Prior Not uniform Assume we have some prior knowledge Conjugate Prior Prior and posterior have same distribution Often makes the maths easier

Noninformative or reference priors Useful when we want inference to be unaffected by information apart from the current data. In many scientific contexts, we would not bother to carry out an experiment unless we thought it was going to increase our knowledge significantly - i.e. we expect and want the likelihood to dominate the prior

Informative priors Elicitation is the process of extracting expert knowledge about some unknown quantity of interest, or the probability of some future event, which can then be used to supplement any numerical data that we may have. If the expert in question does not have a statistical background, as is often the case, translating their beliefs into a statistical form suitable for use in our analyses can be a challenging task. Prior elicitation is an important and yet under researched component of Bayesian statistics

Bayesian Inference As P(X) is a constant, all we need to estimate P(θ | X) are P(θ) and P(X | θ) Bayes’ rule becomes: P(θ | X) is called the posterior distribution Product of the prior and the likelihood We can ignore the constant of proportionality

Posterior distribution The posterior distribution contains all the current information about the unknown parameter All Bayesian inference is based on the posterior distribution: -Estimation Estimating values of unknown parameters that can never be observed or known Testing Prediction Estimating the values of potentially observable but currently unobserved quantities.

Learning The Bayesian approach is often talked about as a learning process As we get more data, we add it to our store of information by multiplying it by our current posterior distribution. It has been argued that this can form the basis of a philosophy of science

Bayesian Inference Estimation Point estimates (mean, mode, median) - Measures of spread Bayesian intervals

Bayesian Inference The posterior variance The posterior variance is one summary of the spread of the posterior distribution The larger the posterior variance, the more uncertainty we still have about the parameter

Bayesian Inference Precisely what information does a p-value provide? Recall the definition of a p-value: The probability of observing a test statistics as extreme as or more extreme than the observed value, assuming that the null hypothesis is true. What is the correct way to interpret a confidence interval? Does a 95% confidence interval provide a 95% probability region for the true parameter value? If not, what is the correct intepretation? “A range of values, which is likely, with a specified degree of certainty, to contain the true population value of a variable drawn from the study sample”

Bayesian Inference Frequentist approach Parameters are considered "fixed but unknown" We can not assign a distribution. Bayesian approach Parameters are considered random and unknown They are random because they are unknown

Bayesian Inference Bayesian intervals Called “posterior intervals” or “credible sets” Recall that the posterior distribution represents our updated subjective probability distribution for the unknown parameter. Thus, for us, the interpretation of the 95% credible set is that the probability is .95 that the true θ is in the interval. Contrast this with the interpretation of a frequentist confidence interval.

Hypothesis testing Frequentist approach H0 vs. H1 (2 hypothesis) α: Type I error (Probability of rejecting the hypothesis when hypothesis is true) 1%, 5% ó 10% ¿β: Type II error (Probability of accepting the hypothesis when hypothesis is false)? p-value  (accept if p-value > 0.05; reject if p-value < 0.05)

Hypthotesis testing Bayesian approach H0 , H1 , H2, etc. (several hypothesis) We can estimate the probability of each event from the posterior distribution of the parameters, f(θ|X). Prob(H0) = Prob(θ ≤ θ0) Prob(H1) = Prob(θ > θ0)

Prediction In many situations, interest focuses on predicting values of a future sample from the same population. -i.e. on estimating values of potentially observable but not yet observed quantities Example: we can be interested in the result of the next blood test. The posterior predictive probability is defined as

Introduction to Bayesian Analysis Some settings in which Bayesian statistics is used today: Marketing Economics and econometrics Social sciences Queues Education Operations & Manufacturing Health policy Quality Medical research Test & Diagnosis Weather Maintenance Environmental

Introduction to Bayesian Analsysis. References Lee, P. (1993) “Bayesian Statistics: An introduction”. Oxford, UK: Oxford University Press, UK. Zellner, A. (1971) “An introduction to Bayesian Inference and Econometrics”. John Wiley & Sons. Chen, M., Shao, Q. e Ibrahim, J.(2000). “Monte Carlo Methods in Bayesian Computation”. Springer-Verlag. NY. Leonard,T. y Hsu, J.S.(1999). “Bayesian Methods. An analysis for statisticians and interdisciplinary researches” Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge. O’Hagan, A.(1994). “Bayesian Inference”. Kendall’s Advanced Theory of Statistics (vol.2b). E. Arnold. University Press. Cambridge. O’Hagan, A.(2003). “A primer on Bayesian Statistics in Health Economics and Outcomes Research”. Centre for Bayesian Statistics in Health Economics.