Transcript bayes-polo-curso

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
Course on Bayesian Methods in Environmental Valuation
 Contents
1. 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;
H1 ,, H k
set of hypothesis
PH i , i  1,...., k
prior probabilities,
P A | H i   i  1,..., k likelihood of the data A
 P H 
i
i
Introduction to Bayesian Analysis
Bayes’ Theorem:
P A | H i PH i 
P H i | A 
 j PA | H j PH j 
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:
1.You have the disease
2.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 |   
P | have disease Phave disease 
P | have disease Phave disease   P | don' t have disease Pdon' t have disease 
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
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 | X   P P X |  
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).
H0 :  
0
H1 :   
0
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




P x | x   p x |  p | x d
*
*
Introduction to Bayesian Analysis
Some settings in which
Bayesian statistics is used
today:
-Marketing
-Social sciences
-Education
-Health policy
-Medical research
-Weather
-Environmental
-Economics and econometrics
-Queues
-Operations & Manufacturing
-Quality
-Test & Diagnosis
-Maintenance
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.