A Practical Course in Graphical Bayesian Modeling; Class 1

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A Practical Course in Graphical Bayesian
Modeling Class 1
Eric-Jan Wagenmakers
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Outline

• A bit of probability theory
• Bayesian foundations
• Parameter estimation A simple example
• WinBUGS and R2WinBUGS

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Probability Theory (Wasserman, 2004)

• The sample space O is the set of possible
  outcomes of an experiment.
• If we toss a coin twice then O HH, HT, TH,
  TT.
• The event that the first toss is heads isA
  HH, HT.

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Probability Theory (Wasserman, 2004)

• denotes intersection A and B
• denotes union A or B

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Probability Theory (Wasserman, 2004)
P is a probability measure when the following
axiomsare satisfied
1. Probabilities are never negative
2. Probabilities add to 1.
2. The probability of the union of
non-overlapping (disjoint) events is its sum
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Probability Theory (Wasserman, 2004)
For any events A and B
O
A
B
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Conditional Probability
The conditional probability of A given B is
O
A
B
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Conditional Probability
You will often encounter this as
O
A
B
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Conditional Probability
From
and
follows Bayes rule.
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Bayes Rule
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The Law of Total Probability
Let A1,,Ak be a partition of O. Then, for any
event B
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The Law of Total Probability
This is just a weighted average of P(B) over
thedisjoint sets A1,,Ak. For instance, when all
P(Ai) areequal, the equation becomes
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Bayes Rule Revisited
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Example (Wasserman, 2004)

• I divide my Email into three categories spam,
  low priority, and high priority.
• Previous experience suggests that the a priori
  probabilities of a random Email belonging to
  these categories are .7, .2, and .1,
  respectively.

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Example (Wasserman, 2004)

• The probabilities of the word free occurring in
  the three categories is .9, .01, .01,
  respectively.
• I receive an Email with the word free. What is
  the probability that it is spam?

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Outline

• A bit of probability theory
• Bayesian foundations
• Parameter estimation A simple example
• WinBUGS and R2WinBUGS

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The Bayesian Agenda

• Bayesians use probability to quantify uncertainty
  or degree of belief about parameters and
  hypotheses.
• Prior knowledge for a parameter ? is updated
  through the data to yield the posterior
  knowledge.

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The Bayesian Agenda
Also note that this equation allows one to learn,
from the probability of what is observed,
something about what is not observed.
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The Bayesian Agenda

• But why would one measure degree of belief by
  means of probability? Couldnt we choose
  something else that makes sense?
• Yes, perhaps we can, but the choice of
  probability is anything but ad-hoc.

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The Bayesian Agenda

• Assume degree of belief can be measured by a
  single number.
• Assume you are rational, that is, not
  self-contradictory or obviously silly.
• Then degree of belief can be shown to follow the
  same rules as the probability calculus.

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The Bayesian Agenda

• For instance, a rational agent would not hold
  intransitive beliefs, such as

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The Bayesian Agenda

• When you use a single number to measure
  uncertainty or quantify evidence, and these
  numbers do not follow the rules of probability
  calculus, you can (almost certainly?) be shown to
  be silly or incoherent.
• One of the theoretical attractions of the
  Bayesian paradigm is that it ensures coherence
  right from the start.

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Coherence Examplea la De Finetti

• There exists a ticket that says If the French
  national soccer team wins the 2010 World Cup,
  this ticket pays 1.
• You must determine the fair price for this
  ticket.
• After you set the price, I can choose to either
  sell the ticket to you, or to buy the ticket from
  you. This is similar to how you would divide a
  pie according to the rule you cut, I choose.
• Please write this number down, you are not
  allowed to change it later!

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Coherence Examplea la De Finetti

• There exists another ticket that says If the
  Spanish national soccer team wins the 2010 World
  Cup, this ticket pays 1.
• You must again determine the fair price for this
  ticket.

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Coherence Examplea la De Finetti

• There exists a third ticket that says If either
  the French or the Spanish national soccer team
  wins the 2010 World Cup, this ticket pays 1.
• What is the fair price for this ticket?

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Bayesian Foundations

• Bayesians use probability to quantify uncertainty
  or degree of belief about parameters and
  hypotheses.
• Prior knowledge for a parameter ? is updated
  through the data to yield posterior knowledge.
• This happens through the use of probability
  calculus.

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Bayes Rule
Prior Distribution
Likelihood
Posterior Distribution
Marginal Probability of the Data
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Bayesian Foundations
This equation allows one to learn, from the
probability of what is observed, something about
what is not observed. Bayesian statistics was
long known as inverse probability.
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Nuisance Variables

• Suppose ? is the mean of a normal distribution,
  and a is the standard deviation.
• You are interested in ?, but not in a.
• Using the Bayesian paradigm, how can you go from
  P(?, a x) to P(? x)? That is, how can you get
  rid of the nuisance parameter a? Show how this
  involves P(a).

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Nuisance Variables
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Predictions

• Suppose you observe data x, and you use a model
  with parameter ?.
• What is your prediction for new data y, given
  that youve observed x? In other words, show how
  you can obtain P(yx).

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Predictions
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Want to Know More?
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Outline

• A bit of probability theory
• Bayesian foundations
• Parameter estimation A simple example
• WinBUGS and R2WinBUGS

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Bayesian Parameter Estimation Example

• We prepare for you a series of 10 factual
  true/false questions of equal difficulty.
• You answer 9 out of 10 questions correctly.
• What is your latent probability ? of answering
  any one question correctly?

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Bayesian Parameter Estimation Example

• We start with a prior distribution for ?. This
  reflect all we know about ? prior to the
  experiment. Here we make a standard choice and
  assume that all values of ? are equally likely a
  priori.

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Bayesian Parameter Estimation Example

• We then update the prior distribution by means of
  the data (technically, the likelihood) to arrive
  at a posterior distribution.

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The Likelihood

• We use the binomial model, in which P(D?) is
  given bywhere n 10 is the number of trials,
  and s9 is the number of successes.

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Bayesian Parameter Estimation Example

• The posterior distribution is a compromise
  between what we knew before the experiment (i.e.,
  the prior) and what we have learned from the
  experiment (i.e., the likelihood). The posterior
  distribution reflects all that we know about ?.

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Mode 0.9 95 confidence interval (0.59, 0.98)
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Bayesian Parameter Estimation Example

• Sometimes it is difficult or impossible to obtain
  the posterior distribution analytically.
• In this case, we can use Markov chain Monte Carlo
  algorithms to sample from the posterior. As the
  number of samples increases, the approximation to
  the analytical posterior becomes arbitrarily
  small.

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Mode 0.89 95 confidence interval (0.59,
0.98) With 9000 samples, almost identical
to analytical result.
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Outline

• A bit of probability theory
• Bayesian foundations
• Parameter estimation A simple example
• WinBUGS and R2WinBUGS

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WinBUGS
Bayesian inference UsingGibbs Sampling
You want to have thisinstalled (plus the
registration key)
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WinBUGS

• Knows many probability distributions
  (likelihoods)
• Allows you to specify a model
• Allows you to specify priors
• Will then automatically run the MCMC sampling
  routines and produce output.

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Want to Know MoreAbout MCMC?
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Models in WinBUGS

• The models you can specify in WinBUGS are
  directed acyclical graphs (DAGs).

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Models in WinBUGS(Spiegelhalter, 1998)
Below, E depends only on C
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Models in WinBUGS(Spiegelhalter, 1998)
If the nodes are stochastic, the
jointdistribution factorizes
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Models in WinBUGS(Spiegelhalter, 1998)
P(A,B,C,D,E) P(A) P(B) P(CA,B)
P(DA,B) P(EC)
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Models in WinBUGS(Spiegelhalter, 1998)
This means we can sometimes performlocal
computations to get what we want
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Models in WinBUGS(Spiegelhalter, 1998)
What is P(CA,B,D,E)?
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Models in WinBUGS(Spiegelhalter, 1998)
P(CA,B,D,E) is proportional to P(CA,B) P(EC)
? D is irrelevant
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WinBUGS R

• WinBUGS produces MCMC samples.
• We want to analyze the output in a nice program,
  such as R.
• This can be accomplished using the R package
  R2WinBUGS

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End of Class 1