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Statistical Decision Theory Bayes’ theorem : For discrete events. For probability density functions. The Bayesian “philosophy” The classical approach (frequentist’s view):
For discrete events
For probability density functions
Frequentist: “What we observe is random, what we do not observe is fixed.”
Prior density: p( )
Likelihood: L( ; x) “as before”
Posterior density: q( | x ) = q( ; x ) The book uses the second notation
Relation through Bayes’ theorem:
Still the posterior is referred to as the distribution of conditional onx
True state of nature: Uncertainty described by the prior p ( )
Data: xobservation of X, whose p.d.f. depends on
(data is thus assumed to be available)
Action: (x) The decision procedure becomes an action when applied to given datax
Loss function: LS ( , (x) ) measures the loss from taking action (x) when holds
Note that the risk function is the expected loss with respect to the simultaneous distribution of X1, … , Xn
Note also that the risk function is for the decision procedure, and not for the particular action
A procedure 1 is inadmissible if there exists another procedure such that R( , 1 ) R( , 2 ) for all values of .
A procedure which is not inadmissible (i.e. no other procedure with lower risk function for any can be found) is said to be admissible
A procedure * is a minimax procedure if
i.e. is chosen to be the “worst” possible value, and under that value the procedure that gives the lowest possible risk is chosen
The minimax procedure uses no prior information about , thus it is not a Bayesian procedure.
Suppose you are about to make a decision on whether you should buy or rent a new TV.
1 = “Buy the TV” 2 = “Rent the TV”
Now, assume is the mean time until the TV breaks down for the first time
Let assume three possible values 6, 12 and 24 months
The cost of the TV is $500 if you buy it and $30 per month if you rent it
If the TV breaks down after 12 months you’ll have to replace it for the same cost as you bought it if you bought it. If you rented it you will get a new TV for no cost provided you proceed with your contract.
Let X be the time in months until the TV breaks down and assume this variable is exponentially distributed with mean
A loss function for an ownership of maximum 24 months may be defined as
LS ( , 1(X ) ) = 500 + 500 H (X – 12) and LS ( , 2(X ) ) = 30 24 = 720
Now compare the risks for the three possible values of
Clearly the risk for the first procedure increases with while the risk for the second in constant. In searching for the minimax procedure we therefore focus on the largest possible value of where 2 has the smallest risk
2 is the minimax procedure
Uses the prior distribution of the unknown parameter
A Bayes procedure is a procedure that minimizes the Bayes risk
Assume the three possible values of (6, 12 and 24) has the prior probabilities 0.2, 0.3 and 0.5.
Thus the Bayes risk is minmized by 1 and therefore 1 is the Bayes procedure
The action is a particular point estimator
State of nature is the true value of
The loss function is a measure of how good (desirable) the estimator is of :
Prior information is quantified by the prior distribution (p.d.f.) p( )
Data is the random sample x from a distribution with p.d.f. f (x ; )
Absolute error loss:
Quadratic (error) loss:
Find the value of that maximizes the expected loss with respect to the sample values, i.e. that maximizes
Then, the particular estimator that minimizes the risk for that value of is the minimax estimator
Not so easy to find!
A Bayes estimator is the estimator that minimizes
For any given value of x what has to be minimized is
The Bayes philosophy is that data (x ) should be considered to be given and therefore the minimization cannot depend on x.
Now minimization with respect to different loss functions will result in measures of location in the posterior distribution of .
Absolute error loss:
Conjugate prior distributions
Example: Assume the parameter of interest is , the proportion of some property of interest in the population (i.e. the probability for this property to occur)
A reasonable prior density for is the Beta density:
Now, assume a sample of size n from the population in which y of the values possess the property of interest.
The likelihood becomes
Thus, the posterior density is also a Beta density with parameters y + and n – y +
Prior distributions that combined with the likelihood gives a posterior in the same distributional family are named conjugate priors.
(prior distribution, sample distribution = likelihood)
In particular, if the sample distribution, i.e. f (x; ) belongs to the k-parameter exponential family of distributions:
we may put
where 1 , … , k + 1 are parameters of this prior distribution and K( )is a function of 1 , … , k + 1 only .
i.e. the posterior distribution is of the same form as the prior distribution but with parameters
Conjugate prior Sample distribution Posterior
Beta Binomial Beta
Normal Normal, known 2 Normal
Gamma Poisson Gamma
Pareto Uniform Pareto
Assume we have a sample x = (x1, … , xn ) from U (0, ) and that a prior density for is the Pareto density
What is the Bayes estimator of under quadratic loss?
The Bayes estimator is the posterior mean.
The posterior distribution is also Pareto with
A prior distribution that gives no more information about than possibly the parameter space is called a non-informative or uninformative prior.
Example: Beta(1,1) for an unknown proportion simply says that the parameter can be any value between 0 and 1 (which coincides with its definition)
A non-informative prior is characterized by the property that all values in the parameter space are equally likely.
Proper non-informative priors:
The prior is a true density or mass function
Improper non-informative priors:
The prior is a constant value over Rk
Example: N ( , ) for the mean of a normal population
Test of H0: = 0 vs. H1: = 1
Decision procedure: C = Use a test with critical region C
Action: C (x) = “Reject H0 if x C , otherwise accept H0 ”
Assume a prior setting p0 = Pr (H0 is true) = Pr ( = 0) and p1 = Pr (H1 is true) = Pr ( = 1)
The prior expected risk becomes
Lemma 6.1: Bayes tests and most powerful tests (Neyman-Pearson lemma) are equivalent in that
every most powerful test is a Bayes test for some values of p0 and p1 and every Bayes test is a most powerful test with
Assume x = (x1, x2 ) is a random sample from Exp( ), i.e.
We would like to test H0: =1 vs. H0: =2 with a Bayes test with losses a = 2 and b = 1 and with prior probabilities p0 and p1
A fixed size gives conditions on p0 and p1, and a certain choice will give a minimized
Suppose that we consider the sampling to be the observation of values in a “stream” x1, x2, … , i.e. we do not consider a sample with fixed size.
We would like to test H0: = 0 vs. H1: = 1
After n observations have been taken we have xn= (x1, … , xn ) , and we put
as the current test statistic.
Specify two numbers K1 and K2 not depending on n such that 0 < K1 < K2 < .
If LR(n) K1 Stop sampling, accept H0
If LR(n) K2 Stop sampling, reject H0
If K1 < LR(n) < K2 Take another observation
Usual choice of K1 and K2 (Property 6.3):
If the size and the power 1 – are pre-specified, put
This gives approximate true size and approximate true power 1 –
The structure is the same, but the choices of K1 and K2 is different.
Let c be the cost of taking one observation, and let as before a and b be the loss values for taking the wring decisions, and p0 and p1 be the prior probabilities of H0 and H1 respectively.
Then the Bayesian choices of K1and K2 are
“Very much lies in the posterior distribution”
Bayesian definition of sufficiency:
A statistic T (x1, … , xn ) is sufficient for if the posterior distribution of given the sample observations x1, … , xn is the same as the posterior distribution of given T, i.e.
The Bayesian definition is equivalent with the original definition (Theorem 7.1)
Let be the parameter space for and let Sx be a subset of .
then Sx is a 1 – credible region for .
For a scalar we refer to it as a credible interval
The important difference compared to confidence regions:
A credible region is a fixed region (conditional on the sample) to which the random variable belongs with probability 1 – .
A confidence region is a random region that covers the fixed with probability 1 –
Equal-tailed credible intervals
An equal-tailed 1 – credible interval is a 1 – credible interval (a , b ) such that
In a consignment of 5000 pills suspected to contain the drug Ecstasy a sample of 10 pills are sampled for chemical analysis. Let be the unknown proportion of Ecstasy pills in the consignment and assume we have a high prior belief that this proportion is 100%.
Such a high prior belief can be modelled with a Beta density
where is set to 1 and is set to a fairly high value, say 20, i.e. Beta (20,1)
Now, suppose after chemical analysis of the 10 pills, all of them showed to contain Ecstasy.
The posterior density for is Beta( + x, + n – x) (conjugate prior with binomial sample distribution as the population is large)
Beta (20 + 10, 1 + 10 – 10) = Beta (30, 1)
Then a lower-tail 99% credible interval for satisfies
Thus with 99% certainty we can state that at least 85.8% of the pills in the consignment consist of Ecstasy
We have used the binomial distribution for the sample. More correct would be to use the hypergeometric distribution, but the binomial is a good approximation.
For a smaller consignment (i.e. population) we can benefit on using the result that the posterior for the number of pills containing Ecstasy in the rest of the consignment after removing the sample is beta-binomial.
This would however give similar results
If a sample consists of 100% of one kind, how would a confidence interval for be obtained?
The issue is to test H0 vs. H1
Without specifying the hypotheses further, we seek to judge upon which of the two hypothesis that, conditional on the sample, is the most probable.
Note the difference compared to the classical approach: There we seek to rejectH0 in favour of H1 and never the opposite.
The aim of Bayesian hypothesis testing is to determine the posterior “odds”
With Q* > 1 we then say that conditional on the sample H0 is Q* times more probable than H1 and with Q* < 1 the expression is reversed.
For two events A and B from a random experiment we have
This result is usually referred to as Bayes theorem on odds form
Now, it is possible to replace A with H0 , A with H1 and B with x (the sample)
where Q is the prior “odds”
where L(H ; x) is the likelihood of H . The concept of likelihood is not restricted to parameters.
Note also that f (x | H0) need not have the same functional form as f (x | H1)
To make the comparison with classical hypothesis testing more transparent the posterior odds may be transformed to posterior probabilities for each of the two hypotheses:
Now, if the posterior probability of H1 is 0.95 this would be a result that could be compared with “H0 is rejected at 5% level of significance”
However, the two approaches cannot be made equal
Assume a crime has been conducted where a blood stain was left at the crime scene. A suspect is identified and a saliva sample is taken from this person. The DNA profiles are compared between the saliva sample and the blood stain and they appear to match (i.e. they are equal).
Put H0: “The blood stain comes from the suspect”
H1: “The blood stain comes from another person than the suspect
The Bayes factor becomes
Now, if laboratory mistakes can be discarded
Pr (Matching DNA profiles | H0 ) = 1
How about the probability in the denominator of B ?
This probability relates to the commonness of the current profile among people in general.
The DNA analysis very strongly supports H0 to be true.
For sake of simplicity we express the parameter as a scalar, but the results also apply to multidimensional parameters
Case 1: H0: = 0 H1: = 1
Example: Let x be an observation from Bin(n, ) and H0: = 0 vs. H1: = 1
Note that the textbook uses the somewhat redundant notation
p0 ( | H0 ) and p1 ( | H1 ) for the prior(s)
It can be shown (Theorem 7.3) that in this case
(As 0 defines the region for H1 the conditioning on H1 in the textbook seems redundant. )
If the parameters involved are two and and the parameter of interest is , then is referred to as a nuisance parameter.
The marginal posterior density for is then obtained by integrating out from the joint posterior density:
where is the parameter space for .
Let xn= ( x1, … , xn ) be a random sample from a distribution with p.d.f. f (x ; )
Suppose we will take a new observation xn + 1and would like to make so-called predictive inference about it. In practice this means that we would like to express the uncertainty about it in terms of a prediction interval.
The marginal p.d.f. of Xn + 1 is the same as that of each variable Xi in the sample, i.e. f (x ; )
However, if we want to make use of the sample we should rather study the simultaneous density of Xn + 1 and | xn .
Xn + 1 and Xn are independent by definition
Xn + 1 and | xn are also independent as the latter is conditional on xn
Now, treating (temporarily) as a nuisance parameter.
The posterior predictive distribution (density) for Xn + 1 given and xn is then
g can be used to
The number of calls to a telephone central during an hour can usually be shown to follow a Poisson distribution with mean . Assume that a prior for is a Gamma (a,b)-distribution, i.e. the prior density is
Now assume that we have observed x1, x2 and x3 calls during each of the three previous hours and we wish to make predictive inference about the number of calls x4 during the current hour.
The posterior distribution is also Gamma with
and e.g. a point prediction under quadratic loss of the number of calls is obtained by
“Something between the Bayesian and the frequentist approach”
Use some of the available data (sample) to estimate the prior for
Use the rest of the available data to make inference about
Let data-point i be represented by the bivariate random variable (Xi , i), i =1, 2, …, k
Let f (x; i) be the p.d.f. for Xi and p( ) be the marginal density of i
f is assumed to be known part from and p is assumed to be unknown.
The marginal density for Xi is
Let p = p( ;) where is a (multidimensional) parameter defining the prior assuming the functional form of p is known.
Method of moments:
Put up the equations
from which an estimate of can be deduced.
is estimated as
Apparently, the textbook is not correct here as the estimation of p should be based on the first k – 1 observations and the kthshould be excluded from that stage.
See further the textbook for simplifications of the MLE procedure.