1. Pattern Classification All materials in these slides were taken from Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons, with the permission of the authors and the publisher

2. Chapter 3: Maximum-Likelihood and Bayesian Parameter Estimation (part 2)
Bayesian Estimation (BE)
Bayesian Parameter Estimation: Gaussian Case
Bayesian Parameter Estimation: General Estimation
Problems of Dimensionality
Computational Complexity
Component Analysis and Discriminants
Hidden Markov Models

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Bayesian Estimation (Bayesian learning to pattern classification problems)
In MLE  was supposed fix
In BE  is a random variable
The computation of posterior probabilities P(i | x) lies at the heart of Bayesian classification
Goal: compute P(i | x,Θ)
Given the sample Θ, Bayes formula can be written
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To demonstrate the preceding equation, use:
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The desired probability density p(x) is unknown
Assume that it has a known parametric form.
The only thing assumed unknown is the value of a parameter vector θ.
We shall express the fact that p(x) is unknown but has known parametric form p(x|θ) .
Any information we might have about θ prior to observing the samples is assumed to be contained in a known prior density p(θ). Observation of the samples converts this to a posterior
We hope density p(θ|D) is sharply peaked about the true value of θ.
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If p(θ|D) peaks very sharply about some value ˆθ, we obtain p(x|D) ~= p(x|ˆθ), i.e., the result we would obtain by
substituting the estimate ˆθ for the true parameter vector.
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Bayesian Parameter Estimation: Gaussian Case
Goal: Estimate  using the a-posteriori density P( | Θ)
The univariate case: P( | Θ)
 is the only unknown parameter
(0 and 0 are known!)
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8. Thus, p(μ|D) is an exponential function of a quadratic function of μ.
If we write

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10. The univariate case P(x |Θ)
hence p(x|D) is normally distributed with mean and variance
Density p(x|D) is the desired class-conditional density and together with the prior probabilities it gives us the probabilistic information needed to design the classifier.
In contrast to maximum likelihood methods that only make points estimates for rather that estimate a distribution for p(x|D).

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P( | Θ) computed
P(x | Θ) remains to be computed!
It provides:
(Desired class-conditional density P(x | Θj, j))
Therefore: P(x | Θj, j) together with P(j)
And using Bayes formula, we obtain the
Bayesian classification rule:
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12. Multivariate case

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Bayesian Parameter Estimation: General Theory
P(x | Θ) computation can be applied to any situation in which the unknown density can be parametrized: the basic assumptions are:
The form of P(x | ) is assumed known, but the value of  is not known exactly
Our knowledge about  is assumed to be contained in a known prior density P()
The rest of our knowledge  is contained in a set Θ of n random variables x1, x2, …, xn that follows P(x)
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The basic problem is:
“Compute the posterior density P( |Θ)” then “Derive P(x | Θ)”
Using Bayes formula, we have:
And by independence assumption:
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Relation to the maximum likelihood solution:
Suppose that p(D|θ) reaches a sharp peak at θ = ˆθ.
If the prior density p(θ) is not zero at θ = ˆθ and does not change much in the surrounding neighborhood, then p(θ|D) also peaks at that point.
Thus, p(x|D) will be approximately p(x|ˆθ), the result one would obtain by using the maximum likelihood estimate as if it were the true value.
If the peak of p(D|θ) is very sharp, then the influence of prior information on the uncertainty in the true value of θ can be ignored.
The Bayesian solution tells us how to use all the available information to compute the desired density p(x|D).
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16. Convergence of p(x|D) to p(x): :
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17. Do Maximum Likelihood and Bayes methods differ?
There are several criteria that will influence our choice:
1- Computational complexity: maximum likelhood methods are often to be prefered.
2- Confidence in the prior information: If such information is reliable, Bayes methods gives better results. Further, general Bayesian methods with a “flat” or uniform prior
are equivalent to maximum likelihood methods.
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18. Problem of Dimensionality
1.How classification accuracy depends upon
the dimensionality (and amount of training data)?
2. How much is the computational complexity of the classifier?
If the features are statistically independent, there are some theoretical results that suggest the possibility of excellent performance. For example, consider the two-class multivariate normal case with the same covariance where
If the a priori probabilities are equal, then the Bayes error rate is given by
In the conditionally independent case,
Each feature contributes to reducing the probability of error.
In the conditionally independent case,

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Most useful features are the ones for which the difference between the means is large relative to the standard deviation.
It has frequently been observed in practice that, beyond a certain point, the inclusion of additional features leads to worse rather than better performance: we have the wrong model !
a)e.g., the Gaussian assumption or conditional assumption are wrong
b) the number of design or training samples is finite and thus the distributions are not estimated accurately.
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Computational Complexity
Our design methodology is affected by the computational difficulty
“big oh” notation
f(x) = O(h(x)) “big oh of h(x)”
If:
(An upper bound on f(x) grows no worse than h(x) for sufficiently large x!)
f(x) = 2+3x+4x2
g(x) = x2
f(x) = O(x2)
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“big oh” is not unique!
f(x) = O(x2); f(x) = O(x3); f(x) = O(x4)
“big theta” notation
f(x) = (h(x))
If:
f(x) = (x2) but f(x)  (x3)
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23. Complexity of the ML Estimation
Gaussian priors in d dimensions classifier with n training samples for each of c classes
For each category, we have to compute the discriminant function
Total = O(d2..n)
Total for c classes = O(cd2.n)  O(d2.n)
Cost increase when d and n are large! 7

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Component Analysis and Discriminants
Combine features in order to reduce the dimension of the feature space
Linear combinations are simple to compute and tractable
Project high dimensional data onto a lower dimensional space
Two classical approaches for finding “optimal” linear transformation
PCA (Principal Component Analysis) “Projection that best represents the data in a least- square sense”
MDA (Multiple Discriminant Analysis) “Projection that best separates the data in a least-squares sense”
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PCA
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37. Multivariate Discriminant Analysis
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Hidden Markov Models:
Markov Chains
Goal: make a sequence of decisions
Processes that unfold in time, states at time t are influenced by a state at time t-1
Applications: speech recognition, gesture recognition, parts of speech tagging and DNA sequencing,
Any temporal process without memory
T = {(1), (2), (3), …, (T)} sequence of states
We might have 6 = {1, 4, 2, 2, 1, 4}
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The system can revisit a state at different steps and not every state need to be visited
First-order Markov models
Our productions of any sequence is described by the transition probabilities
P(j(t + 1) | i (t)) = aij
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 = (aij, T)
P(T | ) = a14 . a42 . a22 . a21 . a14 . P((1) = i)
Example: speech recognition
“production of spoken words”
Production of the word: “pattern” represented by phonemes
/p/ /a/ /tt/ /er/ /n/ // ( // = silent state)
Transitions from /p/ to /a/, /a/ to /tt/, /tt/ to er/, /er/ to /n/ and /n/ to a silent state
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