Statistical Learning Statistical Learning MethodsMethods

Russell and Norvig: Chapter 20 (20.1,20.2,20.4,20.5)

CMSC 421 – Fall 2006

Statistical Approaches

Statistical Learning (20.1)Naïve Bayes (20.2)Instance-based Learning (20.4)Neural Networks (20.5)

Statistical Learning (20.1)

Example: Candy Bags

Candy comes in two flavors: cherry () and lime ()Candy is wrapped, can’t tell which flavor until openedThere are 5 kinds of bags of candy:

H1= all cherry H2= 75% cherry, 25% lime H3= 50% cherry, 50% lime H4= 25% cherry, 75% lime H5= 100% lime

Given a new bag of candy, predict HObservations: D1, D2 , D3, …

Bayesian Learning

Calculate the probability of each hypothesis, given the data, and make prediction weighted by this probability (i.e. use all the hypothesis, not just the single best)

Now, if we want to predict some unknown quantity X

)h(P)h|d(P)d|h(P ii)d(P)h(P)h|d(P

iii

)d|h(P)h|X(P)d|X(P iii

Bayesian Learning cont.

)h(P)h|d(P)d|h(P iii

Calculating P(h|d)

Assume the observations are i.i.d.—independent and identically distributed

likelihood prior

j

iji )h|d(P)h|d(P

Example:

Hypothesis Prior over h1, …, h5 is {0.1,0.2,0.4,0.2,0.1}Data:

Q1: After seeing d1, what is P(hi|d1)?

Q2: After seeing d1, what is P(d2= |d1)?

Making Statistical Inferences

Bayesian – predictions made using all hypothesis, weighted by

their probabilities

MAP – maximum a posteriori uses the single most probable hypothesis to make

prediction often much easier than Bayesian; as we get more

and more data, closer to Bayesian optimal

ML – maximum likelihood assume uniform prior over H when

Naïve Bayes (20.2)

Naïve Bayes

aka Idiot Bayesparticularly simple BNmakes overly strong independence assumptionsbut works surprisingly well in practice…

Bayesian Diagnosis

suppose we want to make a diagnosis D and there are n possible mutually exclusive diagnosis d1, …, dn

suppose there are m boolean symptoms, E1, …, Em )e,...,e|d(P m1i

how do we make diagnosis?

)d(P i )d|e,...,e(P im1we need:

)e,...,e(P)d|e,...,e(P)d(P

m1

im1i

Naïve Bayes Assumption

Assume each piece of evidence (symptom) is independent give the diagnosis then

)d|e,...,e(P im1

what is the structure of the corresponding BN?

m

1kik )d|e(P

Naïve Bayes Example

possible diagnosis: Allergy, Cold and OKpossible symptoms: Sneeze, Cough and Fever

Well Cold Allergy

P(d) 0.9 0.05 0.05

P(sneeze|d)

0.1 0.9 0.9

P(cough|d)

0.1 0.8 0.7

P(fever|d) 0.01 0.7 0.4

my symptoms are: sneeze & cough, what isthe diagnosis?

Learning the Probabilities

aka parameter estimationwe need P(di) – prior P(ek|di) – conditional probability

use training data to estimate

Maximum Likelihood Estimate (MLE)

use frequencies in training set to estimate:

Nn

)d(p ii

i

ikik n

n)d|e(p

where nx is shorthand for the counts of events intraining set

Example:

D Sneeze Cough Fever

Allergy yes no no

Well yes no no

Allergy yes no yes

Allergy yes no no

Cold yes yes yes

Allergy yes no no

Well no no no

Well no no no

Allergy no no no

Allergy yes no no

what is:P(Allergy)?P(Sneeze| Allergy)?P(Cough| Allergy)?

Laplace Estimate (smoothing)

use smoothing to eliminate zeros:

nN1n

)d(p ii

2n1n

)d|e(pi

ikik

where n is number of possible values for dand e is assumed to have 2 possible values

many other smoothing schemes…

Comments

Generally works well despite blanket assumption of independenceExperiments show competitive with decision trees on some well known test sets (UCI)handles noisy data

Learning more complex Bayesian networks

Two subproblems:learning structure: combinatorial search over space of networkslearning parameters values: easy if all of the variables are observed in the training set; harder if there are ‘hidden variables’

Instance-based Learning

Instance/Memory-based Learning

Non-parameteric hypothesis complexity grows with the

data

Memory-based learning Construct hypotheses directly from

the training data itself

Nearest Neighbor Methods

To classify a new input vector x, examine the k-closest training data points to x and assign the object to the most frequently occurring class

x

k=1

k=6

Issues

Distance measure Most common: euclidean Better distance measures: normalize each variable by standard deviation For discrete data, can use hamming distance

Choosing k Increasing k reduces variance, increases bias

For high-dimensional space, problem that the nearest neighbor may not be very close at all!

Memory-based technique. Must make a pass through the data for each classification. This can be prohibitive for large data sets.

Indexing the data can help; for example KD trees

Neural Networks (20.5)

Neural functionBrain function (thought) occurs as the result of the firing of neuronsNeurons connect to each other through synapses, which propagate action potential (electrical impulses) by releasing neurotransmittersSynapses can be excitatory (potential-increasing) or inhibitory (potential-decreasing), and have varying activation thresholdsLearning occurs as a result of the synapses’ plasticicity: They exhibit long-term changes in connection strengthThere are about 1011 neurons and about 1014 synapses in the human brain

Biology of a neuron

Brain structureDifferent areas of the brain have different functions Some areas seem to have the same function in all humans (e.g.,

Broca’s region); the overall layout is generally consistent Some areas are more plastic, and vary in their function; also, the

lower-level structure and function vary greatly

We don’t know how different functions are “assigned” or acquired

Partly the result of the physical layout / connection to inputs (sensors) and outputs (effectors)

Partly the result of experience (learning)

We really don’t understand how this neural structure leads to what we perceive as “consciousness” or “thought”Our neural networks are not nearly as complex or intricate as the actual brain structure

Comparison of computing power

Computers are way faster than neurons…But there are a lot more neurons than we can reasonably model in modern digital computers, and they all fire in parallelNeural networks are designed to be massively parallelThe brain is effectively a billion times faster

Neural networks

Neural networks are made up of nodes or units, connected by linksEach link has an associated weight and activation levelEach node has an input function (typically summing over weighted inputs), an activation function, and an output

Neural unit

Linear Threshold Unit (LTU)

n

W2

0

1

X 1

X 2

X n

X 0 =1

i

n

ii xw

0

otherwise 1-

if 1n

1iii xw

otherwise 1-

0xw...xww if 1)x,...,x(o nn110

n1

Sigmoid Unit

W n

W2

W0W

1

X 1

X 2

X n

X 0 =1

i

n

ii xwnet

0

neteneto

1

1)(σ

-xe11

function sigmoid the is )x(

σ

Neural Computation

McCollough and Pitt (1943)showed how LTU can be use to compute logical functions AND? OR? NOT?

Two layers of LTUs can represent any boolean function

Learning Rules

Rosenblatt (1959) suggested that if a target output value is provided for a single neuron with fixed inputs, can incrementally change weights to learn to produce these outputs using the perceptron learning rule assumes binary valued input/outputs assumes a single linear threshold unit

Perceptron Learning ruleIf the target output for unit j is tj

ijjjiji o)ot(ww

Equivalent to the intuitive rules:If output is correct, don’t change the weightsIf output is low (oj=0, tj=1), increment weights for all the inputs which are 1If output is high (oj=1, tj=0), decrement weights for all inputs which are 1Must also adjust threshold. Or equivalently assume there is a weight wj0 for an extra input unit that has o0=1

Perceptron Learning Algorithm

Repeatedly iterate through examples adjusting weights according to the perceptron learning rule until all outputs are correct

Initialize the weights to all zero (or random) Until outputs for all training examples are correct

for each training example e do compute the current output oj

compare it to the target tj and update weights

each execution of outer loop is an epochfor multiple category problems, learn a separate perceptron for each category and assign to the class whose perceptron most exceeds its thresholdQ: when will the algorithm terminate?

Representation Limitations of a Perceptron

Perceptrons can only represent linear threshold functions and can therefore only learn functions which linearly separate the data, I.e. the positive and negative examples are separable by a hyperplane in n-dimensional space

Perceptron Learnability

Perceptron Convergence Theorem: If there are a set of weights that are consistent with the training data (I.e. the data is linearly separable), the perceptron learning algorithm will converge (Minksy & Papert, 1969)Unfortunately, many functions (like parity) cannot be represented by LTU

Layered feed-forward network

Output units

Hidden units

Input units

Backpropagation Algorithm

i,jji,j

i,ji,ji,j

j,i

koutputsk

h,khhh

kkkkk

x w

where

www

wweight network each Update 4.

w)o1(o

hunit hidden eachFor 3.

)ot)(o1(o

kunit output eachFor 2.

outputs the compute and example training theInput 1.

do example, training eachFor

do satisfied, Until numbers. random small to weights all Initialize

“Executing” neural networks

Input units are set by some exterior function (think of these as sensors), which causes their output links to be activated at the specified levelWorking forward through the network, the input function of each unit is applied to compute the input value

Usually this is just the weighted sum of the activation on the links feeding into this node

The activation function transforms this input function into a final value

Typically this is a nonlinear function, often a sigmoid function corresponding to the “threshold” of that node

Neural Nets for Face Recognition

30x32inputs

left strt rgt up

Typical Input Images

90% accurate learninghead pose, and recognizing1-of-20 faces

Summary: Statistical Learning Methods

Statistical Inference use likehood of data and prob of hypothesis

to predict value for next instance Bayesian MAP ML

Naïve BayesNearest NeighborNeural Networks