Intelligent Systems Statistical Machine Learning

(1)

WS2014/2015, 17.12.2014

Intelligent Systems

Statistical Machine Learning

Carsten Rother, Dmitrij Schlesinger

(2)

17.12.2014

Intelligent Systems: Statistical Machine Learning ²

Our tasks (recap)

The model: two variables are usually present:

- the first one is typically discrete and is called “class”

- the second one is often continuous and is called “observation”

The recognition (inference) task:

Let the joint probability distribution be “given”.

Observe , estimate .

The (statistical) learning task: given a training set

“find” the corresponding probability distribution k 2 K

x 2 X

p(x, k)

x k

L = (x₁, k₁), (x₂, k₂), . . . , (x_l, k_l) p(x, k)

(3)

17.12.2014 Intelligent Systems: Statistical Machine Learning

• Decision making:

• Bayesian Decision Theory

• Non-Bayesian formulation

• Statistical Learning — Maximum Likelihood Principle

3

Outline

(4)

Somebody samples a pair according to a p.d.

He keeps hidden and presents to you

You decide for some according to a chosen decision strategy

Somebody penalizes your decision according to a loss-function, i.e. he compares your decision with the “true” hidden

You know both and the loss-function (how does he compare) Your goal is to design the decision strategy in order to pay as less as possible in average.

4

Bayesian Decision Theory: Idea — a game

(x, k) p(x, k)

k x

k^⇤

k

p(x, k)

(5)

The decision set . Note: it needs not to coincide with !!!

Examples: decisions “I don’t know”, “surely not this class” etc.

Decision strategy (mapping) Loss-function

The Bayesian Risk is the expected loss:

(should be minimized with respect to the decision strategy).

For a particular observation :

5

Bayesian Risk

D K

e : X ! D C : D⇥K ! R

R(e) = X

x

X

k

p(x, k) · C e(x), k ! min

e

R(d) = X

k

p(k|x) · C(d, k) ! min

d

x

(6)

The loss is the simplest one (called delta-function):

i.e. we pay 1 if we are wrong, no matter what error do we make.

From that follows:

6

Maximum A-posteriori Decision (MAP)

C(k, k⁰) = (k6=k⁰)

R(k) = X

k⁰

p(k⁰|x) · (k6=k⁰) =

= X

k⁰

p(k⁰|x) p(k|x) = 1 p(k|x) ! min

k

p(k|x) ! max

k

(7)

The decision set is , i.e. it is extended by a special decision “I don’t know” (rejection). The loss-function is

i.e. we pay a (reasonable) penalty if we are lazy to decide.

Case-by-case analysis:

1. We decide for a class , decision is MAP , the loss for this is

2. We decide to reject , the loss for this is

The decision strategy: Compare with and decide for the variant with the greater value.

7

Decision with rejection

D = K [ {rw}

C(d, k) =

⇢ (k6=k⁰) if d 2 K

" if d = rw

d 2 K d = k^⇤ 1 p(k^⇤|x)

d = rw "

! p(k^⇤|x) 1 "

(8)

Let be given. The conditional probability distribution for observations given classes are Gaussians:

The loss function is , i.e. we want MAP

8

A MAP example

K = {1, 2}, x 2 R², p(k)

p(x|k) = 1

2⇡ _k² exp

 ||x µ_k||² 2 _k²

C(k, k⁰) = (k6=k⁰)

The decision strategy (the mapping ) partitions the input space into two regions: the one corresponding to the first and the one

corresponding to the second class. How does this partition look like ?

e : X ! K

(9)

For a particular we decide for 1, if

Special case for simplicity: and

the decision strategy is (derivations on the board):

a linear classifier — the hyperplane orthogonal to

(more examples at the exercise)

9

A MAP example

p(1) · 1

2⇡ ₁² exp

 ||x µ₁||²

2 ₁² > p(2) · 1

2⇡ ₂² exp

 ||x µ₂||² 2 ₂²

1 = ₂ p(1) = p(2)

!

hx, µ₂ µ₁i > const

µ₂ µ₁

!

x

(10)

10

Outline

(11)

Despite the generality of Bayesian approach, there are many tasks which cannot be expressed within the Bayesian framework:

• It is difficult to establish a penalty function, e.g. it does not assume values from the totally ordered set.

• A priori probabilities are not known or cannot be known because is not a random event.

An example – Russian fairy tales hero

When he turns to the left, he loses his horse, when he turns to the right, he loses his sword, and if he turns back, he loses his beloved girl.

Is the sum of horses and swords is less or more than beloved girls ?

11

Non-Bayesian Decisions

p(k) k

p₁ p₂ p₃

(12)

We have:

— observations (features) measured on a patient — hidden states

— decisions

12

Example: decision while curing a patient

x 2 X

k 2 K = {healthy, seriously sick} d 2 D = {do not cure, apply a drug}

do not cure apply a drug

healthy Correct decision

small health damage

seriously sick death possible

Correct decision

Penalty problem:

how to assign real number to a penalty?

(13)

Observation describes the observed airplane.

Two hidden states:

The conditional probability can depend on the observation in a complicated manner but it exists and describes dependencies of the observation on the situation correctly.

A-priori probabilities are not known and can not be known in principle.

the hidden state is not a random event

13

Example: enemy or allied airplane ?

x

⇢ k = 1 allied airplane k = 2 enemy airplane

p(x|k) x

x k

p(k)

! k

(14)

The strategy (partitioning the input space into two subsets ) is characterized by two numbers:

1. “Probability” of the false positive (false alarm)

2. “Probability” of the false negative (overlooked danger)

Minimize the conditional probability of the false positive subject to the condition that the false negative is bounded:

14

Neyman-Pearson Task (1928, 1933)

!(1) = X

x2X₂

p(x|1)

!(2) = X

x2X₁

p(x|2)

X₁[X₂ = X

!(1) ! min

X₁,X₂ s.t. !(2)  "

(15)

15

Outline

(16)

Let a parameterized class (family) of probability distributions be given, i.e.

Example — the set of Gaussians in

parameterized by the mean and standard deviation Let the training data be given:

One have to decide for a particular probability distribution from the given family, i.e. for a particular parameter (e.g. ) for Gaussians).

16

Learning

p(x; ✓) 2 P

Rⁿ p(x; µ, ) = 1

(p

2⇡ )ⁿ exp

 ||x µ||² 2 ²

µ 2 Rⁿ 2 R

✓^⇤ = (µ^⇤, ^⇤) L = (x₁, x₂, . . . , x_|_L_|)

(17)

Assumption: the training dataset is a realization of the unknown probability distribution – it is sampled according to it.

What is observed should have a high probability

Maximize the probability of the training data with respect to the unknown parameter

17

Maximum Likelihood Principle

p(L; ✓) ! max

✓

!

(18)

We have a Family of probability distributions for :

and a training set

Assumption: the samples are independent identically distributed (iid.), i.e. the probability of the training set is

Take logarithm and maximize it with respect to the unknown parameters:

18

Example: Gaussian

p(x; µ, ) = 1 (p

2⇡ )ⁿ exp

 ||x µ||² 2 ²

x 2 Rⁿ

p(L; µ, ) = Y

l

p(x_l; µ, )

ln p(L; µ, ) = X

l

ln p(x_l; µ, ) ! max

µ,

L = (x₁, x₂, . . . , x_|_L_|)

(19)

Substitute the model:

Assume, we are interested only in the center , i.e. is given. Then the problem can be further simplified to

Take the derivative, set it to zero

and resolve

i.e. the mean value over the dataset.

19

Example: Gaussian

X

l



n ln ||x_l µ||²

2 ² = |L| · n · ln 1 2 ²

X

l

||x_l µ||² ! max

µ,

µ

@

@µ = X

l

(x_l µ) = X

l

x_l |L| · µ = 0

µ =

P

l x_l

|L| X

l

||x_l µ||² ! min

µ

(20)

Maximum Likelihood estimator is not the only estimator – there are many others as well.

Maximum Likelihood is consistent, i.e. it gives the true parameters for infinite training sets.

Consider the following experiment for an estimator: 

1. We generate infinite numbers of training sets each one being finite;

2. For each training set we estimate the parameter; 

3. We average all estimated values.

If the average is the true parameter, the estimator is called unbiased.

Maximum Likelihood is not always unbiased – it depends on the parameter to be estimated. Examples – mean for a Gaussian is unbiased, standard deviation – not.

20

Some comments

(21)

Today: statistical Machine Learning

• Decision making — Bayesian and non-Bayesian formulations

• Learning — Maximum Likelihood Principle

Next two lectures: models — directed and undirected graphical models

Then: discriminative decision making and learning — neural networks Then: unsupervised learning — clustering

21

Intelligent Systems Statistical Machine Learning

WS2014/2015, 17.12.2014