Download - 10a. Markov Chain Monte Carlo · Prof. Daniel Cremers 10a. Markov Chain Monte Carlo. Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Markov Chain Monte

Computer Vision Group Prof. Daniel Cremers

10a. Markov Chain Monte Carlo

Dr. Rudolph TriebelComputer Vision Group

Machine Learning for Computer Vision

Markov Chain Monte Carlo

• In high-dimensional spaces, rejection sampling and importance sampling are very inefficient

• An alternative is Markov Chain Monte Carlo (MCMC)

• It keeps a record of the current state and the proposal depends on that state

• Most common algorithms are the Metropolis-Hastings algorithm and Gibbs Sampling

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Markov Chains Revisited

A Markov Chain is a distribution over discrete-state random variables so that

The graphical model of a Markov chain is this:

We will denote as a row vector

A Markov chain can also be visualized as a state transition diagram.

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x1, . . . ,xM

p(x1, . . . ,xT ) = p(x1)p(x2 | x1) · · · = p(x1)TY

t=2

p(xt | xt�1)

p(xt | xt�1) ⇡t

T



The State Transition Diagram

A33 A33

A11 A11k=1

k=2

k=3

time

t-2 t-1 t

4

states



The Stationary Distribution

The probability to reach state k isOr, in matrix notation:

We say that is stationary if

To find the stationary distribution we need to solve the eigenvector problem .

The stationary distribution is then where is the eigenvector for which the eigenvalue is 1.

This eigenvector needs to be normalized so that it is a valid distribution.

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⇡t = ⇡t�1A

⇡k,t =KX

i=1

⇡i,t�1Aik

⇡t = ⇡t�1⇡t

ATv = v

⇡ = vT v



Existence of a Stationary Distribution

• A Markov chain can have many stationary distributions

• Necessary for having a unique stationary distribution: we can reach every state from any other state in finite steps (the chain is regular)

• A transition distribution satisfies the property of detailed balance if

• The chain is then said to be reversible

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1 2 3 4

0.9

0.90.5 0.5

1.00.10.1

⇡t

⇡iAij = ⇡jAji

⇡1

⇡3 ⇡1A13 + · · ·

⇡3A31 + · · ·



Reversible Chain: Example

A31

t-1 t

7

A13



Existence of a Stationary Distribution

Theorem: If a Markov chain with transition matrix

A is regular and satisfies detailed balance wrt. the distribution , then is a stationary distribution of the chain.

Proof:

it follows .

This is only a sufficient condition, however it is not necessary.

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⇡ ⇡

KX

i=1

⇡iAij =KX

i=1

⇡jAji = ⇡j

KX

i=1

Aji = ⇡j 8j

⇡ = ⇡A



Sampling with a Markov Chain

The main idea of MCMC is to sample state transitions based on a proposal distribution q.

The most widely used algorithm is the Metropolis-Hastings (MH) algorithm.

In MH, the decision whether to stay in a given state is based on a given probability.

If the proposal distribution is , then we stay in state with probability

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q(x0 | x)x

0

min

✓1,

p̃(x0)q(x | x0)

p̃(x)q(x0 | x)

◆

Unnormalized target distribution



The Metropolis-Hastings Algorithm

• Initialize

• for

•define

•sample

•compute acceptance probability

•compute

•sample

•set new sample to

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x

0

s = 0, 1, 2, . . .

x = x

s

x

0 ⇠ q(x0 | x)

↵ =p̃(x0)q(x | x0)

p̃(x)q(x0 | x)r = min(1,↵)

u ⇠ U(0, 1)

x

s+1 =

(x

0 if u < r

x

s if u � r



Why Does This Work?

We have to prove that the transition probability of the MH algorithm satisfies detailed balance wrt the target distribution.

Theorem: If is the transition probability of the MH algorithm, then

Proof:

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pMH(x0 | x)

p(x)pMH(x0 | x) = p(x0)pMH(x | x0)



Why Does This Work?

We have to prove that the transition probability of the MH algorithm satisfies detailed balance wrt the target distribution.

Theorem: If is the transition probability of the MH algorithm, then

Note: All formulations are valid for discrete and for continuous variables!

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pMH(x0 | x)

p(x)pMH(x0 | x) = p(x0)pMH(x | x0)



Choosing the Proposal

• A proposal distribution is valid if it gives a non-zero probability of moving to the states that have a non-zero probability in the target.

• A good proposal is the Gaussian, because it has a non-zero probability for all states.

• However: the variance of the Gaussian is important!

•with low variance, the sampler does not explore sufficiently, e.g. it is fixed to a particular mode

•with too high variance, the proposal is rejected too often, the samples are a bad approximation

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Example

Target is a mixture of 2 1D Gaussians.

Proposal is a Gaussian with different variances.

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0

200

400

600

800

1000 !100

!50

0

50

100

0

0.1

0.2

Samples

MH with N(0,1.0002) proposal

Iterations

0

200

400

600

800

1000 !100

!50

0

50

100

0

0.01

0.02

0.03

Samples


Iterations

0

200

400

600

800

1000 !100

!50

0

50

100

0

0.02

0.04

0.06

Samples


Iterations



Gibbs Sampling

• Initialize

• For

•Sample

•Sample

•...

•Sample

Idea: sample from the full conditional

This can be obtained, e.g. from the Markov blanket in graphical models.

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{zi : i = 1, . . . ,M}⌧ = 1, . . . , T

z(⌧+1)1 ⇠ p(z1 | z(⌧)2 , . . . , z(⌧)M )

z(⌧+1)2 ⇠ p(z2 | z(⌧+1)

1 , . . . , z(⌧)M )

z(⌧+1)M ⇠ p(zM | z(⌧+1)

1 , . . . , z(⌧+1)M�1 )



Gibbs Sampling: Example

• Use an MRF on a binary image with edge potentials (“Ising model”) and node potentials

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(xt) = N (yt | xt,�2)

(xs, xt) = exp(Jxsxt)

xt

yt

xs

xt 2 {�1, 1}



Gibbs Sampling: Example

• Use an MRF on a binary image with edge potentials (“Ising model”) and node potentials

• Sample each pixel in turn

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(xt) = N (yt | xt,�2)

sample 1, Gibbs

−1

−0.5

0

0.5

1sample 5, Gibbs

−1

−0.5

0

0.5

1mean after 15 sweeps of Gibbs

−1

−0.5

0

0.5

1

(xs, xt) = exp(Jxsxt)

After 1 sample After 5 samples Average after 15 samples



Gibbs Sampling for GMMs

• Again, we start with the full joint distribution:

(semi-conjugate prior)

• It can be shown that the full conditionals are:

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p(X,Z,µ,⌃,⇡) = p(X | Z,µ,⌃)p(Z | ⇡)p(⇡)KY

k=1

p(µk)p(⌃k)

p(zi = k | xi,µ,⌃,⇡) / ⇡kN (xi | µk,⌃k)

p(⇡ | z) = Dir({↵k +NX

i=1

zik}Kk=1)

p(µk | ⌃k, Z,X) = N (µk | mk, Vk) (linear-Gaussian)

p(⌃k | µk, Z,X) = IW(⌃k | Sk, ⌫k)



Gibbs Sampling for GMMs

• First, we initialize all variables

• Then we iterate over sampling from each conditional in turn

• In the end, we look at and

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µk ⌃k

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4 6 8 10 12 14 16

"data_gibbs.dat"

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4 6 8 10 12 14 16

class 1class 2

mu1mu2

sigma1sigma2



How Often Do We Have To Sample?

• Here: after 50 sample rounds the values don’t change any more

• In general, the mixing time is related to the eigen gap of the transition matrix:

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0

1

2

3

4

5

6

7

8

0 50 100 150 200

mu1xmu1ymu2xmu2y

� = �1 � �2

⌧✏

⌧✏ O(

1

�log

n

✏)



Gibbs Sampling is a Special Case of MH

• The proposal distribution in Gibbs sampling is

• This leads to an acceptance rate of:

• Although the acceptance is 100%, Gibbs sampling does not converge faster, as it only updates one variable at a time.

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q(x0 | x) = p(x0i | x�i)I(x0

�i = x�i)

↵ =p(x0)q(x | x0)

p(x)q(x0 | x) =p(x0

i | x0�i)p(x

0�i)p(xi | x0

�i)

p(xi | x�i)p(x�i)p(x0i | x�i)

= 1



Summary

• Markov Chain Monte Carlo is a family of sampling algorithms that can sample from arbitrary distributions by moving in state space

• Most used methods are the Metropolis-Hastings (MH) and the Gibbs sampling method

• MH uses a proposal distribution and accepts a proposed state randomly

• Gibbs sampling does not use a proposal distribution, but samples from the full conditionals

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Computer Vision Group Prof. Daniel Cremers

11. Evaluation and Model Selection



Evaluation of Learning Methods

Very often, machine learning tries to find parameters of a model for a given data set.

But: Which parameters give a good model?

Intuitively, a good model behaves well on new, unseen data. This motivates the distinction of

Training data: used to generate different models

Validation data: used to adjust the model parameters so that their performance is optimal

Test data: used to evaluate the models with the optimized parameters

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Loss Function

A common way to evaluate a learning algorithm (e.g. regression, classification) is to define a loss function:

• In the case of regression, a common choice is the squared loss

• In classification, where and t are natural numbers, we use the 0/1-loss:

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L(y, a) = |y � a|q



Some Loss Functions

• Quadratic loss is useful for continuous parameters

• Its minimum is the posterior mean

• Absolute loss is less sensitive to outliers

• Its minimum is the median of the posterior

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!2 !1 0 1 20

1

2|x|0.2

!2 !1 0 1 20

1

2|x|1.0

!2 !1 0 1 20

1

2|x|2.0

a = “action”, e.g. labeling

E[y | x]



Model Selection

Model selection is used to find model parameters to optimize the classification result.

Possible methods:

• Minimizing the training error

• Hold-out testing

• Cross-validation

• Leave-one-out rule

Evaluation is done using an appropriate loss function.

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Minimizing the Training Error

The training error is defined as:

Where are all input/target value pairs from the training data set.

Problem: A model that minimizes the training error does not (necessarily) generalize well. It only behaves well on the data it was trained with.

Example: Polynomial regression with high model complexity (see above)

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Hold-out Testing

Hold-out testing splits the data set up into

• A validation data set of size

• A smaller training data set of size

The model parameters are then selected so that the error

is minimized.

Problems: How big should be ? Which elements should be chosen for evaluation?

Also: Iteration of the model design may lead to overfitting on the validation data

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Hold-out Testing

Training set Validation set

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Cross Validation

Idea of cross validation: Perform hold-out testing times on different evaluation (sub-)sets.

Validation subsets:

Error function:

Minimizing this error function gives good results, but requires huge computational efforts. The training must be done times.

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Cross-Validation


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Cross-Validation

Training set

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Cross-Validation

Training set

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Leave-one-out Rule

Idea: do cross-validation with

• Yields similarly good results compared to cross-validation

• Still requires to do the training times

• Useful if the data set is particularly scarce

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Leave-one-out Rule


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Leave-one-out Rule


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BIC and AIC

Observation: There is a trade-off between the obtained data likelihood (i.e. the quality of the fit) and the complexity M of the model.

Idea: Define a score function that balances both.

Two common examples are:

But: These criteria tend to favor too simple models.

“Akaike Information Criterion”

“Bayesian Information Criterion”

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Classifier Evaluation

Different methods to evaluate a classifier

• True-positive / false-positive rates

• Precision-recall diagram

• Receiving operator characteristics (ROC curves)

• Loss function

The evaluation of a classifier is needed to find the best classification parameters

(e.g. for the kernel function)

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Number of positive examples

Number of all data points:

Classification result = -1

Classification result = 1

Correct label = -1 “True

negative rate”“False

positive rate”

Correct label = 1 “False

negative rate”“True positive

rate”

Number of negative examples40




Terminology

• We define the precision as

intuitively: probability that the real label is positive in case the classifier returns “positive”

• The recall is defined as

intuitively: probability that a “positive” labeled data point is detected as “positive” by the classifier.

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Precision-Recall Curves

Usually, precision and recall are plotted into the same graph.

The optimal classifier parameters are obtained where precision equals recall (break-even-point).

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Receiver Operating Characteristics

Receiving Operating Characteristics (ROC) curves plot the true positive rate vs. the false positive rate.

To find a good classifier we can search for a point where the slope is 1.

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Loss Function

Another method to evaluate a classifier is defined by evaluating its loss function. The simplest loss function is the 0/1-loss:

Where is a classifier, is a feature vector and is the known class label of .

Important: The pair should be taken from an evaluation data set that is different from the training set.

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