Course Calendar Class DATE Contents

1 Sep. 26 Course information & Course overview

2 Oct. 4 Bayes Estimation

3 〃 11 Classical Bayes Estimation - Kalman Filter -

4 〃 18 Simulation-based Bayesian Methods

5 〃 25 Modern Bayesian Estimation ：Particle Filter

6 Nov. 1 HMM(Hidden Markov Model)

Nov. 8 No Class

7 〃 15 Supervised Learning

8 〃 29 Bayesian Decision

9 Dec. 6 PCA(Principal Component Analysis)

10 〃 13 ICA(Independent Component Analysis)

11 〃 20 Applications of PCA and ICA

12 〃 27 Clustering, k-means et al.

13 Jan. 17 Other Topics 1 Kernel machine.

14 〃 22(Tue) Other Topics 2

Lecture Plan

Simulation-based Bayesian Estimation - Prelude to the particle filter -

1. Why simulation-based ? 2. Monte Carlo Sampling Methods Historical example Monte Carlo Approximation 3. Sampling Theory Sample Generation Method Importance Sampling

1. Why simulation-based ？

Go Nonlinear and Non-Gaussian

Kalman Filter

Linear

Gaussian density

Analytic form

Particle Filter

Nonlinear

Non-Gaussian

density

Simulation-based

(Monte Carlo

approaches)

Extended

Kalman Filter

Linearization

of nonlinear

system

(Unscented

Kalman filter)

2. Monte Carlo Methods

3.1 Historical example

- Buffon’s Needle - A needle of length l is dropped at random on

a flat surface ruled with parallel lines a distance d>l apart, what is

the probability that the needle will cross one of the lines.

d l

2Probability =

2

Dropped times, is the number of times

the needle crosses a line.

E Ml

d n

n l

E M d

n M

The experiment by Captain Fox (1864) estimates =3.1416

The Monte Carlo (MC) methods is a collection of techniques

performing estimation through random sampling.

Let perform a numerical integration:

: N-dimensional vector

Factorize where ( ) is interpreted as a proability density

satisfyin

I g x dx x

g x f x p x p x

( )

1

g 0 and 1.

For the new expression

,

(1) Draw N samples : 1, , independently from the density .

(2) Approximate by an empirical density distribution

1ˆ

(3)

i

Ni

i

p x p x dx

I E f x f x p x dx

x i N p x

p x

p x x xN

( )

1 1

1 1ˆ ˆ

N Nii

i i

I E f x f x x x dx f xN N

2.2 Monte Carlo Approximation (Integration)

This approximation is referred to as Monte Carlo integration

Remarks:

1) Numerical integration techniques are not efficient due to;

/ the number of points to be evaluated increases with the

dimensionality of parameter space,

/very small proportion of the samples will make a significant

contribution to the integral

2) Key idea of MC is to represent the target distribution as a set of

random samples.

MC method provides appropriate results in some statistical sense

(Convergence, non-biased etc. ) as far as we generate proper samples.

From the low of large numbers approves the convergence of MC

integration.

3. Sampling Theory

1

1

: Given an input random vector , the PDF transformed by

z which is monotonic, one-to-one, invertible ( ) is given by

where :Jacobian of the transformation .

(whe

Z

Z X

x p z

T x T

xp z p x T z

z

xT

z

Corollary

n and are scalars, opearation means the absolute value)x z

Monte Carlo simulation starts by generating random samples from a

known distribution.

3.1 Sample generation method:

From a given probability density function (PDF) to a target density

: X

x

PDF p x

z

: ZPDF p zT(x)

In practice, it is difficult to generate the samples directly from the

density . Instead, the samples can be generated by , referred

to as importance distribution or proposal distribution, whose PDF is

similar as .

ix

p x q x

Similarity of means:

0 0

In terms of , we have

where : is referred to as the .

q x

p x q x for all x

q x

I f x x q x dx

p xx importance weght

q x

3.2 Importance sampling

p x

The importance sampling approximation of can be written by

N

( i ) ( i )

i

I

ˆ ˆI I f x x q x dx x f xN

1

1

( ) ( )

1

can be evaluated up to a normalization constant, i.e.

( is unknown), q ( is unknown)

Then,

=

1 ,

p qp q

q

p

Nq i i

p i

p x

p x q xp x Z x Z

Z Z

Z p xI f x p x dx f x q x dx

Z q x

Zw x f x

Z N

Case :

( )

( )

( )

i

i

i

p xw x

q x

( )

( )

1

( )

1

Consider the case ( ), the ratio can be evaluated by using the sample

set as follows.

11

1Thus,

1

i

Nq i

p i

q

Np i

i

f x

x

Zp x dx w x

Z N

Z

Zw x

N

( )

( ) ( ) ( )

1 1

( )

( )

( )

1

Therefore,

where :

iN Ni i i

pi i

q

i

i

Nj

j

w xI f x w x f x

Z

Z

w xw x

w x