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Tirza Routtenberg Dept. of ECE, Ben-Gurion University of the NegevPowerPoint Presentation

Tirza Routtenberg Dept. of ECE, Ben-Gurion University of the Negev

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### General Classes of Lower Bounds on Outage Error Probability and MSE in Bayesian Parameter Estimation

Tirza Routtenberg

Dept. of ECE, Ben-Gurion University of the Negev

Supervisor: Dr. Joseph Tabrikian

Outline

Introduction

Derivation of a new class of lower bounds on the probability of outage error

Derivation of a new class of lower bounds on the MSE

Bounds properties: tightness conditions, relation to the ZZLB

Examples

Conclusion

IntroductionBayesian parameter estimation

Goal: to estimate the unknown parameter θ

based on the observation vectorx.

Assumptions:

θ andx are random variables

The observation cdf and

posterior pdf are known

Applications:

Radar/Sonar, Communication, Biomedical, Audio/speech,…

IntroductionParameter estimation criteria

- Mean-square error (MSE)
- Probability of outage error

Advantages of the probability of outage error criterion:

Provides meaningful information in the presence of large errors case.

Dominated by the all error distribution.

Prediction of the operation region.

Large-errors

Threshold

MSE

Small errors

SNR

Large-errors

Probability of outage error

Threshold

Small errors

SNR

IntroductionParameter estimation criteria

IntroductionMMSE estimation

The minimum MSE is attained by MMSE:

Introductionh-MAP estimation

The corresponding minimum probability of h-outage error is

The h-MAP estimator is

bound

PERFORMANCE MEASURE

SNR or number of samples

Performance lower boundsMotivation

- Performance analysis
- Threshold prediction
- System design
- Feasibility study

Performance lower bounds

Threshold

bound

PERFORMANCE MEASURE

SNR or number of samples

Bounds desired features

- Computational simplicity
- Tightness
- Asymptotically coincides with the optimal performance
- Validity: independent of the estimator.

Previous work: probability of outage error bounds

Most of the existing bounds on the probability of outage error are based on the relation to the probability of error in decision procedure (binary/multiple).

Kotelnikov inequality - lower bound for uniformly distributed unknown parameter.

Previous work: Bayesian MSE bounds

Bayesian MSE bounds

Weiss–Weinstein class

The covariance inequality

Ziv-Zakai class

Relation to probability of error

in decision problem

- Bayesian Cramér–Rao (Van Trees, 1968)
- Bayesian Bhattacharyya bound
- (Van Trees 1968)
- Weiss–Weinstein (1985)
- Reuven-Messer (1997)
- Bobrovski–Zakai (1976)

- Ziv–Zakai (ZZLB) (1969)
- Bellini–Tartara (1974)
- Chazan–Zakai–Ziv (1975)
- Extended ZZLB (Bell, Steinberg,
- Ephraim,Van Trees,1997)

General class of outage error probability lower bounds

The probability of outage error

?

(Reverse) Hölder inequality for

Taking

General class of outage error probability lower bounds

Objective: obtain valid bounds, independent of .

A necessary and sufficient condition to obtain a valid bound which is independent of the estimator, is that the function

is periodic in θ with period h, almost everywhere.

General class of outage error probability lower bounds

General class of outage error probability lower bounds

Using Fourier series representation

the general class of bounds is

Example: Linear Gaussian model

The model

The minimum h-outage error probability:

The single coefficient bound:

The tightest subclass of lower bounds

- The bound is maximized w.r.t. for given p

Convergence condition:

There exists l0h(θ,x), α>0such that for all │l│≥│l0h(θ,x)│

This mild condition guaranties that

converges for every p≥1.

The tightest subclass of lower bounds

Repeat for all x

and

Under the convergence condition, the tightest bounds are

h – sampling period

The tightest subclass of lower bounds

Under the convergence condition, the tightest bounds are

Properties:

- The bound exists
- The bound becomes tighter by decreasing p.
- For p→1+, the tightest bound is

h – sampling period

General class of MSE lower bounds

- The probability of outage error and MSE are related via:
- Chebyshev's inequality
- Known probability identity

New MSE lower bounds can be obtained by using

and lower bounding the probability of outage error

General class of MSE lower bounds

- For example:
- General class of MSE bounds:
- The tightest MSE bound:

General class of lower bounds on different cost functions

Arbitrary cost function C(·) that is non-decreasing and differentiable satisfies

Thus, it can be bounded using lower bounds on the probability of outage error

Examples: the absolute error, higher moments of the error.

Properties: Relation to the ZZLB

Theorem

The proposed tightest MSE bound is always tighter than the extended ZZLB.

The extended ZZLB is

The tightest proposed MSE bound can be rewritten as

4

2

2

8

8

1

1

7

7

4

2

1

7

Properties: Relation to the ZZLB

ZZLB

The proposed bound

max out

2

2

1

1

14

6

For any converging sequence of non-negative numbers

Therefore,

Properties: unimodal symmetric pdf

Theorem:

A. If the posterior pdf f θ| x(θ| x) is unimodal, then the proposed tightest outage error probability bound coincides with the minimum probability of outage error for every h>0.

B. If the posterior pdf f θ| x(θ| x) is unimodal and symmetric, then the proposed tightest MSE bound coincides with the minimum MSE.

Example 1

Statistics

Conclusion

The concept of probability of outage error criterion is proposed.

New classes of lower bounds on the probability of outage error and on the MSE in Bayesian parameter estimation were derived.

It is shown that the proposed tightest MSE bound is always tighter than the Ziv-Zakai lower bound.

Tightness of the bounds:

Probability of outage error- condition: Unimodal posterior pdf.

MSE – condition: Unimodal and symmetric posterior pdf.

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