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

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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

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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

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IntroductionBayesian parameter estimation

• Goal to estimate the unknown parameter ?
• based on the observation vector x.
• Assumptions
• ? and x are random variables
• The observation cdf and
• posterior pdf are known

Applications Radar/Sonar, Communication,
Biomedical, Audio/speech,
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IntroductionParameter estimation criteria

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

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IntroductionParameter estimation criteria

• 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.

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IntroductionMMSE estimation

• The minimum MSE is attained by MMSE

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Introductionh-MAP estimation

• The h-MAP estimator is

The corresponding minimum probability of h-outage
error is
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Performance lower bounds

• Motivation
• Performance analysis
• Threshold prediction
• System design
• Feasibility study

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Performance lower bounds

• Bounds desired features
• Computational simplicity
• Tightness
• Asymptotically coincides with the optimal
  performance
• Validity independent of the estimator.

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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.

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Previous work Bayesian MSE bounds
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General class of outage error probability lower
bounds
The probability of outage error
?
(Reverse) Hölder inequality for
Taking
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General class of outage error probability lower
bounds
Objective obtain valid bounds, independent of
.
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General class of outage error probability lower
bounds

• Theorem
• A necessary and sufficient condition to
  obtain a valid bound which is independent of the
  estimator, is that the function
• is periodic in ? with period h, almost
  everywhere.

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General class of outage error probability lower
bounds
Using Fourier series representation the general
class of bounds is
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Example Linear Gaussian model
The model
The minimum h-outage error probability
The single coefficient bound
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The tightest subclass of lower bounds

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

Convergence condition There exists l0h(?,x),
agt0 such that for all ll0h(?,x)
This mild condition guaranties that
converges for every p1.

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The tightest subclass of lower bounds
Under the convergence condition, the tightest
bounds are
h sampling period
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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
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General class of MSE lower bounds

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

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General class of MSE lower bounds

• New MSE lower bounds can be obtained by using
• and lower bounding the probability of outage
  error

• For example
• General class of MSE bounds
• The tightest MSE bound

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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.
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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
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Properties Relation to the ZZLB
ZZLB
The proposed bound
max out
2
2
1
1
6
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For any converging sequence of non-negative
numbers Therefore,
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Properties unimodal symmetric pdf

• Theorem
• A. If the posterior pdf f ? x(? x) is
  unimodal, then the proposed tightest outage
  error probability bound coincides with the
  minimum probability of outage error for every
  hgt0.
• B. If the posterior pdf f ? x(? x) is
  unimodal and symmetric, then the proposed
  tightest MSE bound coincides with the minimum MSE.

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Example 1

Statistics
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Example 2
The model
Statistics
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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.