The Ziv–Zakai bound (named after Jacob Ziv and Moshe Zakai) is used in theory of estimations to provide a lower bound on possible-probable error involving some random parameter X {\displaystyle X} from a noisy observation Y {\displaystyle Y} . The bound work by connecting probability of the excess error to the hypothesis testing. The bound is considered to be tighter than Cramér–Rao bound albeit more involved. Several modern version of the bound have been introduced subsequent of the first version which was published 1969.
Simple Form of the Bound Suppose we want to estimate a random variable X {\displaystyle X} with the probability density f X {\displaystyle f_{X}} from a noisy observation Y {\displaystyle Y} , then for any estimator g {\displaystyle g} a simple form of Ziv-Zakai bound is given by
E [ | X − g ( Y ) | 2 ] ≥ 1 2 ∫ 0 ∞ t ∫ − ∞ ∞ ( f X ( x ) + f X ( x + t ) ) P e ( x , x + t ) d x d t , {\displaystyle {\begin{aligned}&\mathbb {E} {\bigl [}|X-g(Y)|^{2}{\bigr ]}\geq {\frac {1}{2}}\int _{0}^{\infty }t\int _{-\infty }^{\infty }{\bigl (}f_{X}(x)+f_{X}(x+t){\bigr )}\,P_{e}(x,x+t)\,\mathrm {d} x\,\mathrm {d} t,\end{aligned}}}
where
P e ( x , x + t ) {\displaystyle P_{e}(x,x+t)} is the minimum (Bayes) error probability for the binary hypothesis testing problem between
H 0 : Y ∣ X = x H 1 : Y ∣ X = x + t {\displaystyle {\begin{aligned}{\mathcal {H}}_{0}&:Y\mid X=x\\{\mathcal {H}}_{1}&:Y\mid X=x+t\end{aligned}}}
with prior probabilities
Pr ( H 0 ) = f X ( x ) f X ( x ) + f X ( x + t ) {\displaystyle \Pr({\mathcal {H}}_{0})={\frac {f_{X}(x)}{f_{X}(x)+f_{X}(x+t)}}} and
Pr ( H 1 ) = 1 − Pr ( H 0 ) {\displaystyle \Pr({\mathcal {H}}_{1})=1-\Pr({\mathcal {H}}_{0})} .
Generalization The original lower bound can be tightened by introducing a notion of the valley-filling function, which for a function f {\displaystyle f}
… excerpt ends here. Continue reading the full article.
