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Ali-Silvey Distances

Module by: Don Johnson

Ali-Silvey distances comprise a family of quantities that depend on the likelihood ratio Λr Λ r and on the model-describing densities p 0 p 0 , p 1 p 1 in the following way.

d p 0 p 1 =fE0cΛr d p 0 p 1 f 0 c Λ r (1)
Here, f· f · is an increasing function, c· c · is a convex function, and E0· 0 · means expectation with respect to p 0 p 0 . Where applicable, π 0 π 0 , π 1 π 1 denote the a priori probabilities of the models. Basseville is a good reference on distances in this class and many others. In all cases, the observations consist of LL IID random variables.

Ali-Silvey Distances and Relation to Detection Performance
Name c· c · Performance Comment
Kullback-Leibler p 1 ∥ p 0 p 1 p 0 · log · · · limL→∞-1Llog P F =d p 0 p 1 L 1 L P F d p 0 p 1 Neyman-Pearson error rate under both fixed and exponentially decaying constraints on P M P M ( P F P F )
Kullback-Leibler p 0 ∥ p 1 p 0 p 1 -log · · limL→∞-1Llog P M =d p 0 p 1 L 1 L P M d p 0 p 1
J J-Divergence · -1log · · 1 · π 0 π 1 ⅇ-d p 0 p 1 2≤ P e π 0 π 1 d p 0 p 1 2 P e J p 0 p 1 = p 0 ∥ p 1 + p 1 ∥ p 0 J p 0 p 1 p 0 p 1 p 1 p 0
Chernoff · s · s , s∈01 s 0 1 max{limL→∞-1Llog P e }=infs{d p 0 p 1 |s∈01} L 1 L P e s s 0 1 d p 0 p 1 Independent of a priori probabilities
M M-Hypothesis Chernoff · s · s , s∈01 s 0 1 max{limL→∞-1Llog P e }=min{infs{d p i p j |s∈01}|i≠j} L 1 L P e i j s s 0 1 d p i p j
Bhattacharyya · 12 · 1 2 π 0 π 1 d2 p 0 p 1 ≤ P e ≤ π 0 π 1 d p 0 p 1 π 0 π 1 d p 0 p 1 2 P e π 0 π 1 d p 0 p 1 Minimizing d p 0 p 1 d p 0 p 1 will tend to minimize P e P e
Orsak | π 1 ·- π 0 | π 1 · π 0 P e =12-12d p 0 p 1 P e 1 2 1 2 d p 0 p 1 Exact formula for average error probability
Kolmogorov 12| · -1| 1 2 · 1 If π 0 = π 1 π 0 π 1 , P e =12-12d p 0 p 1 P e 1 2 1 2 d p 0 p 1
Hellinger · 12-12 · 1 2 1 2

References

  1. M. Basseville. (1989). Distance measures for signal processing and pattern recognition. Signal Processing, 18, 349-369.

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