New error bounds for M-testing and estimation of source location with subdiffractive error

Abstract

I present new lower and upper bounds on the minimum probability of error (MPE) in Bayesian multihypothesis testing that follow from an exact integral of a version of the statistical entropy of the posterior distribution, or equivocation. I also show that these bounds are exponentially tight and thus achievable in the asymptotic limit of many conditionally independent and identically distributed measurements. I then relate the minimum mean-squared error (MMSE) and the MPE by means of certain elementary error probability integrals. In the second half of the paper, I compare the MPE and MMSE for the problem of locating a single point source with subdiffractive uncertainty. The source-strength threshold needed to achieve a desired degree of source localization seems to be far more modest than the well established threshold for the different optical super-resolution problem of disambiguating two point sources with subdiffractive separation.

Asymptotics of Bayesian Error Probability and Rotating-PSF-Based Source Super-Localization in Three Dimensions

“…The MPE metric also provides a useful relationship between Bayesian inference and statistical information via the Fano bound and its generalizations [18]. The present work extends our earlier study [19] of localization accuracy based on both the MPE and MMSE metrics, which were shown to be related closely for a highly sensitive Bayesian detector, from a few sensor pixels to the asymptotic domain of many sensor pixels. It differs, however, from the more standard asymptotic analyses of MHT [20] in which one assumes that many statistically identical data frames are present, by addressing the experimentally more realistic context of many image pixels that sample a spatially varying PSF, typically with a single peak so the pixels progressively farther away from the peak contain progressively less signal.…”

“…In particular, we showed a quadratic (M 2 ⊥ ) dependence of the minimum source strength needed to achieve a transverse localization improvement factor of M ⊥ at a statistical confidence limit of 95% or better. The agreement in the predictions of MSE and MPE based analyses in the high-sensitivity asymptotic limit is a consequence of the equivalence of the two metrics in that limit [19].…”

“…An alternative, minimum mean-squared error (MMSE) based analysis may also be given to describe this trade-off, but the two Bayesian error analyses are expected to produce similar conclusions, at least in the highly-sensitive detection limit, as we showed in Ref. [19].…”

“…(N − 1) times to integrate over the (N − 1) nonzero, mutually orthogonal components of the vector x ⊥ . The remaining integral in expression (19) may be evaluated in terms of the complementary error function as…”

Bayesian Error-Based Sequences of Statistical Information Bounds

“…For the special choice, s m (X) = p(θ m |X), the RHS of inequality (14) reduces to −H(Θ|X), which is the FM bound. A more general choice involves the generalized posterior PMF, p n (θ m |X), defined earlier in (7), that becomes sharper the larger its order n,…”