Multi-hypothesis Testing Research Articles

Optimal signal processing for continuous qubit readout

The measurement of a quantum two-level system, or a qubit in modern terminology, often involves an electromagnetic field that interacts with the qubit, before the field is measured continuously and the qubit state is inferred from the noisy field measurement. During the measurement, the qubit may undergo spontaneous transitions, further obscuring the initial qubit state from the observer. Taking advantage of some well known techniques in stochastic detection theory, here we propose a novel signal processing protocol that can infer the initial qubit state optimally from the measurement in the presence of noise and qubit dynamics. Assuming continuous quantum-nondemolition measurements with Gaussian or Poissonian noise and a classical Markov model for the qubit, we derive analytic solutions to the protocol in some special cases of interest using It\={o} calculus. Our method is applicable to multi-hypothesis testing for robust qubit readout and relevant to experiments on qubits in superconducting microwave circuits, trapped ions, nitrogen-vacancy centers in diamond, semiconductor quantum dots, or phosphorus donors in silicon.

Asymptotics of Bayesian error probability and source super-localization in three dimensions.

We present an asymptotic analysis of the minimum probability of error (MPE) in inferring the correct hypothesis in a Bayesian multi-hypothesis testing (MHT) formalism using many pixels of data that are corrupted by signal dependent shot noise, sensor read noise, and background illumination. We perform our analysis for a variety of combined noise and background statistics, including a pseudo-Gaussian distribution that can be employed to treat approximately the photon-counting statistics of signal and background as well as purely Gaussian sensor read-out noise and more general, exponentially peaked distributions. We subsequently evaluate both the exact and asymptotic MPE expressions for the problem of three-dimensional (3D) point source localization. We focus specifically on a recently proposed rotating-PSF imager and compare, using the MPE metric, its 3D localization performance with that of conventional and astigmatic imagers in the presence of background and sensor-noise fluctuations.

Asymptotics of Bayesian error probability and 2D pair superresolution.

This paper employs a recently developed asymptotic Bayesian multi-hypothesis testing (MHT) based error analysis to treat the problem of superresolution imaging of a pair of closely spaced, equally bright point sources. The analysis exploits the notion of the minimum probability of error (MPE) in discriminating between two competing equi-probable hypotheses, a single point source of a certain brightness at the origin vs. a pair of point sources, each of half the brightness of the single source and located symmetrically about the origin, as the distance between the source pair is changed. For a Gaussian point-spread function (PSF), the analysis makes predictions on the scaling of the minimum source strength, expressed in units of photon number, required to disambiguate the pair as a function of their separation in both the signal-dominated and background-dominated regimes. Certain logarithmic corrections to the quartic scaling of the minimum source strength with respect to the degree of superresolution characterize the signal-dominated regime, while the scaling is purely quadratic in the background-dominated regime. For the Gaussian PSF, general results for arbitrary strengths of the signal, background, and sensor noise levels are also presented.

IMPROVING THE SPACE SURVEILLANCE TELESCOPE'S PERFORMANCE USING MULTI-HYPOTHESIS TESTING

The Space Surveillance Telescope (SST) is a Defense Advanced Research Projects Agency program designed to detect objects in space like near Earth asteroids and space debris in the geosynchronous Earth orbit (GEO) belt. Binary hypothesis test (BHT) methods have historically been used to facilitate the detection of new objects in space. In this paper a multi-hypothesis detection strategy is introduced to improve the detection performance of SST. In this context, the multi-hypothesis testing (MHT) determines if an unresolvable point source is in either the center, a corner, or a side of a pixel in contrast to BHT, which only tests whether an object is in the pixel or not. The images recorded by SST are undersampled such as to cause aliasing, which degrades the performance of traditional detection schemes. The equations for the MHT are derived in terms of signal-to-noise ratio (S/N), which is computed by subtracting the background light level around the pixel being tested and dividing by the standard deviation of the noise. A new method for determining the local noise statistics that rejects outliers is introduced in combination with the MHT. An experiment using observations of a known GEO satellite are used to demonstrate the improved detection performance of the new algorithm over algorithms previously reported in the literature. The results show a significant improvement in the probability of detection by as much as 50% over existing algorithms. In addition to detection, the S/N results prove to be linearly related to the least-squares estimates of point source irradiance, thus improving photometric accuracy.

Controlled Sensing for Multihypothesis Testing

The problem of multiple hypothesis testing with observation control is considered in both fixed sample size and sequential settings. In the fixed sample size setting, for binary hypothesis testing, the optimal exponent for the maximal error probability corresponds to the maximum Chernoff information over the choice of controls, and a pure stationary open-loop control policy is asymptotically optimal within the larger class of all causal control policies. For multihypothesis testing in the fixed sample size setting, lower and upper bounds on the optimal error exponent are derived. It is also shown through an example with three hypotheses that the optimal causal control policy can be strictly better than the optimal open-loop control policy. In the sequential setting, a test based on earlier work by Chernoff for binary hypothesis testing, is shown to be first-order asymptotically optimal for multihypothesis testing in a strong sense, using the notion of decision making risk in place of the overall probability of error. Another test is also designed to meet hard risk constrains while retaining asymptotic optimality. The role of past information and randomization in designing optimal control policies is discussed.

Decision-making out of neural events: from discrimination information to psychometric power laws

The strength/intensity of the stimulus in the random dot motion task (RDMT) [1] is determined by the percentage of dots in the kinematogram moving towards a saccadic target, a. Due to the uncertainty in the stimuli, neurons in sensory systems have evolved to transform environmental information, comprising evidence upon which a decision can be made (e.g. saccading to a). The neurons in the middle-temporal area (MT) appear to produce such evidence during the RDMT, given their tuning to a 'preferred' direction of visual motion. If the dots move predominantly in the preferred direction of an MT neuron, it generates inter-spike intervals (ISI) supporting a saccade to a. These ISIs seem randomly sampled from a distribution, fa, with mean, μa. Otherwise, the ISIs follow another distribution, fb, with mean, μb, where μb is larger than μa and this difference increases with stimulus strength. The accuracy vs motion-strength function of an ideal observer provided with empirical distributions like fa and fb, from a single MT neuron, approximates the subject's psychometric function (at the behavioral level) [1]. The distributions fa and fb are non-negative, positively skewed and have a mode larger than 0 (figure ​(figure1A),1A), as is typical for neural events recorded in many brain areas. Here we investigate why this is advantageous for decision-making. As theoretical decision-making units, we produced 5 new instantiations of the multi-hypothesis sequential probability ratio test (MSPRT) [2]. Each unit assumes its stream of input evidence to follow 1 of 5 probability density functions (PDF) whose compatibility with the empirical distribution of ISIs varies (figure ​(figure1A).1A). These include the Inverse Gaussian, Lognormal, Gamma, Inverse-Gamma and Exponential PDFs (the latter is the distribution of the inter-event times in the oft-used Poisson process). Under equal and appropriate conditions, we then compared their mean decision sample with that of an MSPRT instantiation that assumes Gaussian inputs proposed in [3] and discussed in general in [4], as exemplified in figure ​figure1B.1B. The mean decision sample is the mean number of observations required by a unit to identify which of N parallel information sources supports saccading to a, with a given accuracy. This decision sample is a model of the 'neural decision time'; the psychophysical reaction time also includes sensory and motor delays. The pattern of our results is explicable using a measure of the discrimination information between fa and fb, i.e. the Kullback-Leibler divergence (KLD). We found that, the mean decision sample decreases with increasing fa to fb KLD and, crucially, this follows a power law (figure ​(figure1C).1C). At the behavioral level, Pieron [5] reported the mean reaction time to the presentation of a stimulus (go/no-go decision-making) being shorter for more intense stimuli, and that a power law relates these measures. The universality of Pieron's law indicates that it can inform us of something fundamental about sensorimotor decision-making. Our results suggest that its explanation could lie in the power law relationship between the mean neural decision time and the discrimination information (KLD) among the distributions of sensory evidence. Figure 1 (A) PDFs of interest fitted to ISIs (grey bars) recorded in [1]from the MT during the RDMT. (B) mean decision sample for each MSPRT realisation (accuracy 95% over 1000 trials). (C) the values in (B) vs the fa to fb KLD. The dashed line is a fitted power ...

Enumeration of Base Station Antennas in a Cognitive Receiver by Exploiting Pilot Patterns

This letter deals with the identification of the number of transmitting antennas at a base station. This estimation is achieved by exploiting the orthogonality between the pilot patterns from each transmitting antenna. This orthogonality allows us to use multi-hypothesis testing in order to enumerate the number of transmitting antennas using only one antenna at the receiver. As a side product, the proposed algorithm allows us to estimate the channel gains from all transmitting antennas and the noise variance.

Implementation of MIMO Radar for Multiple Target Detection

This paper focuses on the early detection problem of multiple moving targets in statistical MIMO radar systems using TBD techniques. At first, assuming prior knowledge of the number of targets, a binary generalized likelihood ratio test (GLRT) is derived, which shows that the optimal implementation of the GLRT requires multidimensional joint search. To reduce the implementation complexity, a suboptimum multitarget TBD algorithm using successive-target cancellation and polar Hough transform (STC-PHT) is proposed. In addition to low complexity, the new proposed algorithm doesn't need the prior information of the number of targets, and can avoid the implementation of multi-hypothesis test when the number of targets is unknown.

A Multistage Procedure for Decentralized Sequential Multi-Hypothesis Testing Problems

We studied the problem of sequentially testing M ≥ 2 hypotheses with a decentralized sensor network system. In such a system, the local sensors observe raw data and then send quantized observations to a fusion center, which makes a final decision regarding hypothesis is true. Motivated by the two-stage tests in Wang and Mei (2011), we propose a multistage decentralized sequential test that provides multiple opportunities for the local sensors to adjust to the optimal local quantizers. It is demonstrated that when the hypothesis testing problem is asymmetric, the multistage test is second-order asymptotically optimal. Even though this result constitutes an interesting theoretical improvement over two-stage tests that can enjoy only first-order asymptotic optimality, the corresponding practical merits seem to be only marginal. Indeed, performance gains over two-stage procedures with carefully selected thresholds are small.

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

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.

Hybrid likelihood ratio bit-rate detectors for variable-gain multiple-access systems in unknown noise variance

The authors assume a linear equidistance antenna array as the receiver for a fixed frame-length multiple-access system, which employs variable-gain receiver power and repetition encoding. They propose a robust maximum a posteriori probability (MAP) blind bit-rate detector (BBRD). This detector considers the rate detection (RD) as a multi-hypothesis test and maximises the likelihood functions (LFs) to find the true bit-rate, whereas the complex amplitude of the received signal, the noise variance and the direction of arrival are unknown parameters. First, assuming that the location parameter is known and the information sequence are independent and uniformly distributed random variables, the authors propose a hybrid likelihood ratio test (HLRT). The proposed HLRT requires to solve a set of non-linear equations that have no closed-form solution. Thus, an iterative numerical algorithm is proposed. In addition, a quasi-HLR detector which has a significantly lower computational complexity is also proposed. In the case of unknown location parameter, the authors develop two quasi-HLR methods. They use fast Fourier transform and search to estimate the unknown location. In Q-HLRT-Method-I, the non-linear equation similar to the one in known location parameter is iteratively solved. In Q-HLRT-Method-II, a low complexity solution is proposed. Simulation examples evaluate and compare the performances of the proposed BBRDs.

A Generalized Poor-Verdú Error Bound for Multihypothesis Testing

A lower bound on the minimum error probability for multihypothesis testing is established. The bound, which is expressed in terms of the cumulative distribution function of the tilted posterior hypothesis distribution given the observation with tilting parameter , generalizes an earlier bound due the Poor and Verdú (1995). A sufficient condition is established under which the new bound (minus a multiplicative factor) provides the exact error probability asymptotically in . Examples illustrating the new bound are also provided.

Asymptotic Optimality Theory for Decentralized Sequential Multihypothesis Testing Problems

The Bayesian formulation of sequentially testing $M \ge 3$ hypotheses is studied in the context of a decentralized sensor network system. In such a system, local sensors observe raw observations and send quantized sensor messages to a fusion center which makes a final decision when stopping taking observations. Asymptotically optimal decentralized sequential tests are developed from a class of "two-stage" tests that allows the sensor network system to make a preliminary decision in the first stage and then optimize each local sensor quantizer accordingly in the second stage. It is shown that the optimal local quantizer at each local sensor in the second stage can be defined as a maximin quantizer which turns out to be a randomization of at most $M-1$ unambiguous likelihood quantizers (ULQ). We first present in detail our results for the system with a single sensor and binary sensor messages, and then extend to more general cases involving any finite alphabet sensor messages, multiple sensors, or composite hypotheses.

Sequential tests and biologically grounded multi-alternative decision making

Recently, Bogacz and Gurney [1] proposed that the cortico-basal-ganglia system implements asymptotically optimal decision making between several alternatives, based on sensory evidence, through a statistical algorithm known as the multi-hypothesis sequential probability ratio test (MSPRT). The original programme of work focused on architectural features of the system and made simplifying assumptions as to its physiology. Here, we extend that work to include more biologically-realistic properties and pathways, and explain their impact on the MSPRT performance. One assumption in [1] was that ‘noise’ on sensory signals was Gaussian distributed, which poses the problem of interpreting the signals in the negative tail of the distribution in terms of neural events. We addressed this issue in previous work [2] by deducing a new MSPRT that proposed these signals are Inverse Gaussian distributed (wholly positive and skewed). Using realistic parameters we showed this MSPRT requires about a tenth of the samples of its Gaussian predecessor to reach a decision with the same proportion of errors. A further simplification in [1] was that the processing elements were dynamics-free and the inter-element delays were zero. In reality neural membranes display non-trivial dynamics and significant inter-neuronal/synaptic processing delays are present, both having an effect on the performance. Hence we developed a system of first order differential equations with the time constants for membranes at rest available in the literature. We found that such a system had decision sample sizes roughly twice that of the non-dynamic MSPRT. However, active membranes have reduced time constants [3]. Using values one half of those at rest gave a 30% reduction in performance, and values of one fifth of those at rest, only a 4% decrease. Furthermore, following a closer examination of the anatomy, we added new pathways to give the architecture cortex→basal-ganglia→thalamus↔cortex (arrows indicating information flow), which completes the mapping of a ‘Bayesian structure’ onto it and represents the full decision/inference circuit. Closing such loop recursively feeds back the posterior inference from sampling step t as prior ‘knowledge’ for further steps t + Δt. By doing this we also constructed another recursive statistical test (also a MSPRT) that differs from the non-recursive (feedback-less) one proposed in [1]. To test the effect on performance of including these pathways, we incorporated them into a non-dynamic MSPRT with Inverse Gaussian inputs from [2], including the main delays. The recursion increased the number of samples to reach a decision by around 30% compared with its original non-recursive counterpart. This is nevertheless a small sacrifice considering that in return the system becomes more general and ultimately able to bias further decisions by adjusting its prior knowledge. Finally, in figure ​figure11 we show the evolution of several representative elements of the loop during a decision. These results suggest that the closed loop, delayed MSPRT might hold the key to explain the time course and magnitude of the firing rates observed, in macaque electrophysiological single unit recordings, during a decision task [4]. Figure 1 Time course of firing-rate-like signals on selected stages, during a single decision trial up to a threshold, using the closed loop, delayed MSPRT (1 sample every 15 ms). Compare with figure 7 in [4].

Biomarker studies: a call for a comprehensive biomarker study registry

Tumor biomarker studies may generate insights into the biological characteristics that drive the clinical behavior of a cancer. Publication bias and hidden multiple hypotheses testing distort the assessment of the true value of biomarkers. Publication bias from preferential reporting of 'positive' findings is well recognized. Hidden multihypothesis testing arises from several biomarkers being tested by different teams using the same samples. The more hypotheses (that is, biomarker association with outcome) tested, the greater the risk of false-positive findings. These biases inflate the potential clinical validity and utility of published biomarkers while negative results often remain hidden. Trial registries have been developed where all phase II and phase III trials should be listed regardless of study outcome. However, such steps have not been taken to reduce such bias in tumor biomarker research. We propose that a registry should be created for biomarker studies initially focused on studies that use specimens from randomized trials. Further development could include nonrandomized studies and deposition of raw data similar to existing genomic data repositories. The benefits of a comprehensive biomarker study registry include more balanced evaluation of proposed markers, fewer false positive leads in research, and hopefully more rapid identification of promising candidate biomarkers.

Accelerating FAB-MAP With Concentration Inequalities

We outline an approach for using concentration inequalities to perform rapid approximate multi-hypothesis testing. In a scenario where multiple hypotheses are ranked according to a large set of features, our scheme improves the efficiency of selecting the best hypothesis by providing a “bail-out threshold” at which unpromising hypotheses can be excluded from further evaluation. We show how concentration inequalities can be used to derive principled bail-out thresholds, subject to a user-specified error tolerance. The technique is similar to the sequential probability ratio test, but is applicable in more general conditions. We apply the technique to improve the speed of the fast-appearance-based mapping system for appearance-based place recognition and mapping. The speed increase provided by the new approach is data dependent, but we demonstrate speed improvements of between 25x - 50x on real data, with only a slight degradation in accuracy.

A Method for Designing Three-Hypothesis Test Problems and Sequential Schemes

In applications, a two-sided hypothesis test problem sometimes needs to be changed to a three-hypothesis one with the two alternative hypotheses properly selected. In this article, we obtain the hypothesis design and the three-hypothesis sequential test scheme under the Koopman–Darmois distribution by solving a system of equations that meet requirements on the error rates and average sample number. This method provides a useful guide for practitioners to design hypotheses in multihypothesis test problems with controlled error rates and sampling cost. Formulas of the scheme's error rates and average sample number are obtained using numerical quadrature for the discrete-time situation.

Optimal decision making with biologically realistic neural signals

Action selection in animals requires rapid decision making that can discriminate the most salient requests for behavioral expression. The basal ganglia (BG) are believed to play a critical role in resolving competition between these requests [1] and, recently it has been proposed that the BG and cortex, taken together, implement a decision algorithm known as the multi-hypothesis sequential probability ratio test (MSPRT) [2]. Here, the cortex first integrates noisy ‘evidence’ indicating salience of action requests. The BG then examine this integrated evidence, and report the channel with maximal mean salience. The MSPRT is optimal in the sense that it guarantees the smallest decision time for a given error. The signals used in [2] assume cortical evidence - as neural firing rates - is supplied by Gaussian distributed signals. However, it is known that such rates are not distributed normally. To remedy this we have deduced three new forms for MSPRT that are based on more biologically realistic input signals. They include the Inverse Gaussian, Gamma and Lognormal distributions. All implementations preserve a form which could be mapped, in principle, onto the anatomy of the BG. Figure ​Figure11 shows an example of the performance of the MSPRT models across all distributions, together with the so called race model. All variants used the same input statistics (mean and variance). Counterintuitively, the relative performance of the models did not change with base level of salience or evidence. Figure 1 Performance of original Gaussian MSPRT[2], new implementations of MSPRT, and Race model. Performance is measured as decision time against number of alternatives for selection. The selection criterion was 1% error rate. The results are averaged across ... A critical parameter in the algorithm is the time step Δt used to sample the cortical input. The Gaussian, Inverse Gaussian, and Gamma cases are generated as stable Levy processes and may be scaled indefinitely with decreasing Δt. However, unlike the MSPRT with Gaussian signals, we show that decision time for the inverse Gaussian, and Gamma decreases with Δt. Figure ​Figure11 use a Δt = 1 ms for all models and clearly shows a performance advantage for the cases with more realistic distributions (the lognormal case is shown with similar scaling even though it has not an associated Levy process). Indefinitely small decision times may appear unrealistic, but we present an interpretation of cortical sampling that relates Δt to the inter-spike intervals of an ensemble of neural afferents impinging on the cortical ‘integrators’. This and, other mechanisms, provide a natural lower bound for Δt. We conclude that neurons may take advantage of the properties of natural spike trains to enhance decision making in the cortex-BG complex.

Robust and distributed stochastic localization in sensor networks

We present a robust localization system allowing wireless sensor networks to determine the physical location of their nodes. The coverage area is partitioned into regions and we seek to identify the region of a sensor based on observations by stationary clusterheads. Observations (e.g., signal strength) are assumed random. We pose the localization problem as a composite multihypothesis testing problem, develop the requisite theory, and address the problem of optimally placing clusterheads. We show that localization decisions can be distributed by appropriate in-network processing. The approach is validated in a testbed yielding promising results.

Use of molecular classification combined with p53 and topoisomerase IIa expression to identify tumors highly responsive to FEC regimen: A tissue microarray

546 Background: Identification of highly responsive tumors to specific regimen is needed in order to better tailor post-operative chemotherapy regimen. We previously reported using a tissue microarrays that basal-like and HER2-overexpressing tumors derive higher benefit from adjuvant chemotherapy than luminal-like tumors. Previous studies have reported that p53+/basal-like tumors, and HER2+/topoisomerase IIa + tumors were highly sensitive to alkylating agents and anthracyclines respectively. In the present study, we have evaluated whether p53+/basal-like and HER2+TopIIa+ tumors derive high benefit from an alkylating/anthracyclines based chemotherapy. Methods: A tissue microarrays was built with tumor samples from 823 patients randomized between FEC/FAC regimen and no chemotherapy. Basal-like breast cancers were defined as HER2-/ER-/EGFR+ and/or CK5/6+ tumors. HER2-overexpressing were defined using HER2 expression by IHC (3+ score). P53 and topoisomerase IIa expression were assessed by IHC. Treatment effect was assessed by Cox model that included prognostic parameters and stratification variables. Predictive value for treatment effect were assessed by interaction test and test for heterogeneity. Results: 823 patients were included. 12.5% and 12.9% patients presented a basal-like and HER2-overexpressing breast cancer respectively. P53 and topoisomerase IIa were expressed in 20% and 35% patients. Hazard ratio for relapse associated with FEC/FAC regimen was 0.79 (95%CI: 0.62–1). Hazard ratio associated with FEC/FAC were 0.31 (0.1–0.97 ) and 1.19 (0.38–3.7 ) in patients with p53+/basal-like and p53- basal-like breast cancer. A p53/basal-based classification was associated with a trend for heterogeneity (p = 0.15). Hazard ratio associated with FEC/FAC were 0.34 (0.11–1.0 ) and 1.27(0.32–4.9) in patients with topIIa+/HER2+ and topIIa-/HER2+ breast cancer. A HER2/TopIIa classification was associated with a trend for heterogeneity (p = 0.07). Conclusions: The present study suggest that p53+/basal-like cancer and TopIIA+/HER2 cancers derive high benefit from adjuvant FEC/FAC regimen. This finding requires further validation given multihypothesis testing and subgroup analysis. No significant financial relationships to disclose.