Optimal Discrimination Research Articles

Optimal Discrimination of Two Nonorthogonal States by Continuous Probing and Feedback Operation

AbstractFor a quantum system prepared probabilistically on two or more non‐orthogonal states, the observer cannot discriminate the initial preparation perfectly. Kurt Jacobs [Quantum Inf. Comput. 2007, 7, 127] introduced a measurement operator that could increase the rate of information obtained using a continuous measurement scheme. However, the better effect happens at the expense of reducing the total information from the quantum system. To address this problem, an optimal operator that could yield the maximal value of mutual information for a long‐time measurement is found. Particularly, it turns out that the error probability is minimized for any measurement moment by measuring the optimal operator. Furthermore, for a given finite measurement time, a measurement scheme to maximize the obtained mutual information is presented. It is shown that while the Jacobs' operator should be used for a short‐time case, for a long‐time limit, the proposed optimal operator should be employed. For a given finite duration, the proposal could determine the optimal measurement operator that maximizes the final value of mutual information.

Experimental multi-state quantum discrimination through optical networks

Developing strategies to effectively discriminate between different quantum states is a fundamental issue in quantum information and communication. The actual realization of generally optimal protocols in this task is often limited by the need of supplemental resources and very complex receivers. We have experimentally implemented two discrimination schemes in a minimum-error scenario based on a receiver featured by a network structure and a dynamical processing of information. The first protocol implemented in our experiment, directly inspired to a recent theoretical proposal, achieves binary optimal discrimination, while the second one provides a novel approach to multi-state quantum discrimination, relying on the dynamical features of the network-like receiver. This strategy exploits the arrival time degree of freedom as an encoding variable, achieving optimal results, without the need for supplemental systems or devices. Our results further reveal the potential of dynamical approaches to quantum state discrimination tasks, providing a possible starting point for efficient alternatives to current experimental strategies.

Worry Modifies the Relationship between Locus Coeruleus Activity and Emotional Mnemonic Discrimination.

Background: The locus coeruleus (LC) plays a critical role in modulating emotional memory performance via widespread connections to the medial temporal lobe (MTL). Interestingly, both the LC and MTL are affected during aging. Therefore, we aimed to investigate whether worry during cognitive aging changes the relationship between memory performance and the neural activity patterns during an emotional memory task. Methods: Twenty-eight participants aged 60–83 years from the Maastricht Aging study conducted an emotional mnemonic discrimination task during a 7T fMRI-scan. We performed a robust multiple linear regression to examine the association between worry and mnemonic memory performance under different levels of arousal. Subsequently, we examined if worry modifies the relationship between neuronal activity and mnemonic memory performance. Results: We observed that under low arousal, only participants with low compared to high levels of worry benefitted from additional LC activity. Under high arousal, additional LC activity was associated with lower mnemonic memory performance. Conclusion: Our results suggest there might be an optimal involvement of the NA-system for optimal memory discrimination performance, as we observed that under low levels of worry and with lower levels of arousal, higher LC activity might be needed to achieve similar levels of optimal memory performance as achieved under higher arousal when LC activity remained lower.

Bound on local minimum-error discrimination of bipartite quantum states

We consider the optimal discrimination of bipartite quantum states and provide an upper bound for the maximum success probability of optimal local discrimination. We also provide a necessary and sufficient condition for a measurement to realize the upper bound. We further establish a necessary and sufficient condition for this upper bound to be saturated. Finally, we illustrate our results using an example.

Locking and unlocking of quantum nonlocality without entanglement in local discrimination of quantum states

The phenomenon of nonlocality without entanglement (NLWE) arises in discriminating multi-party quantum separable states. Recently, it has been found that the post-measurement information about the prepared subensemble can lock or unlock NLWE in minimum-error discrimination of non-orthogonal separable states. Thus It is natural to ask whether the availability of the post-measurement information can influence on the occurrence of NLWE even in other state-discrimination strategies. Here, we show that the post-measurement information can be used to lock as well as unlock the occurrence of NLWE in terms of optimal unambiguous discrimination. Our results can provide a useful application for hiding or sharing information based on non-orthogonal separable states.

Optimal Bayesian design for model discrimination via classification

Performing optimal Bayesian design for discriminating between competing models is computationally intensive as it involves estimating posterior model probabilities for thousands of simulated data sets. This issue is compounded further when the likelihood functions for the rival models are computationally expensive. A new approach using supervised classification methods is developed to perform Bayesian optimal model discrimination design. This approach requires considerably fewer simulations from the candidate models than previous approaches using approximate Bayesian computation. Further, it is easy to assess the performance of the optimal design through the misclassification error rate. The approach is particularly useful in the presence of models with intractable likelihoods but can also provide computational advantages when the likelihoods are manageable.

Diagnostic of Injury Risk in the Anterior Cruciate Ligament Based on Shape Context Description of the Intercondylar Notch Curvature

Abstract An estimated number of 300,000 new anterior cruciate ligament (ACL) injuries occur each year in the United States. Recent studies have pointed out a correlation between the curvature of the femur intercondylar notch and the risk of noncontact ACL injury. Although several magnetic resonance (MR) imaging-based ACL diagnostics methods have already been proposed in the literature, most of them are based on machine learning or deep learning strategies, which are computationally expensive. In this paper, we propose a diagnostics framework for the risk of injury in the ACL based on the application of the inner-distance shape context (IDSC) to describe the curvature of the intercondylar notch from MR images. First, the contours of the intercondylar notch curvature from 91 MR images of the distal end of the femur (70 healthy and 21 with confirmed ACL injury) were extracted manually using standard image processing tools. Next, the IDSC was applied to calculate the similarity factor between the extracted contours and reference standard curvatures. Finally, probability density functions of the similarity factor data were obtained through parametric statistical inference, and the accuracy of the ACL injury risk diagnostics framework was assessed using receiver operating characteristic analysis (ROC). The overall results for the area under the curve (AUC) showed that the method reached a maximum accuracy of about 66%. Furthermore, the sensitivity and specificity results showed that an optimum discrimination threshold value for the similarity factor can be pursued that minimizes the incidence of false positives and false negatives simultaneously.

Bronchial Aspirate-Based Profiling Identifies MicroRNA Signatures Associated With COVID-19 and Fatal Disease in Critically Ill Patients.

BackgroundThe pathophysiology of COVID-19-related critical illness is not completely understood. Here, we analyzed the microRNA (miRNA) profile of bronchial aspirate (BAS) samples from COVID-19 and non-COVID-19 patients admitted to the ICU to identify prognostic biomarkers of fatal outcomes and to define molecular pathways involved in the disease and adverse events.MethodsTwo patient populations were included (n = 89): (i) a study population composed of critically ill COVID-19 and non-COVID-19 patients; (ii) a prospective study cohort composed of COVID-19 survivors and non-survivors among patients assisted by invasive mechanical ventilation (IMV). BAS samples were obtained by bronchoaspiration during the ICU stay. The miRNA profile was analyzed using RT-qPCR. Detailed biomarker and bioinformatics analyses were performed.ResultsThe deregulation in five miRNA ratios (miR-122-5p/miR-199a-5p, miR-125a-5p/miR-133a-3p, miR-155-5p/miR-486-5p, miR-214-3p/miR-222-3p, and miR-221-3p/miR-27a-3p) was observed when COVID-19 and non-COVID-19 patients were compared. In addition, five miRNA ratios segregated between ICU survivors and nonsurvivors (miR-1-3p/miR-124-3p, miR-125b-5p/miR-34a-5p, miR-126-3p/miR-16-5p, miR-199a-5p/miR-9-5p, and miR-221-3p/miR-491-5p). Through multivariable analysis, we constructed a miRNA ratio-based prediction model for ICU mortality that optimized the best combination of miRNA ratios (miR-125b-5p/miR-34a-5p, miR-199a-5p/miR-9-5p, and miR-221-3p/miR-491-5p). The model (AUC 0.85) and the miR-199a-5p/miR-9-5p ratio (AUC 0.80) showed an optimal discrimination value and outperformed the best clinical predictor for ICU mortality (days from first symptoms to IMV initiation, AUC 0.73). The survival analysis confirmed the usefulness of the miRNA ratio model and the individual ratio to identify patients at high risk of fatal outcomes following IMV initiation. Functional enrichment analyses identified pathological mechanisms implicated in fibrosis, coagulation, viral infections, immune responses and inflammation.ConclusionsCOVID-19 induces a specific miRNA signature in BAS from critically ill patients. In addition, specific miRNA ratios in BAS samples hold individual and collective potential to improve risk-based patient stratification following IMV initiation in COVID-19-related critical illness. The biological role of the host miRNA profiles may allow a better understanding of the different pathological axes of the disease.

Discrimination of basmati rice adulteration using colorimetric sensor array system

This study involved the fabrication of colorimetric sensor array system for rapid discrimination of adulterants in premium grade basmati rice. Purposely, low-grade white rice was used to adulterate the pure basmati rice at 5%, 10%, 15%, 25%, 50%, and 75% weight ratios. Sensor array system was used to capture the odors of prepared samples, resulting in color difference map based on chemical environment. The principal component analysis (PCA), hierarchical cluster analysis (HCA), and k-nearest neighbors (kNN) algorithm were subsequently used to identify the similarity between authentic and adulterated samples. A decent discrimination between authentic and adulterated rice samples was observed in the scatter plot of PCA and HCA dendrogram. The multilayered kNN models were able to effectively discriminate the prepared rice samples. The study concludes that fabricated sensor array system may be used as an effective tool for rapid discrimination of authentic and adulterated samples. • Colorimetric sensor array was fabricated for discrimination of rice adulteration. • Volatile compounds of basmati rice and adulterant were identified using GC-MS. • PCA and HCA were able to discriminate different levels of adulteration. • Optimal discrimination results were achieved with multilayered kNN algorithm. • Fabricated system promotes the development and application of sensor arrays.

Establishment and Validation of a Nomogram to Predict Hospital-Acquired Infection in Elderly Patients After Cardiac Surgery.

BackgroundHospital-acquired infection (HAI) after cardiac surgery is a common clinical concern associated with adverse prognosis and mortality. The objective of this study is to determine the prevalence of HAI and its associated risk factors in elderly patients following cardiac surgery and to build a nomogram as a predictive model.MethodsWe developed and internally validated a predictive model from a retrospective cohort of 6405 patients aged ≥70 years, who were admitted to our hospital and underwent cardiac surgery. The primary outcome was HAI. Multivariable logistic regression analysis was used to identify independent factors significantly associated with HAI. The performance of the established nomogram was assessed by calibration, discrimination, and clinical utility. Internal validation was achieved by bootstrap sampling with 1000 repetitions to reduce the overfit bias.ResultsIndependent factors derived from the multivariable analysis to predict HAI were smoking, myocardial infarction, cardiopulmonary bypass use, intraoperative erythrocytes transfusion, extended preoperative hospitalization days and prolonged duration of mechanical ventilation postoperatively. The derivation model showed good discrimination, with a C-index of 0.706 [95% confidence interval 0.671–0.740], and good calibration [Hosmer–Lemeshow test P = 0.139]. Internal validation also maintained optimal discrimination and calibration. The decision curve analysis revealed that the nomogram was clinically useful.ConclusionsWe developed a predictive nomogram for postoperative HAIs based on routinely available data. This predictive tool may enable clinicians to achieve better perioperative management for elderly patients undergoing cardiac surgery but still requires further external validation.

Qutrit state tomography via optimal unambiguous state discrimination and mutually unbiased measurements

We propose an efficient scheme for qutrit state tomography based on optimal unambiguous discrimination of three nonorthogonal pure qutrit states. In our scheme, we first introduce an ancillary qubit initially prepared in a known state and apply a joint unitary transformation on the whole system composed of a qutrit state and an ancillary qubit state. Afterwards, we make a local von Neumann measurement (VNM) on the ancillary qubit. If it is projected onto the state |0〉, we finally perform mutually unbiased measurements (MUMs) on the collapsed qutrit state arbitrarily chosen from a complete set of four mutually unbiased bases (MUBs). With only six properly-chosen projectors, the real and imaginary parts of non-diagonal elements can be entirely obtained. Together with the diagonal elements obtained directly from one VNM in the computational bases, the density matrix of an arbitrary unknown qutrit state can be fully determined. Compared with the previous work (R. Salazar and A. Delgado 2012), our scheme consumes much less resources. In comparison with MUBs-based tomography scheme (H. Yuan, et al., 2016), our scheme requires more measurements and less projectors, but pays the price of a loss of certain fraction identically prepared states. Our proposal can be extended to the case of other d-dimensional quantum states if their d MUBs are exactly known.

Qubit state discrimination using post-measurement information

We consider the optimal discrimination of nonorthogonal qubit states with post-measurement information and provide an analytic structure of the optimal measurements. We also show that there is always a null optimal measurement when post-measurement information is given. Further, in discriminating four states using post-measurement information, we analytically provide the optimal probability of correct guessing and show that the uniqueness of optimal measurement is equivalent to the non-existence of non-null optimal measurement with post-measurement information.

Machine learning approach to predict acute kidney injury after liver surgery.

BACKGROUNDAcute kidney injury (AKI) after surgery appears to increase the risk of death in patients with liver cancer. In recent years, machine learning algorithms have been shown to offer higher discriminative efficiency than classical statistical analysis.AIMTo develop prediction models for AKI after liver cancer resection using machine learning techniques.METHODSWe screened a total of 2450 patients who had undergone primary hepatocellular carcinoma resection at Changzheng Hospital, Shanghai City, China, from January 1, 2015 to August 31, 2020. The AKI definition used was consistent with the Kidney Disease: Improving Global Outcomes. We included in our analysis preoperative data such as demographic characteristics, laboratory findings, comorbidities, and medication, as well as perioperative data such as duration of surgery. Computerized algorithms used for model development included logistic regression (LR), support vector machine (SVM), random forest (RF), extreme gradient boosting (XGboost), and decision tree (DT). Feature importance was also ranked according to its contribution to model development.RESULTSAKI events occurred in 296 patients (12.1%) within 7 d after surgery. Among the original models based on machine learning techniques, the RF algorithm had optimal discrimination with an area under the curve value of 0.92, compared to 0.87 for XGBoost, 0.90 for DT, 0.90 for SVM, and 0.85 for LR. The RF algorithm also had the highest concordance-index (0.86) and the lowest Brier score (0.076). The variable that contributed the most in the RF algorithm was age, followed by cholesterol, and surgery time.CONCLUSIONMachine learning algorithms are highly effective in discriminating patients at high risk of developing AKI. The successful application of machine learning models may help guide clinical decisions and help improve the long-term prognosis of patients.

High-frequency ultrasound for differentiation between high-risk basal cell carcinoma and cutaneous squamous cell carcinoma.

The similar visual appearance of high-risk basal cell carcinoma (BCC) andcutaneous squamous cell carcinoma (cSCC) may cause confusion for diagnosis. High-frequency ultrasound (HFUS) may provide additional intralesional information and thus help to distinguish them. Inthis retrospective study, we analyzed the clinical characteristics, HFUS grayscale, and color Doppler flow imaging (CDFI) features of pathologically confirmed high-risk BCC andcSCC lesions (n=65vs n=68). Subsequently, discrimination models based on the significant HFUS features were established. Between high-risk BCC and cSCC lesions,the HFUS grayscalefeatures of the lesion size(10.0mm vs 17.4mm),thickness(3.1mm vs 5.9mm), internal hyperechoic spots(80.0%vs 23.5%), and posterior acoustic shadowing (16.9%vs 66.2%)were statistically different(all p<0.001).As for the CDFI features,high-risk BCC lesions mainly appeared as pattern II (47.7%), while cSCC lesions mainly appeared as pattern III (66.2%). Based on the above five features, an optimal discrimination modelwasestablished with a sensitivity of 91.2%, a specificity of 87.7%, and an accuracy of 89.5%. HFUS features, including size, thickness, internal hyperechoic spots, posterior acoustic shadowing, and Doppler vascularity pattern, are useful for differential diagnosis between high-risk BCC and cSCC.

Driver Speed Profiles Index Warning Signs of MCI

GoalUse driver behavior profiles to screen and index early warnings of cognitive decline and Alzheimer’s disease (AD). Hypothesis: Real-world driver speed behavior profiles discriminate mild cognitive impairment (MCI).MethodsSensors were installed in personal vehicles of 74 legally-licensed, active drivers (age: 65-90 years, μ = 75.85) who completed 2, 3-month real-world driving assessments, including demographic and cognitive assessments, 1 year apart (244,564 miles driven). MCI status was indexed using 8 neuropsychological tests (spanning executive function, visuospatial skills, processing speed, and memory), relevant to MCI and driving. Driving environment was indexed from state speed limit (SL; roadway type: residential, commercial, interstate) and sunrise-sunset databases (time of day: day vs. night). Models: Data were randomly split into training (66%) and validation (33%) sets. An optimal mixed effects logistic regression model was determined from validation data AUC values.ResultsMCI drivers drove slower with optimal discrimination (estimated for every 5 mph decrease in speeding) in 1) residential roads (SL 25-35 mph; MCI odds increased by 6% [95% CI: 2-11%]), 2) interstate roads (SL >55 mph; MCI odds increased by 14% [95% CI: 8-20%]), and 3) night environments (MCI odds increased by 7% [95% CI: 2-12%]).ConclusionQuantitative indices of real-world driver data provide “ground truth” for screening and indexing phenotypes of cognitive decline, in line with ongoing efforts to link driver behavior with age-related cognitive decline and AD biomarkers. Behavioral biomarkers for diagnosing early warnings of dementia could ultimately bolster our ability to detect and intervene in early AD.

Differentiating atypical meningioma from anaplastic meningioma using diffusion weighted imaging

PurposeTo explore the value of MRI conventional features and apparent diffusion coefficient (ADC) on the differential diagnosis of atypical meningioma (AtM) and anaplastic meningioma (AnM). Materials and methodsThis retrospective study analyzed the preoperative clinical data, MRI conventional features, and DWI data of 55 AtM and 25 AnM confirmed by pathology in our hospital. The clinical features, MRI conventional features, ADCmean, ADCmin, and relative ADC (rADC) values were compared between the two tumors by Chi-square test or an independent sample t-test. Receiver operating characteristic curve (ROC) and binary logistic regression analysis were used to evaluate the diagnostic efficacy of each parameter to differentiate between these tumors. ResultsThe MRI conventional features had a certain ability to distinguish AnM and AtM, with an area under the curve value (AUC) of 0.824 (95% CI, 0.723–0.900). The ADCmean, ADCmin, and rADC values were significantly higher in AtM compared to AnM (all P < 0.05). ADCmean had the best identification effect with an AUC of 0.867 (95% CI, 0.772–0.933) among them, at an cut-off of 0.817 × 10−3 mm2/s, the sensitivity and specificity of distinguishing AtM from AnM were 78.18% and 88.00%, respectively. A combination of ADCmean and MRI conventional features showed the optimum discrimination ability for the two tumors, the AUC, sensitivity, specificity, and accuracy were 0.918 (95% CI, 0.835–0.967), 80.00%, 94.55%, and 90.00%, respectively. ConclusionMRI conventional features combined with ADCmean, as a non-invasive method, has potential clinical value in the preoperative diagnosis of AtM and AnM.

Prognostic Stratification Based on HIF-1 Signaling for Evaluating Hypoxic Status and Immune Infiltration in Pancreatic Ductal Adenocarcinomas.

Pancreatic ductal adenocarcinoma (PDAC) has a hypoxic and desmoplastic tumor microenvironment (TME), leading to treatment failure. We aimed to develop a prognostic classifier to evaluate hypoxia status and hypoxia-related molecular characteristics of PDAC. In this study, we classified PDAC into three clusters based on 16 known hypoxia-inducible factor 1 (HIF-1)-related genes. Nine differentially expressed genes were identified to construct an HIF-1 score system, whose predictive efficacy was evaluated. Furthermore, we investigated oncogenic pathways and immune-cell infiltration status of PDAC with different scores. The C-index of the HIF-1score system for OS prediction in the meta-PDAC cohort and the other two validation cohorts were 0.67, 0.63, and 0.65, respectively, indicating that it had a good predictive value for patient survival. Furthermore, the area under the curve (AUC) of the receiver operating characteristic (ROC) curve of the HIF-1α score system for predicting 1-, 3-, and 4-year OS indicated the HIF-1α score system had an optimal discrimination of prognostic prediction for PDAC. Importantly, our model showed superior predictive ability compared to previous hypoxia signatures. We also classified PDAC into HIF-1 scores of low, medium, and high groups. Then, we found high enrichment of glycolysis, mTORC1 signaling, and MYC signaling in the HIF-1 score high group, whereas the cGMP metabolic process was activated in the low score group. Of note, analysis of public datasets and our own dataset showed a high HIF-1 score was associated with high immunosuppressive TME, evidenced by fewer infiltrated CD8+ T cells, B cells, and type 1 T-helper cells and reduced cytolytic activity of CD8+ T cells. In summary, we established a specific HIF-1 score system to discriminate PDAC with various hypoxia statuses and immune microenvironments. For highly hypoxic and immunosuppressive tumors, a combination treatment strategy should be considered in the future.

A Clinical Predictive Nomogram for Traumatic Brain Parenchyma Hematoma Progression.

IntroductionAcute traumatic intraparenchymal hematoma (tICH) expansion is a major cause of clinical deterioration after brain contusion. Here, an accurate prediction tool for acute tICH expansion is proposed.MethodsA multicenter hospital-based study for multivariable prediction model was conducted among patients (889 patients in a development dataset and 264 individuals in an external validation dataset) with initial and follow-up computed tomography (CT) imaging for tICH volume evaluation. Semi-automated software was employed to assess tICH expansion. Two multivariate predictive models for acute tICH expansion were developed and externally validated.ResultsA total of 198 (22.27%) individuals had remarkable acute tICH expansion. The novel Traumatic Parenchymatous Hematoma Expansion Aid (TPHEA) model retained several variables, including age, coagulopathy, baseline tICH volume, time to baseline CT time, subdural hemorrhage, a novel imaging marker of multihematoma fuzzy sign, and an inflammatory index of monocyte-to-lymphocyte ratio. Compared with multihematoma fuzzy sign, monocyte-to-lymphocyte ratio, and the basic model, the TPHEA model exhibited optimal discrimination, calibration, and clinical net benefits for patients with acute tICH expansion. A TPHEA nomogram was subsequently introduced from this model to facilitate clinical application. In an external dataset, this device showed good predicting performance for acute tICH expansion.ConclusionsThe main predictive factors in the TPHEA nomogram are the monocyte-to-lymphocyte ratio, baseline tICH volume, and multihematoma fuzzy sign. This user-friendly tool can estimate acute tICH expansion and optimize personalized treatments for individuals with brain contusion.Supplementary InformationThe online version contains supplementary material available at 10.1007/s40120-021-00306-8.

The need for nutritional assessment and interventions based on the prognostic nutritional index for patients with femoral fractures: a retrospective study

BackgroundThe incidence of adverse perioperative outcomes in surgery for femoral fractures is high and associated with malnutrition. Here, we identified independent factors and assessed the predictive value of the prognostic nutritional index (PNI) for perioperative adverse outcomes in patients with femoral fractures.MethodsThis retrospective study included 343 patients who underwent surgery for a single femur fracture. Demographic characteristics, surgery and anaesthesia records and blood test results at admission, 1 day postoperatively and before discharge were evaluated using logistic regression analysis. The discriminatory ability of the independent factors was assessed using the receiver operating characteristic curve analysis, and DeLong’s test was used to compare the area under the curve (AUC).ResultsOverall, 159 patients (46.4%) experienced adverse perioperative outcomes. Amongst these, 123 (35.9%) had lower limb vein thrombus, 68 (19.8%) had hospital-acquired pneumonia, 6 (1.7%) were transferred to the postoperative intensive care unit, 4 (1.2%) had pulmonary embolism, 3 (0.9%) died during hospitalisation and 9 (2.6%) had other adverse outcomes, including incision disunion, renal and liver function impairment, acute heart failure, acute cerebral infarction and stress gastroenteritis. The PNI at admission, age, postoperative hospital stay, time to admission, hypertension, combined injures and surgery type were independent factors for adverse perioperative outcomes. Based on the AUC (PNI at admission: 0.772 [0.723–0.821], P < 0.001; age: 0.678 [0.622–0.734], P < 0.001; postoperative hospital stay: 0.608 [0.548–0.668], P = 0.001; time to admission: 0.585 [0.525–0.646], P = 0.006), the PNI at admission had optimal discrimination ability, indicating its superiority over other independent factors (age vs. PNI at admission, P = 0.002; postoperative hospital stay vs. PNI at admission, P < 0.001; time to admission vs. PNI at admission, P < 0.001).ConclusionsPatients with femoral fractures require a nutritional assessment and appropriate nutritional intervention at admission, and that the PNI value at admission may be a good nutritional assessment indicator.

Third-degree price discrimination versus uniform pricing

We compare the profit of the optimal third-degree price discrimination policy against a uniform pricing policy. A uniform pricing policy offers the same price to all segments of the market. Our main result establishes that for a broad class of third-degree price discrimination problems with concave profit functions (in the price space) and common support, a uniform price is guaranteed to achieve one half of the optimal monopoly profits. This profit bound holds for any number of segments and prices that the seller might use under third-degree price discrimination. We establish that these conditions are tight and that weakening either common support or concavity can lead to arbitrarily poor profit comparisons even for regular or monotone hazard rate distributions.