Likelihood-free hypothesis testing

Consider the problem of testing $Z \sim \mathbb P^{\otimes m}$ vs $Z \sim \mathbb Q^{\otimes m}$ from $m$ samples. Generally, to achieve a small error rate it is necessary and sufficient to have $m \asymp 1/\epsilon^2$, where $\epsilon$ measures the separation between $\mathbb P$ and $\mathbb Q$ in total variation ($\mathsf{TV}$). Achieving this, however, requires complete knowledge of the distributions $\mathbb P$ and $\mathbb Q$ and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem, which we call likelihood-free (or simulation-based) hypothesis testing, where access to $\mathbb P$ and $\mathbb Q$ (which are a priori only known to belong to a large non-parametric family $\mathcal P$) is given through $n$ iid samples from each. We demostrate existence of a fundamental trade-off between $n$ and $m$ given by $nm \asymp n^2_\mathsf{GoF}(\epsilon,\mathcal P)$, where $n_\mathsf{GoF}$ is the minimax sample complexity of testing between the hypotheses $H_0: \mathbb P= \mathbb Q$ vs $H_1: \mathsf{TV}(\mathbb P,\mathbb Q) \ge \epsilon$. We show this for three non-parametric families $\cal P$: $\beta$-smooth densities over $[0,1]^d$, the Gaussian sequence model over a Sobolev ellipsoid, and the collection of distributions $\mathcal P$ on a large alphabet $[k]$ with pmfs bounded by $c/k$ for fixed $c$. The test that we propose (based on the $L^2$-distance statistic of Ingster) simultaneously achieves all points on the tradeoff curve for these families. In particular, when $m\gg 1/\epsilon^2$ our test requires the number of simulation samples $n$ to be orders of magnitude smaller than what is needed for density estimation with accuracy $\asymp \epsilon$ (under $\mathsf{TV}$). This demonstrates the possibility of testing without fully estimating the distributions.Comment: 48 pages, 1 figur

Kernel-Based Tests for Likelihood-Free Hypothesis Testing

Given $n$ observations from two balanced classes, consider the task of labeling an additional $m$ inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with complete knowledge of class distributions ($n=\infty$) the problem is solved optimally by the likelihood-ratio test; when $m=1$ it corresponds to binary classification; and when $m\approx n$ it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-off between $m$ and $n$: increasing the data sample $m$ reduces the amount $n$ of training/simulation data needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detection of the Higgs boson and detection of planted DDPM generated images amidst CIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric $m$ vs $n$ trade-off.Comment: 36 pages, 6 figure

Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP solvers. This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems.Comment: 48 pages, 8 figure

Early Intubation in Endovascular Therapy for Basilar Artery Occlusion:A Post Hoc Analysis of the BASICS Trial

BACKGROUND: The optimal anesthetic management for endovascular therapy (EVT) in patients with posterior circulation stroke remains unclear. Our objective was to investigate the impact of early intubation in patients enrolled in the BASICS trial (Basilar Artery International Cooperation Study). METHODS: BASICS was a multicenter, randomized, controlled trial that compared the efficacy of EVT compared with the best medical care alone in patients with basilar artery occlusion. In this post hoc analysis, early intubation within the first 24 hours of the estimated time of basilar artery occlusion was examined as an additional covariate using regression modeling. We estimated the adjusted relative risks (RRs) for favorable outcomes, defined as modified Rankin Scale scores of 0 to 3 at 90 days. An adjusted common odds ratio was estimated for a shift in the distribution of modified Rankin Scale scores at 90 days. RESULTS: Of 300 patients in BASICS, 289 patients were eligible for analysis (151 in the EVT group and 138 in the best medical care group). compared with medical care alone, EVT was related to a higher risk of early intubation (RR, 1.29 [95% CI, 1.09–1.53]; P&lt;0.01), and early intubation was negatively associated with favorable outcome (RR, 0.61 [95% CI, 0.45–0.84]; P=0.002). Whereas there was no overall treatment effect of EVT on favorable outcome (RR, 1.22 [95% CI, 0.95–1.55]; P=0.121), EVT was associated with favorable outcome (RR, 1.34 [95% CI, 1.05–1.71]; P=0.018) and a shift toward lower modified Rankin Scale scores (adjusted common odds ratio, 1.63 [95% CI, 1.04–2.57]; P=0.033) if adjusted for early intubation. CONCLUSIONS: In this post hoc analysis of the neutral BASICS trial, early intubation was linked to unfavorable outcomes, which might mitigate a potential benefit from EVT by indirect effects due to an increased risk of early intubation. This relationship may be considered when assessing the efficacy of EVT in patients with basilar artery occlusion in future trials.</p

Sex-related differences in risk factors, type of treatment received and outcomes in patients with atrial fibrillation and acute stroke: Results from the RAF-study (Early Recurrence and Cerebral Bleeding in Patients with Acute Ischemic Stroke and Atrial Fibrillation)

Introduction: Atrial fibrillation is an independent risk factor of thromboembolism. Women with atrial fibrillation are at a higher overall risk for stroke compared to men with atrial fibrillation. The aim of this study was to evaluate for sex differences in patients with acute stroke and atrial fibrillation, regarding risk factors, treatments received and outcomes. Methods Data were analyzed from the “Recurrence and Cerebral Bleeding in Patients with Acute Ischemic Stroke and Atrial Fibrillation” (RAF-study), a prospective, multicenter, international study including only patients with acute stroke and atrial fibrillation. Patients were followed up for 90 days. Disability was measured by the modified Rankin Scale (0–2 favorable outcome, 3–6 unfavorable outcome). Results: Of the 1029 patients enrolled, 561 were women (54.5%) (p &lt; 0.001) and younger (p &lt; 0.001) compared to men. In patients with known atrial fibrillation, women were less likely to receive oral anticoagulants before index stroke (p = 0.026) and were less likely to receive anticoagulants after stroke (71.3% versus 78.4%, p = 0.01). There was no observed sex difference regarding the time of starting anticoagulant therapy between the two groups (6.4 ± 11.7 days for men versus 6.5 ± 12.4 days for women, p = 0.902). Men presented with more severe strokes at onset (mean NIHSS 9.2 ± 6.9 versus 8.1 ± 7.5, p &lt; 0.001). Within 90 days, 46 (8.2%) recurrent ischemic events (stroke/TIA/systemic embolism) and 19 (3.4%) symptomatic cerebral bleedings were found in women compared to 30 (6.4%) and 18 (3.8%) in men (p = 0.28 and p = 0.74). At 90 days, 57.7% of women were disabled or deceased, compared to 41.1% of the men (p &lt; 0.001). Multivariate analysis did not confirm this significance. Conclusions: Women with atrial fibrillation were less likely to receive oral anticoagulants prior to and after stroke compared to men with atrial fibrillation, and when stroke occurred, regardless of the fact that in our study women were younger and with less severe stroke, outcomes did not differ between the sexes

Prediction of early recurrent thromboembolic event and major bleeding in patients with acute stroke and atrial fibrillation by a risk stratification schema: the ALESSA score study

Background and Purposes—This study was designed to derive and validate a score to predict early ischemic events and major bleedings after an acute ischemic stroke in patients with atrial fibrillation. Methods—The derivation cohort consisted of 854 patients with acute ischemic stroke and atrial fibrillation included in prospective series between January 2012 and March 2014. Older age (hazard ratio 1.06 for each additional year; 95% confidence interval, 1.00–1.11) and severe atrial enlargement (hazard ratio, 2.05; 95% confidence interval, 1.08–2.87) were predictors for ischemic outcome events (stroke, transient ischemic attack, and systemic embolism) at 90 days from acute stroke. Small lesions (≤1.5 cm) were inversely correlated with both major bleeding (hazard ratio, 0.39; P=0.03) and ischemic outcome events (hazard ratio, 0.55; 95% confidence interval, 0.30–1.00). We assigned to age ≥80 years 2 points and between 70 and 79 years 1 point; ischemic index lesion &#62;1.5 cm, 1 point; severe atrial enlargement, 1 point (ALESSA score). A logistic regression with the receiver-operating characteristic graph procedure (C statistic) showed an area under the curve of 0.697 (0.632–0.763; P=0.0001) for ischemic outcome events and 0.585 (0.493–0.678; P=0.10) for major bleedings. Results—The validation cohort consisted of 994 patients included in prospective series between April 2014 and June 2016. Logistic regression with the receiver-operating characteristic graph procedure showed an area under the curve of 0.646 (0.529–0.763; P=0.009) for ischemic outcome events and 0.407 (0.275–0.540; P=0.14) for hemorrhagic outcome events. Conclusions—In acute stroke patients with atrial fibrillation, high ALESSA scores were associated with a high risk of ischemic events but not of major bleedings

Borreliosis-associated orofacial pain: A case report

Lyme borreliosis is a prevalent disease with a wide variety of clinical manifestations. However, only a few pertinent references are found in the dental literature. Considering the diversity of differential diagnoses, borreliosis-associated orofacial pain may be a challenge for treatment providers and patients alike. The aim of this case report is to emphasize the importance of a sound, structured medical history and assessment to identify severe cases early and to start interdisciplinary therapy if indicated