Probabilistic Version Research Articles

Entanglement and coherence in the Bernstein-Vazirani algorithm

Quantum algorithms can outperform their classical counterparts in various tasks, the most prominent example being Shor's algorithm for efficient prime factorization on a quantum computer. It is clear that one of the reasons for the speedup is the superposition principle of quantum mechanics, which allows a quantum processor to be in a superposition of different states at the same time. While such a superposition can lead to entanglement across different qubits of the processors, there also exist quantum algorithms that outperform classical ones using superpositions of individual qubits without entangling them. As an example, the Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. While the classical version of the algorithm requires multiple calls of the oracle to learn the bit string, a single query of the oracle is enough in the quantum case. In this article, we analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. For this, we introduce and study its probabilistic version, where the goal is to guess the bit string after a single call of the oracle. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state. We further demonstrate that a large amount of entanglement in the initial state prevents the algorithm from achieving optimal performance. We also apply our methods to quantum computation with mixed states, proving that pseudopure states achieve optimal performance for a given purity in the Bernstein-Vazirani algorithm. We further investigate quantum resources in the one clean qubit model, showing that the model can exhibit speedup over any known classical algorithm even with an arbitrarily little amount of multipartite entanglement, general quantum correlations, and coherence.

Corrigendum: Visualizing uncertainty for non-expert end users: The challenge of the deterministic construal error

Users: The Challenge of the Deterministic Construal Error.Front. Comput. Sci. 2:590232. doi: 10.3389/fcomp.2020.590232 In the published article, there was an error in Figure 1b and Figure 1f as published. The definition in Figure 1b read "There is an 80% chance that the daytime high will be equal to or greater than 44F and equal to or less than or equal to 38F." The definition was partly in error and should have read, "80% chance that the daytime high will be between 44F and 38F." The nighttime low temperature bar was colored yellow. It should have been blue.The definition in Figure 1f was a frequency version reading, "On 8 out of 10 days like this, the observed temperature will fall between these two values". It should have been a probabilistic version, "80% chance that the daytime high will be between 44F and 38F" to align to the other probabilistic versions used in this figure.In addition, we added a Note to the caption to inform readers that small changes in font and figure sizes were made to the versions published here to conform to journal requirements.The corrected Figure 1 and its caption [Figure 1. Predictive intervals, each accompanied by a key (shown in a) describing "41°F" as the "best forecast." Note: Font and figure sizes are slightly smaller than they were in the original experiments to conform to journal requirements] appear below.

OP29 Lifecycle Assessment Of Machine Learning-Derived Early Warning System. An Early Economic Evaluation Of An Intraoperative Hypotension Prediction Index

IntroductionAn iterative, life-cycle approach to the evaluation of healthcare technologies requires that clinical and economic evidence is collected since the initial stages of diffusion. Nevertheless, early cost-effectiveness models are challenging mainly due to the difficulties in estimating model parameters and faithfully characterizing parameter uncertainty. This is especially true with AI-based diagnostics, where attribution of effects on costs and patient-relevant outcomes is more challenging. Empirical applications of early-models are useful to identify the main challenges of iterative modelling and provide recommendations on best-practices. Here, we reported on a case study on a machine learning-derived hypotension predictive index (HPI), that predicts the onset of intraoperative hypotension and trigger corrective measures.MethodsA hybrid decision-tree/Markov model was developed comparing an HPI-based intervention protocol to standard-of-care intervention protocol during gynecological procedures. A short-term component of the model was populated using data from individual patients collected at one hospital in Italy. An historical control group was also defined using propensity score matching. Long-term costs and consequences of HPI were modelled using secondary data. A probabilistic version of the headroom approach was used to determine the maximum achievable price of HPI based on available evidence. Value of Information analysis was also conducted to identify the parameters that contribute the most to the overall uncertainty, and to identify optimal future study designs. Extensive deterministic and probabilistic sensitivity analyses were conducted to characterize the uncertainty over the cost-effectiveness of HPI.ResultsThe preliminary results of the analysis suggest that HPI has potential to improve patients’ outcomes and generate efficiency gains by reducing hypotension events and permanent complications, such as acute kidney injury. The link between reduction in hypotension and the rate of complications, or the long-term effects on healthcare costs and patients’ quality of life are the parameters that contribute the most to model uncertainty.ConclusionsEarly cost-effectiveness models are a valuable tool to inform further product development and evidence requirements, but characterization of uncertainty and transparency in modelling assumptions are key.

Radiative transfer as a Bayesian linear regression problem

ABSTRACT Electromagnetic radiation plays a crucial role in various physical and chemical processes. Hence, almost all astrophysical simulations require some form of radiative transfer model. Despite many innovations in radiative transfer algorithms and their implementation, realistic radiative transfer models remain very computationally expensive, such that one often has to resort to approximate descriptions. The complexity of these models makes it difficult to assess the validity of any approximation and to quantify uncertainties on the model results. This impedes scientific rigour, in particular, when comparing models to observations, or when using their results as input for other models. We present a probabilistic numerical approach to address these issues by treating radiative transfer as a Bayesian linear regression problem. This allows us to model uncertainties on the input and output of the model with the variances of the associated probability distributions. Furthermore, this approach naturally allows us to create reduced-order radiative transfer models with a quantifiable accuracy. These are approximate solutions to exact radiative transfer models, in contrast to the exact solutions to approximate models that are often used. As a first demonstration, we derive a probabilistic version of the method of characteristics, a commonly-used technique to solve radiative transfer problems.

A new stochastic method based on the Taylor expansion to compute response probability densities of uncertain systems

AbstractIn this article, a new stochastic approach is proposed for the analysis of structural systems with uncertainties. This novel strategy method combines the probability transformation method, expressed in terms of characteristic function with a probabilistic version of the Taylor expansion. Some notes about the convergence and the accuracy of the proposed method are reported. The results of these notes and several examples here reported evidence an optimum level of accuracy, coupled with great simplicity of application of this approach, even for relatively high levels of uncertainties. Hence, this method can be considered a valuable tool for the solution of stochastic analyses of engineering systems. Several examples are performed to show the capability of the results established. These applications show how the method is a useful technique for engineering analysis.

How do Chinese-English Bilinguals and Tibetan-Chinese-English Trilinguals Differ in Explicit and Implicit Aptitude?

This study seeks to empirically explore the relation between multilingual learning experiences and language aptitude. Through employing LLAMA aptitude test battery (Meara, 2005) and a probabilistic version of the serial reaction time (SRT) task (Kaufman et al., 2010), scores from 24 Chinese-English bilinguals and 24 Tibetan-Chinese-English trilinguals were analyzed with One-way Analyses of Variance (ANOVA). LLAMA-B, E, and F sub-tests measured explicit language aptitude, while LLAMA-D sub-test and SRT task measured implicit language aptitude, a cutting-edge that has gained sway in aptitude research. Qualitative and quantitative results showed that trilingual group performed better on the LLAMA-E sub-test than bilingual group, whereas bilingual group outperformed trilingual group on the SRT task. These findings suggested that trilinguals might possess higher explicit aptitude but lower implicit aptitude than bilinguals. Thus, prior language learning experiences might be positively (for explicit aptitude) or negatively (for implicit aptitude) correlated with language aptitude. Additionally, explicit aptitude and implicit aptitude might have a competitive relationship. Possible implications were discussed in this article.

A Bayesian Deep Learning Approach to Near‐Term Climate Prediction

AbstractSince model bias and associated initialization shock are serious shortcomings that reduce prediction skills in state‐of‐the‐art decadal climate prediction efforts, we pursue a complementary machine‐learning‐based approach to climate prediction. The example problem setting we consider consists of predicting natural variability of the North Atlantic sea surface temperature on the interannual timescale in the pre‐industrial control simulation of the Community Earth System Model. While previous works have considered the use of recurrent networks such as convolutional LSTMs and reservoir computing networks in this and other similar problem settings, we currently focus on the use of feedforward convolutional networks. In particular, we find that a feedforward convolutional network with a Densenet architecture is able to outperform a convolutional LSTM in terms of predictive skill. Next, we go on to consider a probabilistic formulation of the same network based on Stein variational gradient descent and find that in addition to providing useful measures of predictive uncertainty, the probabilistic (Bayesian) version improves on its deterministic counterpart in terms of predictive skill. Finally, we characterize the reliability of the ensemble of machine learning models obtained in the probabilistic setting by using analysis tools developed in the context of ensemble numerical weather prediction.

The mesoscopic geometry of sparse random maps

We investigate the structure of large uniform random maps with $n$ edges, $\mathrm{f}_n$ faces, and with genus $\mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: the planar case $(\mathrm{f}_n, 2\mathrm{g}_n) = (\mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(\mathrm{f}_n, 2 \mathrm{g}_n) = (1, \mathrm{s}_n-1)$, both when $1 \ll \mathrm{s}_n \ll n$. Albeit different at first sight, these two models can be treated in a unified way using a probabilistic version of the classical core-kernel decomposition. In particular, we show that the number of edges of the core of such maps, obtained by iteratively removing degree $1$ vertices, is concentrated around $\sqrt{n \mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the vertices of the core with degree $2$, is such that the sum of the degree of its vertices exceeds that of a trivalent map by a term of order $\sqrt{\mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high probability when $\mathrm{s}_{n} \ll n^{1/3}$. This enables us to identify a mesoscopic scale $\sqrt{n/\mathrm{s}_n}$ at which the scaling limits of these random maps can be seen as the local limit of their kernels, which is the dual of the UIPT in the planar case and the infinite three-regular tree in the unicellular case, where each edge is replaced by an independent (biased) Brownian tree with two marked points.

Dependency-length minimization and its limits: A possible role for a probabilistic version of the final-over-final condition

Dependency-length minimization and its limits: A possible role for a probabilistic version of the final-over-final condition

Dependency-length minimization and its limits: A possible role for a probabilistic version of the final-over-final condition: Supplementary Material

Dependency-length minimization and its limits: A possible role for a probabilistic version of the final-over-final condition: Supplementary Material

Probabilistic deep learning for real-time large deformation simulations

For many novel applications, such as patient-specific computer-aided surgery, conventional solution techniques of the underlying nonlinear problems are usually computationally too expensive and are lacking information about how certain can we be about their predictions. In the present work, we propose a highly efficient deep-learning surrogate framework that is able to accurately predict the response of bodies undergoing large deformations in real-time. The surrogate model has a convolutional neural network architecture, called U-Net, which is trained with force–displacement data obtained with the finite element method. We propose deterministic and probabilistic versions of the framework. The probabilistic framework utilizes the Variational Bayes Inference approach and is able to capture all the uncertainties present in the data as well as in the deep-learning model. Based on several benchmark examples, we show the predictive capabilities of the framework and discuss its possible limitations.

A rare scare: The role of intolerance of uncertainty in startle responses and event-related potentials in anticipation of unpredictable threat

Intolerance of uncertainty (IU) represents a transdiagnostic risk factor for internalizing psychopathology. However, little is known regarding its psychophysiological correlates. IU is thought to render individuals hypersensitive to threatening events, even if the occurrence probability is low. To test this, we recruited 90 students who completed two NPU-threat tests separating temporal unpredictability from probabilistic unpredictability (i.e., probability of occurrence): First, a NPU version in which the timing of threat (i.e., shock) was either predictable or unpredictable but the shock probability was 100 % in each trial. Second, a probabilistic NPU version in which the timing of the shock was also either predictable or unpredictable, but the shock probability was 33 % in each trial. Startle reflex, event-related potentials N1 and P3 locked to auditory startle probes, and anxiety ratings were analyzed. As expected, temporally unpredictable threat modulated startle, N1, and anxiety ratings. IU predicted increased startle to lower shock probability, regardless of the temporal predictability. No evidence emerged for IU affecting N1 or P3. However, IU predicted anxiety ratings to threat conditions regardless of temporal or probabilistic predictability. Individual differences in IU shape startle responses to eventual threat: IU increases automatic defensive responding to unpredictable and potentially threatening situations if the threat probability is low. Individuals with higher IU may tend to “err on the side of caution” with a mismatch between physiological preparation and threat probability, such that the effects of IU are particularly observable in low threat probability conditions.

Cost-Risk Analysis Reconsidered—Value of Information on the Climate Sensitivity in the Integrated Assessment Model PRICE

Infeasible solutions or negative expected values of future climate information are undesired problems if climate policies are adopted under Cost-Effectiveness Analysis (CEA) to reach uncertain temperature targets. Cost-Risk Analysis (CRA) was developed to resolve these issues. It allows for a trade-off between expected welfare losses of mitigation and avoided risk of transgressing a climate target with a certain probability of compliance (Safety). Some of the significant contributions of this paper are: (i) It updates the Probabilistic Integrated model of Climate and the Economy (PRICE) as a probabilistic version of the latest version of the Dynamic Integrated Climate-Economy model (DICE) 2016, and it extends the model to run welfare-maximizing decision analytic frameworks readily. (ii) It highlights that the standard method of applying CRA (Old CRA) leads to an extra welfare cost. (iii) It proposes revised instruction on how to use CRA. (iv) It simulates and compares welfare-maximizing decision analytic frameworks on the level of risk, damages, and carbon prices. (v) It measures the value of information using risk-based methods and compares them with the value of information calculated using the damage-based method. (vi) It measures the carbon prices for the CRA scenarios for the first time. The results show that the choice of the disutility function governs the magnitude of the value of information. Using a damage function or Old CRA, the value of information is significantly high for new information arriving between 2020 and 2060. If the New CRA is applied, however, such benefits are negligible.

Large deviations analysis for random combinatorial partitions with counter terms

In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. The large deviation rate functions are normalised limiting free energies and the main focus is to study their minimiser for various Gibbsian ensembles with respect to the reference measure which is a probabilistic version of the ideal Bose gas. Scaling limits of similar models have been studied recently (Fatkullin and Slastikov 2018 arXiv:1801.00812v2; Fatkullin and Xue 2021 J. Stat. Phys. 183 22) going back to (Vershik 1996 Func. Anal. Appl. 30 90–105). After studying the reference model, we provide a complete analysis of two mean field models, one of which is well-know (Benfatto et al 2005 J. Math. Phys. 46 033303) and the other one is the cycle mean field model. Both models show critical behaviour despite their rate functions having unique minimiser. The main focus is then a model with negative counter term, the probabilistic version of the so-called Huang–Yang–Luttinger model (van den Berg et al 1988 Commun. Math. Phys. 118 61–85). Criticality in this model is the existence of a critical parameter for which two simultaneous minimiser exists. At criticality an order parameter is introduced as the double limits for the density of cycles with diverging length, and as such it extends recent work in (Adams and Dickson 2021 Ann. Henri Poincaré 22 1535–60).

An uncertainty-quantification framework for assessing accuracy, sensitivity, and robustness in computational fluid dynamics

Combining different existing uncertainty quantification (UQ) techniques, a framework is obtained to assess a set of metrics in computational physics problems, in general, and computational fluid dynamics (CFD), in particular. The metrics include accuracy, sensitivity and robustness of the simulator’s outputs with respect to uncertain inputs and parameters. These inputs and parameters are divided into two groups: based on the variation of the first group (e.g. numerical/computational parameters such as grid resolution), a computer experiment is designed, the data of which may become uncertain due to the parameters of the second group (e.g. finite time-averaging). To construct a surrogate model based on uncertain data, Gaussian process regression (GPR) with observation-dependent (heteroscedastic) noise is used. To estimate the propagated uncertainties in the simulator’s outputs from the first group of parameters, a probabilistic version of the polynomial chaos expansion (PCE) is employed Global sensitivity analysis is performed using probabilistic Sobol indices. To illustrate its capabilities, the framework is applied to the scale-resolving simulations of turbulent channel and lid-driven cavity flows using the open-source CFD solver Nek5000. It is shown that at wall distances where the time-averaging uncertainty is high, the quantities of interest are also more sensitive to numerical/computational parameters. In particular for high-fidelity codes such as Nek5000, a thorough assessment of the results’ accuracy and reliability is crucial. The detailed analyses and the resulting conclusions can enhance our insight into the influence of different factors on physics simulations, in particular the simulations of high-Reynolds-number turbulent flows including wall turbulence.

Instance-based learning using the half-space proximal graph

• We propose a parameter-free instance-based classifier , the HSP classifier. • Our accuracy outperforms kNN while keeping its simplicity and complexity. • We also present a probabilistic and asymptotic versions of the HSP improving both the time and the original proposal’s accuracy. • The same weigthing methods that improves kNN, also improves the HSP and its asymptotic variants The primary example of instance-based learning is the k -nearest neighbor rule (kNN), praised for its simplicity and the capacity to adapt to new unseen data and toss away old data. The main disadvantages often mentioned are the classification complexity , which is O ( n ) , and the estimation of the parameter k , the number of nearest neighbors to be used. The use of indexes at classification time lifts the former disadvantage, while there is no conclusive method for the latter. This paper presents a parameter-free instance-based learning algorithm using the Half-Space Proximal (HSP) graph. The HSP neighbors simultaneously possess proximity and variety concerning the center node. To classify a given query, we compute its HSP neighbors and apply a simple majority rule over them. In our experiments, the resulting classifier bettered K N N for any k in a battery of datasets. This improvement sticks even when applying weighted majority rules to both kNN and HSP classifiers. Surprisingly, when using a probabilistic index to approximate the HSP graph and consequently speeding-up the classification task , our method could improve its accuracy in stark contrast with the kNN classifier, which worsens with a probabilistic index.

Comparing MaxEnt and Noisy Harmonic Grammar

MaxEnt grammar is a probabilistic version of Harmonic Grammar in which the harmony scores of candidates are mapped onto probabilities. It has become the tool of choice for analyzing phonological phenomena involving probabilistic variation or gradient acceptability, but there is a competing proposal for making Harmonic Grammar probabilistic, Noisy Harmonic Grammar, in which variation is derived by adding random ‘noise’ to constraint weights. In this paper these grammar frameworks, and variants of them, are analyzed by reformulating them all in a format where noise is added to candidate harmonies, and the differences between frameworks lie in the distribution of this noise. This analysis reveals a basic difference between the models: in MaxEnt the relative probabilities of two candidates depend only on the difference in their harmony scores, whereas in Noisy Harmonic Grammar it also depends on the differences in the constraint violations incurred by the two candidates. This difference leads to testable predictions which are evaluated against data on variable realization of schwa in French (Smith & Pater 2020). The results support MaxEnt over Noisy Harmonic Grammar.

Macroscale behavior of random lower triangular matrices

We analyze the macroscale behavior of random lower (and therefore upper) triangular matrices with entries drawn i.i.d. from a distribution with nonzero mean and finite variance. We show that such a matrix behaves like a probabilistic version of a Riemann sum and therefore in the limit behaves like the Volterra operator. Specifically, we analyze certain SOT-like and WOT-like modes of convergence for random lower triangular matrices to a scaled Volterra operator. We close with a brief discussion of moments.

Robust Bilinear Probabilistic Principal Component Analysis

Principal component analysis (PCA) is one of the most popular tools in multivariate exploratory data analysis. Its probabilistic version (PPCA) based on the maximum likelihood procedure provides a probabilistic manner to implement dimension reduction. Recently, the bilinear PPCA (BPPCA) model, which assumes that the noise terms follow matrix variate Gaussian distributions, has been introduced to directly deal with two-dimensional (2-D) data for preserving the matrix structure of 2-D data, such as images, and avoiding the curse of dimensionality. However, Gaussian distributions are not always available in real-life applications which may contain outliers within data sets. In order to make BPPCA robust for outliers, in this paper, we propose a robust BPPCA model under the assumption of matrix variate t distributions for the noise terms. The alternating expectation conditional maximization (AECM) algorithm is used to estimate the model parameters. Numerical examples on several synthetic and publicly available data sets are presented to demonstrate the superiority of our proposed model in feature extraction, classification and outlier detection.

On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

<p>A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ) type non-expansive mappings is established.</p>