Numerical Probability Research Articles

Interpretable machine learning models for predicting probabilistic axial buckling strength of steel circular hollow section members considering discreteness of geometries and material

Steel circular hollow section (CHS) members are widely utilized as axial force-resisting structural members in civil engineering structures. The buckling strength under axial loads is one of the critical parameters to determine the performance of the steel CHS members, which is significantly affected by the discreteness introduced by geometries, material, and initial imperfections. However, the reduction factor employed in the modern design codes (i.e. Chinese codes and EC3) only accounts for the reduction caused by all kinds of discreteness and does not reflect the impacts of every single discreteness and imperfection. To fill the gap, this paper proposed an interpretable machine-learning method to provide the probabilistic axial buckling strength of steel CHS members prediction result in a distribution form with the consideration of detailed discreteness. The model to predict the nominal axial buckling strength of steel CHS members was first developed utilizing ten machine learning algorithms after sufficient numerical simulations, where the numerical model was verified using test results. The artificial neural network (ANN) was selected for developing the prediction model due to its highly reliable performance in testing. The developed ANN models were further interpreted utilizing Shapley Additive exPlanations (SHAP) to determine the interrelationship of different parameters. Then, the probabilistic axial bucking strength prediction model was established based on the developed ANN models, where the Latin hypercube sampling method was applied to address the discreteness of geometries, material, and initial imperfections. The generated probabilistic axial bucking strength prediction model’s effectiveness was verified by the evidence that the machine learning prediction results can highly match the numerical results' probability density function and the result from codes while significantly reducing the computation time. Finally, the design parameters’ impact on the axial buckling strength’s discreteness was evaluated using the global sensitivity analysis (GSA) method. The result shows that the discreteness of design parameters substantially influences the distribution of the axial buckling strength of the steel CHS members and the proposed prediction model can provide an accurate probabilistic distribution prediction.

Finite-size effect in Kuramoto oscillators with higher-order interactions.

Finite-size systems of a Kuramoto model display intricate dynamics, especially in the presence of multi-stability where both coherent and incoherent states coexist. We investigate such a scenario in globally coupled populations of Kuramoto phase oscillators with higher-order interactions and observe that fluctuations inherent to finite-size systems drive the transition to the synchronized state before the critical point in the thermodynamic limit. Using numerical methods, we plot the first exit-time distribution of the magnitude of a complex order parameter and obtain numerical transition probabilities across various system sizes. Furthermore, we extend this study to a two-population oscillator system, and, using the velocity field of the associated order parameters, show the emergence of a new fixed point corresponding to a partially synchronized state arising due to the finite-size effect, which is absent in the thermodynamics limit.

Subjective Probability Increases Across Communication Chains: Introducing the Probability Escalation Effect.

A severity effect has previously been documented, whereby numerical translations of verbal probability expressions are higher for severe outcomes than for non-severe outcomes. Recent work has additionally shown the same effect in the opposite direction (translating numerical probabilities into words). Here, we aimed to test whether these effects lead to an escalation of subjective probabilities across a communication chain. In four ‘communication chain’ studies, participants at each communication stage either translated a verbal probability expression into a number, or a number into a verbal expression (where the probability to be translated was yoked to a previous participant). Across these four studies, we found a general Probability Escalation Effect, whereby subjective probabilities increased with subsequent communications for severe, non-severe and positive events. Having ruled out some alternative explanations, we propose that the most likely explanation is in terms of communications directing attention towards an event's occurrence. Probability estimates of focal outcomes increase across communication stages.

Severity influences categorical likelihood communications: A case study with Southeast Asian weather forecasters

Risk assessments are common in multiple domains, from finance to medicine. They require evaluating an event’s potential severity and likelihood. We investigate the possible dependence of likelihood and severity within the domain of impact-based weather forecasting (IBF), following predictions derived from considering asymmetric loss functions. In a collaboration between UK psychologists and partners from four meteorological organisations in Southeast Asia, we conducted two studies (N = 363) eliciting weather warnings from forecasters. Forecasters provided warnings denoting higher likelihoods for high severity impacts than low severity impacts, despite these impacts being described as having the same explicit numerical likelihood of occurrence. This ‘Severity effect’ is pervasive, and we find it can have a continued influence even for an updated forecast. It is additionally observed when translating warnings made on a risk matrix to numerical probabilities.

Evaluating the accuracy of a state-of-the-art large language model for prediction of admissions from the emergency room.

Artificial intelligence (AI) and large language models (LLMs) can play a critical role in emergency room operations by augmenting decision-making about patient admission. However, there are no studies for LLMs using real-world data and scenarios, in comparison to and being informed by traditional supervised machine learning (ML) models. We evaluated the performance of GPT-4 for predicting patient admissions from emergency department (ED) visits. We compared performance to traditional ML models both naively and when informed by few-shot examples and/or numerical probabilities. We conducted a retrospective study using electronic health records across 7 NYC hospitals. We trained Bio-Clinical-BERT and XGBoost (XGB) models on unstructured and structured data, respectively, and created an ensemble model reflecting ML performance. We then assessed GPT-4 capabilities in many scenarios: through Zero-shot, Few-shot with and without retrieval-augmented generation (RAG), and with and without ML numerical probabilities. The Ensemble ML model achieved an area under the receiver operating characteristic curve (AUC) of 0.88, an area under the precision-recall curve (AUPRC) of 0.72 and an accuracy of 82.9%. The naïve GPT-4's performance (0.79 AUC, 0.48 AUPRC, and 77.5% accuracy) showed substantial improvement when given limited, relevant data to learn from (ie, RAG) and underlying ML probabilities (0.87 AUC, 0.71 AUPRC, and 83.1% accuracy). Interestingly, RAG alone boosted performance to near peak levels (0.82 AUC, 0.56 AUPRC, and 81.3% accuracy). The naïve LLM had limited performance but showed significant improvement in predicting ED admissions when supplemented with real-world examples to learn from, particularly through RAG, and/or numerical probabilities from traditional ML models. Its peak performance, although slightly lower than the pure ML model, is noteworthy given its potential for providing reasoning behind predictions. Further refinement of LLMs with real-world data is necessary for successful integration as decision-support tools in care settings.

State inference for low-observable distribution system based on graph convolutional network

Distribution network state inference refers to the process of calculating the state variables of each node by using measurement data and network models in the operation of the distribution system. However, the uneven measurement layout and insufficient measurement accuracy in the distribution network have brought great challenges to the state inference of the distribution network. This paper proposes a low-observable distribution network state inference method based on a graph convolution network (GCN), which uses sparse measurement data to infer missing measurement information. Firstly, the observability of the distribution network is analyzed by the numerical probability analysis method. Secondly, the GCN is employed to extract feature information from measurement data and integrate these features. The state inference model of the distribution network based on the GCN is established. Subsequently, power flow constraints of the distribution network are incorporated into the GCN training process to enhance the precision of the generated data. Ultimately, the efficacy of the proposed method is validated using the IEEE 33-node distribution system.

Do numerical probabilities promote informed stated preference responses under inherent outcome uncertainty? Insight from a coastal adaptation choice experiment

Stated preference (SP) valuation provides insight into individuals’ preferences for environmental goods and assigns a monetary value to these preferences. It employs survey questions that elicit information on choices or behaviors in response to hypothetical future scenarios involving changes to these goods. Outcome uncertainty (OU) within SP studies refers to uncertainty concerning whether the effects communicated in these hypothetical scenarios would actually occur, were the scenario to be implemented as indicated in the questionnaire. Inherent OU may be further defined as OU that is invariant across scenarios. For example, the effect of installing coastal flood defenses may depend on a probability of severe storms that is fixed in the study area over the relevant time horizon. Inherent OU has received little attention in the environmental SP literature. For example, it is unknown whether numerical, percentage probabilities—a ubiquitous means of communicating uncertainty in SP questionnaires—are an effective risk communication tool for inherent OU in environmental valuation scenarios. This article evaluates two treatments in a discrete choice experiment survey related to coastal climate change adaptation in Connecticut, USA: one provides only raw frequencies as a risk communication tool, while the other provides implied percentage probabilities in addition to the same raw frequencies. Results show that the use of percentage probabilities to communicate inherent OU has no additional effect on average welfare estimates or the choice behavior of respondents. These findings suggest that percentage probabilities may not provide useful information to SP respondents, beyond that conveyed by past event frequencies.

Unit Maxwell-Boltzmann Distribution and Its Application to Concentrations Pollutant Data

In the vast statistical literature, there are numerous probability distribution models that can model data from real-world phenomena. New probability models, nevertheless, are still required in order to represent data with various spread behaviors. It is a known fact that there is a great need for new models with limited support. In this study, a flexible probability model called the unit Maxwell-Boltzmann distribution, which can model data values in the unit interval, is derived by selecting the Maxwell-Boltzmann distribution as a base-line model. The important characteristics of the derived distribution in terms of statistics and mathematics are investigated in detail in this study. Furthermore, the inference problem for the mentioned distribution is addressed from the perspectives of maximum likelihood, method of moments, least squares, and maximum product space, and different estimators are obtained for the unknown parameter of the distribution. The derived distribution outperforms competitive models according to different fit tests and information criteria in the applications performed on four actual air pollutant concentration data sets, indicating that it is an effective model for modeling air pollutant concentration data.

The impact of probabilistic tornado warnings on risk perceptions and responses.

Many warnings issued to members of the public are deterministic in that they do not include event likelihood information. This is true of the current polygon-based tornado warning used by the American National Weather Service, although the likelihood of a tornado varies within the boundaries of the polygon. To test whether adding likelihood information benefits end users, two experimental studies and one in-person interview study were conducted. The experimental studies compared five probabilistic formats, two with color and three with numeric probabilities alone, to the deterministic polygon. In both experiments, probabilistic formats led to better understanding of tornado likelihood and higher trust than the polygon alone, although color-coding led to several misunderstandings. When the polygon boundary was drawn at 10% chance, those using probabilistic formats made fewer correct shelter decisions at low probabilities and more correct shelter decisions at high probabilities compared to those using the deterministic warning, although overall decision quality, operationalized as expected value, did not differ. However, when the polygon boundary was drawn around 30%, participants with probabilistic forecasts had higher expected value. The interview study revealed that, although tornado-experienced individuals would not shelter at 10% chance, they would take intermediate actions, such as information-seeking and sharing. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Finite Additivity, Complete Additivity, and the Comparative Principle

AbstractIn the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) Existence: Settings where no countably additive probability exists though finitely additive probabilities do. (2) Complete Additivity: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the Comparative principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the Comparative principle [$${\mathcal{C}}$$ C ]. For numerical probabilities, principle $${\mathcal{C}}$$ C contrasts pairs, P1 and P2, defined over a common partition $$\prod$$ ∏ = {ai: i ∈ I} of measurable events. $${\mathcal{C}}$$ C requires that no P1 may be pointwise dominated, i.e., no (finitely additive) probability P2 exists such that for each i ∈ I, P2(ai) > P1(ai). By design, the cardinality of $$\prod$$ ∏ is not limited in $${\mathcal{C}}$$ C , which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle $${\mathcal{C}}$$ C in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle $${\mathcal{C}}$$ C is subject to each of the above concerns, (1) and (2). Also, Easwaran considers principle $${\mathcal{C}}$$ C with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle $${\mathcal{C}}$$ C with qualitative probabilities and show that $${\mathcal{C}}$$ C restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle $${\mathcal{C}}$$ C .

A New Sine-Based Probabilistic Approach: Theory and Monte Carlo Simulation with Reliability Application

Data modeling is a very crucial stage for decision making in applied sectors. Probability distributions are considered important tools for decision making. So far, numerous probability distributions have been developed and implemented. Most of these distributions are developed by introducing from one to eight additional parameters. Sometimes, the addition of new parameters leads to re-parameterization problems. To avoid such issues, we introduce a novel probabilistic approach. The proposed approach may be termed as a new weighted sine-G method. The beauty and key advantage of the new weighted sine-G method are that it has no additional parameters. Through using the new weighted sine-G method, a new weighted sine-Weibull distribution is introduced, which is a modification of the Weibull distribution. The estimators of the new model are also derived. Furthermore, a simulation study is carried out to evaluate the estimators of the new weighted sine-Weibull distribution. Finally, a practical application from the reliability sector is considered to evaluate the new weighted sine-Weibull distribution. Based on certain decision tools, it is observed that the proposed model is the best competing distribution for applying it in the reliability sector.

Generalisation of Companion of Ostrowski’s Type Inequality Via Riemann-Liouville Fractional Integral for Mappings whose 1st Derivatives are Bounded with Applications

We apply Riemann-Liouville fractional integral to get generalisation of companion of Ostrowski’s type integral inequality for differentiable mappings whose 1st derivatives are bounded. The present article recapture all results of M. W. Alomari’s article and also for one more article of different authors. Applications are also deduced for numerical integration, probability theory and special means.

Banyuwangi Bamboo Puppets from an Ethnomathematics Perspective

Ethnomathematics denotes mathematical phenomena entrenched within cultural contexts. Bamboo puppetry emerges as an intriguing cultural enclave ripe for investigation, distinguished by unique material composition, crafting techniques, and character portrayal in contrast to conventional puppetry forms. An exploratory inquiry unfolded within the precincts of Gintangan village, Banyuwangi, facilitated by a selective triad of informants. Employing qualitative methodologies, data accrual entailed a blended approach integrating observational scrutiny and in-depth interviews. Insights gleaned from the investigation unveil that the craftsmanship of bamboo puppets entails a gamut of mathematical undertakings, encompassing enumeration, categorization, measurement, and strategic planning. Furthermore, the genesis of bamboo puppetry production intricately intertwines with ethnomathematical dimensions, elucidating the utilization of numerical operations, geometric principles, statistical analysis, and probability considerations. These findings portend the transformative potential to transmute into a bespoke educational resource for contextualized mathematics pedagogy, leveraging the craft of bamboo puppetry as a conduit for experiential learning.

The discovery of Wupus agilis in South Korea and a new quantitative analysis of intermediate ichnospecies between non-avian theropods and birds

Recently, Wupus agilis-like footprint (NHCG a10955) was discovered in South Korea. Morphological comparison with Wupus agilis showed that NHCG a10955 could be assigned to the first occurrence of Wupus agilis from South Korea, representing the largest bird footprint in South Korea. In addition, this study performed a subsequent quantitative analysis to find a better solution to identify the trackmaker of intermediate ichnospecies. Principal component analysis (PCA) and novel probabilistic method relative bird-likeness percentage (RBP) in FL-included condition and FL-excluded condition was performed between unwebbed Mesozoic bird, theropod, and intermediate ichnospecies. The result showed that intermediate ichnospecies generally show high morphological similarity with Mesozoic bird ichnospecies if foot length was excluded from the analysis. Besides Wupus agilis, some large intermediate ichnospecies (foot length over 200 mm), such as Magnoavipes lowei and Laramie Saurexallopus, tend to have higher RBP in FL-excluded conditions that the avian affinity of these intermediate ichnospecies is hard to dismiss. Furthermore, this study also found that FL/FW ratio is an essential criterion for identifying intermediate ichnospecies since it is tied to the intrinsic foot width of the trackmaker. Although the existence of large Mesozoic birds is relatively difficult to prove due to the scarcity of skeletal records, the high bird-likeness of large intermediate ichnospecies could represent the ichnological record of large Mesozoic birds. Adding a numerical probability along with qualitative identification could be more practical rather than making a solid conclusion (e.g., 100% dinosaur or bird footprint) for objectively identifying trackmakers until the proper skeletal evidence is discovered.

Decision aids as tools to facilitate shared decision making in neonatal care: A standardization analysis

Background Neonatal practice involves complex decision-making that prioritizes different ethical principles than adult care, with a particular focus on beneficence and the best interests standard, while respecting parental autonomy. Prioritizing autonomy and best interests are facilitated through shared decision-making (SDM). Decision aids (DA)s are educational, evidence-based tools designed to facilitate SDM between patients, caregivers, and healthcare professionals. The development and evaluation of existing neonatal DAs have been variable, with as yet unestablished effectiveness and generalizability. Methods Standardized frameworks allow neonatal DAs to be evaluated for completeness and elucidate areas of opportunity to better promote the ethical goals of SDM. DAs were included in analysis based on a comprehensive search strategy focusing on neonatal topics, and then evaluated for compliance with both the Standards for UNiversal reporting of patient Decision Aid Evaluations checklist (SUNDAE) and the Systematic Development Process (SDP). Results Compliance with SUNDAE and SDP were inconsistent in currently published neonatal DAs. SUNDAE evaluation revealed gaps in visual and numerical probability factors, values clarification exercises, and provision of tailored information to meet parental needs, overall limiting the potential of informed and adaptable SDM. SDP evaluation showed gaps in longitudinal engagement of steering committees, a lack of preliminary alpha testing with clinicians and a lack of beta testing with both clinicians and parents. Conclusions In order to maximize SDM and support ethical decision-making honoring parental autonomy and best interests standard in neonates, a holistic framework for DA development and reporting is needed to maximize their clinical impact.

Symbolic control for stochastic systems via finite parity games

We consider the problem of computing the maximal probability of satisfying an ω-regular specification for stochastic, continuous-state, nonlinear systems evolving in discrete time. The problem reduces, after automata-theoretic constructions, to finding the maximal probability of satisfying a parity condition on a (possibly hybrid) state space. While characterizing the exact satisfaction probability is open, we show that a lower bound on this probability can be obtained by (I) computing an under-approximation of the qualitative winning region, i.e., states from which the parity condition can be enforced almost surely, and (II) computing the maximal probability of reaching this qualitative winning region.The heart of our approach is a technique to symbolically compute the under-approximation of the qualitative winning region in step (I) via a finite-state abstraction of the original system as a 212-player parity game. Our abstraction procedure uses only the support of the probabilistic evolution; it does not use precise numerical transition probabilities. We prove that the winning set in the abstract 212-player game induces an under-approximation of the qualitative winning region in the original synthesis problem, along with a policy to solve it. By combining these contributions with (a) a symbolic fixpoint algorithm to solve 212-player games and (b) existing techniques for reachability policy synthesis in stochastic nonlinear systems, we get an abstraction-based algorithm for finding a lower bound on the maximal satisfaction probability.We have implemented the abstraction-based algorithm in Mascot-SDS, where we combined the outlined abstraction step with our tool Genie (Majumdar et al., 2023) that solves 212-player parity games (through a reduction to Rabin games) more efficiently than existing algorithms. We evaluated our implementation on the nonlinear model of a perturbed bistable switch from the literature. We show empirically that the lower bound on the winning region computed by our approach is precise, by comparing against an over-approximation of the qualitative winning region. Moreover, our implementation outperforms a recently proposed tool for solving this problem by a large margin.

A novel immunogenic cell death-related genes signature for predicting prognosis, immune landscape and immunotherapy effect in hepatocellular carcinoma.

Immunogenic cell death (ICD) has emerged as a promising strategy to activate the adaptive immune response, modulate the tumor microenvironment (TME) and enhance the efficacy of immune therapy. However, the relationship between ICD and TME reprogramming in hepatocellular carcinoma (HCC) remains poorly understood. Transcriptional profiles and clinical spectrum of 486 HCC patients were obtained from TCGA and GEO databases. We utilized consensus clustering analysis to construct two distinct molecular subtypes and established an ICD-based scoring system (named ICD score) via WGCNA and LASSO Cox regression to predict the prognosis of the HCC cohort. Then we employed CIBERSORT and ESTIMATE methods to analyze the immune landscape of ICD score in HCC. Subsequently, the immunophenoscore (IPS) and tumor immune dysfunction and rejection (TIDE) analyses were performed to determine whether the ICD score could influence the immune therapeutic effect. Based on the ICD scoring system, a novel nomogram was generated to provide a numerical probability of HCC patients' overall survival (OS). We identified two independent ICD clusters (cluster A/B), and cluster B possessed a worse prognosis and higher immune cell infiltration. Using ICD scoring system, the HCC patients were divided into high- and low-ICD-score groups. Through integrative analyses, the high-ICD cohort owned advanced TNM stage, high pathologic grade and increased suppressive immune cell enrichment. We developed a nomogram containing the ICD score, demonstrating a high predictive accuracy with a C-index of 0.703. We further discovered that PSMD2 and PSMD14 could serve as ICD-associated prognostic biomarkers and therapeutic targets in HCC. The ICD score exhibits a high degree of reliability for predicting prognosis and may provide valuable guidance for the selection of immunotherapy for HCC patients. This novel scoring system enables the estimation of clinical immunotherapy response for HCC patients, offering new opportunities for personalized immunotherapy.

Patient-Led Approaches to a Vaginal Birth After Cesarean Delivery Calculator.

To describe patient approaches to navigating their probability of a vaginal birth after cesarean (VBAC) within the context of prediction scores generated from the original Maternal-Fetal Medicine Units' VBAC calculator, which incorporated race and ethnicity as one of six risk factors. We invited a diverse group of participants with a history of prior cesarean delivery to participate in interviews and have their prenatal visits recorded. Using an open-ended iterative interview guide, we queried and observed these individuals' mode-of-birth decisions in the context of their VBAC calculator scores. We used a critical and feminist approach to analyze thematic data gleaned from interview and visit transcripts. Among the 31 participants who enrolled, their self-identified racial and ethnic categories included: Asian or South Asian (2); Black (4); Hispanic (12); Indigenous (1); White (8); and mixed-Black, -Hispanic, or -Asian background (4). Predicted VBAC success probabilities ranged from 12% to 95%. Participants completed 64 interviews, and 14 prenatal visits were recorded. We identified four themes that demonstrated a range of patient-led approaches to interpreting the probability generated by the VBAC calculator: 1) rejecting the role of race and ethnicity; 2) reframing failure, finding success; 3) factoring the physical experience of labor; and 4) modifying the probability for VBAC. Our findings demonstrate that a numeric probability for VBAC may not be highly valued or important to all patients, especially those who have strong intentions for VBAC. Black and Hispanic participants challenged the VBAC calculator's incorporation of race and ethnicity as a risk factor and resisted the implication it produced, especially that their bodies were less capable of achieving a vaginal birth. Our findings suggest that patient-led approaches to assessing and interpreting VBAC probability may be an untapped resource for achieving a more person-centered, equitable approach to counseling.

Bootstrapped low complex iterative LDPC decoding algorithms for free-space optical communication links

In recent years, much research has been devoted to free-space optical communication (FSO). The unregulated spectrum, low implementation costs, and robust security of FSO systems all are of great importance which lead to a wide range of applications for FSO links, from terrestrial communications to satellite communications. However, the fundamental limitation with FSO links is atmospheric turbulence (AT) caused by fading, significantly reducing link performance. Random phenomena are the best characteristic of atmospheric turbulence caused by changes in the air’s refractive index over time. Numerous probability density functions of the AT models were presented to model the randomness in AT channels. The Log-Normal (LN) channel model is for weak atmospheric turbulence, while the Gamma–Gamma (G–G) channel is selected for moderate and strong atmospheric turbulence. The impacts of geometric losses, attenuation due to weather, and errors due to misalignment are addressed using LN and G–G channels. Channel coding is one of the possible solutions for mitigating such FSO channel impairments as the low-density parity check (LDPC) codes. In this article, the Weighted Bit Flipping (Algorithm (1)), Implementation Efficient Reliability Ratio Weighted Bit Flipping (Algorithm (2)), and Min-Sum (Algorithm (3)) algorithms are compared and evaluated against FSO atmospheric turbulence channels. In addition, two novel algorithms are proposed to enhance the complexity or Bit Error Rate performance of LDPC decoding over FSO channels. The results showed an impressive improvement of the coded FSO system by employing the proposed algorithms compared to the existing LDPC decoders for FSO communications from the point of all comparison parameters.

Probability and informed consent.

In this paper, we illustrate some serious difficulties involved in conveying information about uncertain risks and securing informed consent for risky interventions in a clinical setting. We argue that in order to secure informed consent for a medical intervention, physicians often need to do more than report a bare, numerical probability value. When probabilities are given, securing informed consent generally requires communicating how probability expressions are to be interpreted and communicating something about the quality and quantity of the evidence for the probabilities reported. Patients may also require guidance on how probability claims may or may not be relevant to their decisions, and physicians should be ready to help patients understand these issues.