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Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior from related sources of information or past data. However, in its absence, Bayesian methods need to rely on some \"objective\" or \"default\" priors, and the resulting posterior inference can still be quite valuable. Not surprisingly, over the years, the catalog of objective priors also has become prohibitively large, and one has to set some specific criteria for the selection of such priors. Our aim is to review some of these criteria, compare their performance, and illustrate them with some simple examples. While for very large sample sizes, it does not possibly matter what objective prior one uses, the selection of such a prior does influence inference for small or moderate samples. For regular models where asymptotic normality holds, Jeffreys' general rule prior, the positive square root of the determinant of the Fisher information matrix, enjoys many optimality properties in the absence of nuisance parameters. In the presence of nuisance parameters, however, there are many other priors which emerge as optimal depending on the criterion selected. One new feature in this article is that a prior different from Jeffreys' is shown to be optimal under the chi-square divergence criterion even in the absence of nuisance parameters. The latter is also invariant under one-to-one reparameterization.

Machine learning (ML) precipitation forecasting models typically employ a mean squared error (MSE) loss function for optimization. However, due to the well‐known “double penalty” effect, MSE‐based losses often lead to overly smoothed prediction fields and a systematic underestimation of the frequency of heavy rainfall. To address this limitation, we propose a novel probability‐matching (PM) based loss for ML precipitation nowcasting and short‐term forecasting, comparing its performance with other classical losses. Comprehensive skill evaluation demonstrates that PM‐based loss offers relatively more balanced and consistent performance across metrics, particularly its lower forecast bias from light to heavy rainfall intensities. Spectral power analysis further indicates that PM‐based loss better preserves small‐scale precipitation variability throughout the forecast period. Additionally, it results in a forecast frequency distribution of precipitation that more closely aligns with the observed distribution. These findings indicate consistent improvements in predictive skill and reliability of ML precipitation forecasts when trained with the PM‐based loss.

People often probability match: they select choices based on the probability of outcomes. For example, when predicting 10 individual results of a spinner with 7 green and 3 purple sections, many people choose green mostly but not always, even though they would be better off always choosing it (i.e., maximizing). This behavior has perplexed cognitive scientists for decades. Why do people make such an obvious error? Here, we provide evidence that this difficulty may often arise from statistical naïveté: Even when shown the optimal strategy of maximizing, many people fail to recognize that it will produce better payouts than other strategies. In 3 preregistered experiments (N = 907 Americans tested online), participants made 10 choices in a spinner game and estimated the payout for each of 3 strategies: probability matching, maximizing, and 50/50 guessing. The key finding across experiments is that while most maximizers recognize that maximizing results in higher payouts than matching, probability matchers predict similar payouts for each .

We consider two inverse Gaussian populations with a common mean but different scale-like parameters, where all parameters are unknown. We construct noninformative priors for the ratio of the scale-like parameters to derive matching priors of different orders. Reference priors are proposed for different groups of parameters. The Bayes estimators of the common mean and ratio of the scale-like parameters are also derived. We propose confidence intervals of the conditional error rate in classifying an observation into inverse Gaussian distributions. A generalized variable-based confidence interval and the highest posterior density credible intervals for the error rate are computed. We estimate parameters of the mixture of these inverse Gaussian distributions and obtain estimates of the expected probability of correct classification. An intensive simulation study has been carried out to compare the estimators and expected probability of correct classification. Real data-based examples are given to show the practicality and effectiveness of the estimators.

The landmarks encountered by a flight vehicle in its scene matching navigation are insufficient and distributed unevenly, thus being unable to effectively assist in its inertial navigation system (INS). Therefore, this paper proposes a novel algorithm for sea area landmark usability evaluation based on the probability model. The algorithm defines three types of sea area landmarks and gives their matching method and strategy. The visible range of each sea area landmark is determined according to its relative relations between each position and all landmarks in flight area, flight altitude, flight speed, camera field of view angle and INS drift error. The matching probability of different types of landmarks at different flight altitudes is calculated. Then the observable probability of each landmark is reckoned to give the probability cloud of the sea area landmark usability. The simulation results verify the effectiveness of the algorithm. The sea area landmark usability evaluation results can provide bases for INS to effectively utilize landmarks to perform route planning and realize long-time flight and high-precision navigation. 针对飞行器在海域景象匹配导航(SMN)中可遇到的地标点较少且分布不均匀, 不能有效辅助惯性导航(INS)的问题, 提出了基于概率模型的地标点可用性评估算法。该算法定义了三类海域地标点, 给出了三类地标点的匹配方法和策略; 根据飞行区内每个位置与所有地标点之间的相对关系、飞行高度、飞行速度、相机的视场角以及INS漂移误差确定每个地标点的可见范围, 并结合不同类型地标点在不同飞行高度下的匹配概率, 计算得到每个位置的地标点可观测概率, 最终给出海域地标点可用性的概率云图。仿真结果验证了该算法的有效性。海域地标点可用性评估结果可为INS有效利用地标点进行航路规划, 实现长航时、高精度导航提供依据。

In this article we address two related issues on the learning of probabilistic sequences of events. First, which features make the sequence of events generated by a stochastic chain more difficult to predict. Second, how to model the procedures employed by different learners to identify the structure of sequences of events. Playing the role of a goalkeeper in a video game, participants were told to predict step by step the successive directions—left, center or right—to which the penalty kicker would send the ball. The sequence of kicks was driven by a stochastic chain with memory of variable length. Results showed that at least three features play a role in the first issue: (1) the shape of the context tree summarizing the dependencies between present and past directions; (2) the entropy of the stochastic chain used to generate the sequences of events; (3) the existence or not of a deterministic periodic sequence underlying the sequences of events. Moreover, evidence suggests that best learners rely less on their own past choices to identify the structure of the sequences of events.

Moving-target detection under strict sensing constraints is a recurring subproblem in surveillance, search-and-rescue, and autonomous robotics. We study a canonical one-dimensional finite grid in which a sensor probes one location per time step with binary observations while the target follows reflecting random-walk dynamics. The objective is to minimize the expected time to detection using transparent, training-free decision rules defined on the belief state of the target location. We compare two belief-driven heuristics with purely online implementation: a greedy rule that always probes the most probable location and a belief-proportional sampling (BPS, probability matching) rule that samples sensing locations according to the belief distribution (i.e., posterior probability of the target location). Repeated Monte Carlo simulations quantify the exploitation–exploration trade-off and provide a self-comparison between the two policies. Across tested grid sizes, the greedy policy consistently yields the shortest expected time to detection, improving by roughly 17–20% over BPS and uniform random probing in representative settings. BPS trades some average efficiency for stochastic exploration, which can be beneficial under model mismatch. This study provides an interpretable baseline and quantitative reference for extensions to noisy sensing and higher-dimensional search.

A diversity of decision-making systems has been observed in animal collectives. In some species, choices depend on the differences of the numbers of animals that have chosen each of the available options, whereas in other species on the relative differences (a behavior known as Weber’s law), or follow more complex rules. We here show that this diversity of decision systems corresponds to a single rule of decision making in collectives. We first obtained a decision rule based on Bayesian estimation that uses the information provided by the behaviors of the other individuals to improve the estimation of the structure of the world. We then tested this rule in decision experiments using zebrafish (Danio rerio), and in existing rich datasets of argentine ants (Linepithema humile) and sticklebacks (Gasterosteus aculeatus), showing that a unified model across species can quantitatively explain the diversity of decision systems. Further, these results show that the different counting systems used by animals, including humans, can emerge from the common principle of using social information to make good decisions.

This paper investigates the estimation of the stress–strength reliability parameter ρ=P(X≤Y), where stress (X) and strength (Y) are independently modeled by geometric distributions. Objective Bayesian approaches are employed by developing Jeffreys, reference, and probability-matching priors for ρ, and their effects on the resulting Bayes estimates are examined. Posterior inference is carried out using the random-walk Metropolis–Hastings algorithm. The performance of the proposed Bayesian estimators is assessed through extensive Monte Carlo simulations based on average estimates, root mean squared errors, and frequentist coverage probabilities of the highest posterior density credible intervals. Furthermore, the applicability of the methodology is demonstrated using two real data sets.

Honeybees forage on diverse flowers which vary in the amount and type of rewards they offer, and bees are challenged with maximizing the resources they gather for their colony. That bees are effective foragers is clear, but how bees solve this type of complex multi-choice task is unknown. Here, we set bees a five-comparison choice task in which five colours differed in their probability of offering reward and punishment. The colours were ranked such that high ranked colours were more likely to offer reward, and the ranking was unambiguous. Bees' choices in unrewarded tests matched their individual experiences of reward and punishment of each colour, indicating bees solved this test not by comparing or ranking colours but by basing their colour choices on their history of reinforcement for each colour. Computational modelling suggests a structure like the honeybee mushroom body with reinforcement-related plasticity at both input and output can be sufficient for this cognitive strategy. We discuss how probability matching enables effective choices to be made without a need to compare any stimuli directly, and the use and limitations of this simple cognitive strategy for foraging animals.