No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Optimal Statistical Decisions.
... The prior belief of firm i about the remaining firms' mean output level µ i is given by the conjugate normal distribution [29] prior: ...
... The hyperparameters ν i,0 , τ i,0 are known and represent the prior beliefs of firm i about its rivals' behavior at the start of the Cournot game, at time 0. As the output choices are revealed in the game across time, Bayesian posterior estimation is used to update these hyperparameters, as detailed in [29], using ...
... Recall that the parameters τ i,0 and ν i,0 encode the initial priors that each firm has about the behavior of its rivals. If we assume that the firms have no prior information about their rivals at the start of the game, we can use an improper distribution with τ i,0 approaching zero [29]. Then, the update equations take the simpler form ...
A number of learning models have been suggested to analyze the repeated interaction of boundedly rational agents competing in oligopolistic markets. The agents form a model of the environment that they are competing in, which includes the market demand and price formation process, as well as their expectations of their rivals’ actions. The agents update their model based on the observed output and price realizations and then choose their next period output levels according to an optimization criterion. In previous works, the global dynamics of price movement have been analyzed when risk-neutral agents maximize their expected rewards at each round. However, in many practical settings, agents may be concerned with the risk or uncertainty in their reward stream, in addition to the expected value of the future rewards. Learning in oligopoly models for the case of risk-averse agents has received much less attention. In this paper, we present a novel learning model that extends fictitious play learning to continuous strategy spaces where agents combine their prior beliefs with market price realizations in previous periods to learn the mean and the variance of the aggregate supply function of the rival firms in a Bayesian framework. Next, each firm maximizes a linear combination of the expected value of the profit and a penalty term for the variance of the returns. Specifically, each agent assumes that the aggregate supply of the remaining agents is sampled from a parametric distribution employing a normal-inverse gamma prior. We prove the convergence of the proposed dynamics and present simulation results to compare the proposed learning rule to the traditional best response dynamics.
... Optimal Bayesian decisions [32] are then defined by the solution to the prior expected utility: ...
... Decision problems under uncertainty are characterized by a utility function U(d, y, θ) defined over decisions, d ∈ D, signals y ∈ Y and parameters, θ ∈ Θ. The a priori expected utility is defined by [32] as u(d) = E y,θ (U(d, y, θ)) = U(d, y, θ)dΠ(y, θ). ...
Generative Bayesian Computation (GBC) methods are developed to provide an efficient computational solution for maximum expected utility (MEU). We propose a density-free generative method based on quantiles that naturally calculates expected utility as a marginal of posterior quantiles. Our approach uses a deep quantile neural estimator to directly simulate distributional utilities. Generative methods only assume the ability to simulate from the model and parameters and as such are likelihood-free. A large training dataset is generated from parameters, data and a base distribution. Then, a supervised learning problem is solved as a non-parametric regression of generative utilities on outputs and base distribution. We propose the use of deep quantile neural networks. Our method has a number of computational advantages, primarily being density-free and an efficient estimator of expected utility. A link with the dual theory of expected utility and risk taking is also described. To illustrate our methodology, we solve an optimal portfolio allocation problem with Bayesian learning and power utility (also known as the fractional Kelly criterion). Finally, we conclude with directions for future research.
... i , and the prior is the uninformative prior in this case. For a normal distribution with unknown variance, there is a normal-gamma conjugate prior (see DeGroot [2005]). ...
... Remark. For independent sampling and prior distributions, the most frequently used conjugate models, including the two models introduced in Section III-B, satisfy the conditions in Theorem 2 (see DeGroot [2005]). In the proof, we can see that under mild regularity conditions, every alternative will be sampled infinitely often following the optimal A&S policy as the simulation budget goes to infinity, which in turn leads to the conclusion of the theorem. ...
Under a Bayesian framework, we formulate the fully sequential sampling and selection decision in statistical ranking and selection as a stochastic control problem, and derive the associated Bellman equation. Using value function approximation, we derive an approximately optimal allocation policy. We show that this policy is not only computationally efficient but also possesses both one-step-ahead and asymptotic optimality for independent normal sampling distributions. Moreover, the proposed allocation policy is easily generalizable in the approximate dynamic programming paradigm.
... In this scenario, the player is confronted with a "how to gamble if you must" situation [17] and must make a series of probabilistic choices under ambiguity [15,19]. This situation lies at the core of the classic multi-armed bandit problem [16]. The player's goal is to sequentially determine how much to play on each machine, with the objective of maximizing the expected gain (or, more appropriately, minimizing the expected loss). ...
... A common formulation of the armed-bandit problem is the Bernoulli-armed bandit, where each round's outcome is binary: either a win (= 1) or a loss (= 0). Theoretical solutions to the multi-armed bandit problems, which are beyond the scope of this research, typically focus on the efficiency of the sampling process [1,4,16]. ...
Conventional wisdom in casino gaming research suggests that gamblers are unable to identify slot machines with better odds. However, this study challenges that notion by examining whether experienced players, with years of casino play, can make informed decisions to optimize their chances of winning. With real gambling data, our findings reveal that seasoned players tend to favor slot machines with better odds. Interestingly, they refine their choices during less crowded hours when availability constraints are eased. Furthermore, consistent with the tradeoff between exploitation and exploration in reinforcement learning, more experienced players exhibit higher consistency in slot machine selection over time, suggesting genuine knowledge of which machines offer better odds. These results have significant implications for understanding casino player behavior and the potential for human learning to optimize complex decisions in a large-scale multi-armed bandit problem.
... (2.2)) which tells us that incorporating more evidence is on average better than not incorporating it. Previous proofs of various versions of this statement relied on auxiliary constructions with proper scoring rules [8,7,6,11,5,9,28]. ...
... In [28], Vergeer ranked various "LR systems" for two hypotheses including score based methods, the common source scenario and the specific source scenario. He also argues against the claim of [20] that certain methods should not be used, albeit in a rather indirect way using (strictly) proper scoring rules ((S)PSRs) [8,7,6,11,5,9]. Using PSRs, it is also possible to obtain the results that we derived directly in section 2.2. Although strictly speaking not necessary for this paper, we briefly discuss SPRSs for completeness. ...
We show that the incorporation of any new piece of information allows for improved decision making in the sense that the expected costs of an optimal decision decrease (or, in boundary cases where no or not enough new information is incorporated, stays the same) whenever this is done by the appropriate update of the probabilities of the hypotheses. Versions of this result have been stated before. However, previous proofs rely on auxiliary constructions with proper scoring rules. We, instead, offer a direct and completely general proof by considering elementary properties of likelihood ratios only. We do point out the relation to proper scoring rules. We apply our results to make a contribution to the debates about the use of score based/feature based and common/specific source likelihood ratios. In the literature these are often presented as different ``LR-systems''. We argue that deciding which LR to compute is simply a matter of the available information. There is no such thing as different ``LR-systems'', there are only differences in the available information. In particular, despite claims to the contrary, scores can very well be used in forensic practice and we illustrate this with an extensive example in DNA kinship context.
... In the field of built environment engineering, Bayesian decision theory has been introduced by Benjamin and Cornell [10]. Significant other works in relation to decision theory can also be found during this period, see e.g., [12,13]. Since the 1970 s, the Bayesian decision theory has been sparsely used in built environment engineering; however, it has been extensively researched on in the context of optimization of inspection and repair planning of offshore structures, partly with value of information considerations for inspections, see e.g., [14][15][16]. ...
... Beside the distinct theoretical basis, the methodological bases for information value quantification encompass (1) general probability and utility theory-based approaches extending the modelling as introduced by [10], (2) Markov property-based approaches, i.e., a) Markov decision processes [13,48], b) partially observable Markov decision processes as introduced by Åström [49], see e.g., [50][51][52][53][54][55], and (3) methods for information entropy-based approaches, e.g., [56]. In the following, we focus on general probability and utility-theory-based approaches and their adaptation to built environment engineering. ...
... where p (M j | y) denotes the posterior probability of the model the variance of the parameters, furthermore, 2 K is the total number of all linear combinations in the regression model. The calculation of the posterior probability of the model and the estimation of parameters in the linear regression model is a well-known topic in the Bayesian statistics literature, To simplify the calculations, a natural normal-Gamma conjugate is used before the regression parameters (DeGroot, 1970;Koop, 2003). Therefore, the non-informative standard priors for the α-intercept are assumed, which are common parameters in all regression models: ...
The Latin American region has been characterized by low growth in recent decades, for this, different empirical studies have been developed to identify both microeconomic and macroeconomic elements that influence its performance. The heterogeneity of the results, from an empirical point of view, generates a problem of uncertainty, due to the large number of suggested determining factors. Thus, in order to reduce uncertainty, the Bayesian Model Average (BMA) methodology is proposed. Twenty-seven possible determinants are considered in a sample of 19 Latin American countries covering the period 1996-2021. In the same way, the BMA with instrumental variables (IVBMA) is used to consider possible endogeneity problems that have their origin in the reverse causality of some explanatory variables. The results show some economic and institutional factors significant to understand economic growth in Latin America. Additionally, a non-linear relationship of corruption with economic growth is found.
... Customer interprets m(ȳ̄t t ) as a proxy for the quality q and approximates E[ | Φ n,k,t ] with ̃t , where ̃t is defined The above formula and the derivations are taken from(DeGroot 2005;Berger 2013). The corresponding demand function can be rewritten as ...
The article considers the assortment planning problem with respect to trustworthy reviewers. The trustworthiness of the product review and rating is based on whether the reviewer is rational or irrational. The average review ratings and comments are affected by the average trustworthiness. On the other hand, consumers have attention spans, i.e., the maximum number of products they are willing to view and inspect sequentially before purchasing a product or leaving the platform empty-handed when the attention span gets exhausted. The current paper provides an assortment planning model that maximizes revenue while considering trustworthy online product reviews as the quality by considering a threshold number of reviews and prices. This, in turn, is expected to minimize the information asymmetry within the decision-making process and identify the paid reviewers or manipulative reviews.
... We express the well-known concavity property of the Bayes loss [DeGroot (1970), Section 8.4] as follows. ...
We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss. Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other. Although Tops\oe described this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case. We here generalize this theory to apply to arbitrary decision problems and loss functions. We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context. This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts. We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions. This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback-Leibler divergence (the ``redundancy-capacity theorem'' of information theory). For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions.
... The Bayesian sequential selection is very efficient using the Bayesian model in (1) and (2). This is due to the conjugacy property of the normal-inverse-Wishart distribution in (1), which allows us to update the prior information after each new sample in a computationally tractable way [4]. Specifically, suppose that the parameters in (1) and (2) have been updated to θ n , B n , q n and b n at the n-th step, i.e., ...
We study the conjugacy approximation method in the context of Bayesian ranking and selection with unknown correlations. Under the assumption of normal-inverse-Wishart prior distribution, the posterior distribution remains a normal-inverse-Wishart distribution thanks to the conjugacy property when all alternatives are sampled at each step. However, this conjugacy property no longer holds if only one alternative is sampled at a time, an appropriate setting when there is a limited budget on the number of samples. We propose two new conjugacy approximation methods based on the idea of moment matching. Both of them yield closed-form Bayesian prior updating formulas. This updating formula can then be combined with the knowledge gradient algorithm under the "value of information" framework. We conduct computational experiments to show the superiority of the proposed conjugacy approximation methods, including applications in wind farm placement and computer model calibration.
... whereas the rational planners used computations similar to backward induction DeGroot (2005). Yet, some of the strategies mentioned above had certain predictions that were not consistent with the experimental data. ...
We designed a grid world task to study human planning and re-planning behavior in an unknown stochastic environment. In our grid world, participants were asked to travel from a random starting point to a random goal position while maximizing their reward. Because they were not familiar with the environment, they needed to learn its characteristics from experience to plan optimally. Later in the task, we randomly blocked the optimal path to investigate whether and how people adjust their original plans to find a detour. To this end, we developed and compared 12 different models. These models were different on how they learned and represented the environment and how they planned to catch the goal. The majority of our participants were able to plan optimally. We also showed that people were capable of revising their plans when an unexpected event occurred. The result from the model comparison showed that the model-based reinforcement learning approach provided the best account for the data and outperformed heuristics in explaining the behavioral data in the re-planning trials.
... Likewise, it is by the requirement of rationality in such a predicament that we impose a fixed prior ν i (·) on every agent i and carry it through for all times t. Indeed, it is the grand tradition of Bayesian statistics, as advocated in the prominent and influential works of [76], [77], [78], [79] and many others, to argue on normative grounds that rational behavior in a decision theoretic framework forces individuals to employ Bayes rule and appropriate it to their personal priors. ...
We analyze a model of learning and belief formation in networks in which agents follow Bayes rule yet they do not recall their history of past observations and cannot reason about how other agents' beliefs are formed. They do so by making rational inferences about their observations which include a sequence of independent and identically distributed private signals as well as the actions of their neighboring agents at each time. Successive applications of Bayes rule to the entire history of past observations lead to forebodingly complex inferences: due to lack of knowledge about the global network structure, and unavailability of private observations, as well as third party interactions preceding every decision. Such difficulties make Bayesian updating of beliefs an implausible mechanism for social learning. To address these complexities, we consider a Bayesian without Recall model of inference. On the one hand, this model provides a tractable framework for analyzing the behavior of rational agents in social networks. On the other hand, this model also provides a behavioral foundation for the variety of non-Bayesian update rules in the literature. We present the implications of various choices for the structure of the action space and utility functions for such agents and investigate the properties of learning, convergence, and consensus in special cases.
... These questions ask for decisions that have some (minor to drastic) consequences. We usually want to make "good" decisions, where the quality is measured in terms of some reward (money, life expectancy) or loss [Fer67,DeG70,Jef83]. In order to compute this reward as a function of our decision, we need to predict the environment: whether there will be rain or sunshine today, whether the market will go up or down, whether doomsday is tomorrow, or which type of cancer the patient has. ...
Specialized intelligent systems can be found everywhere: finger print, handwriting, speech, and face recognition, spam filtering, chess and other game programs, robots, et al. This decade the first presumably complete mathematical theory of artificial intelligence based on universal induction-prediction-decision-action has been proposed. This information-theoretic approach solidifies the foundations of inductive inference and artificial intelligence. Getting the foundations right usually marks a significant progress and maturing of a field. The theory provides a gold standard and guidance for researchers working on intelligent algorithms. The roots of universal induction have been laid exactly half-a-century ago and the roots of universal intelligence exactly one decade ago. So it is timely to take stock of what has been achieved and what remains to be done. Since there are already good recent surveys, I describe the state-of-the-art only in passing and refer the reader to the literature. This article concentrates on the open problems in universal induction and its extension to universal intelligence.
... Bayesian Optimisation: A third solution method is an adaptive strategy where the policy is derived from Bayesian decision theory (Berger, 1985;DeGroot, 1970;Robert, 2007) and formalises the approach to decision-making under uncertainty with respect to the unknown objective function. bo, which is the principal subject of Chapter 6 and plays a major role in Chapter 4, has recently achieved notable and widely-publicised success as a component of AlphaGo (Chen et al., 2018) as well as across applications including chemical reaction optimisation (Shields et al., 2021), robotics (Calandra et al., 2016), and machine learning hyperparameter optimisation (Cowen-Rivers et al., 2022;Turner et al., 2021). ...
In many areas of the observational and experimental sciences data is scarce. Data observation in high-energy astrophysics is disrupted by celestial occlusions and limited telescope time while data derived from laboratory experiments in synthetic chemistry and materials science is time and cost-intensive to collect. On the other hand, knowledge about the data-generation mechanism is often available in the sciences, such as the measurement error of a piece of laboratory apparatus. Both characteristics, small data and knowledge of the underlying physics, make Gaussian processes (GPs) ideal candidates for fitting such datasets. GPs can make predictions with consideration of uncertainty, for example in the virtual screening of molecules and materials, and can also make inferences about incomplete data such as the latent emission signature from a black hole accretion disc. Furthermore, GPs are currently the workhorse model for Bayesian optimisation, a methodology foreseen to be a guide for laboratory experiments in scientific discovery campaigns. The first contribution of this thesis is to use GP modelling to reason about the latent emission signature from the Seyfert galaxy Markarian 335, and by extension, to reason about the applicability of various theoretical models of black hole accretion discs. The second contribution is to extend the GP framework to molecular and chemical reaction representations and to provide an open-source software library to enable the framework to be used by scientists. The third contribution is to leverage GPs to discover novel and performant photoswitch molecules. The fourth contribution is to introduce a Bayesian optimisation scheme capable of modelling aleatoric uncertainty to facilitate the identification of material compositions that possess intrinsic robustness to large scale fabrication processes.
... Effort Enhancement is quite evident in Machine Learning and Deep Learning owing to the need for precise predictions and improved accuracies in the developed models and hence, the research-oriented analysis was performed towards the same [6] with some works involving even ensemble-and heuristic-based learning approaches [7]. For Statistical and numerical measurement of prediction availabilities for a model, there is some effort laid towards cross-validated choice and assessment [8] with some numerical techniques like linear equation least square analysis [9], Discriminatory Analysis Error Rae Determination [10], Optimal Statistical Decisions [11], etc. More unconventional numerical measures of effort do exist with examples like person-hours, person-months [12], expertprovided software effort [13], Latent Dirichlet Allocation (LDA) [14], Use Case Points (UCP) [15], etc. ...
Machine Learning Research often involves the use of diverse libraries, modules, and pseudocodes for data processing, cleaning, filtering, pattern recognition, and computer intelligence. Quantization of Effort Required for the above cumulative processes is rarely discussed in the existing works & The time to reach the desired level of the model’s functionality is essential to gauge the environment training for the model training & pre-deployment testing. In this study, we empirically defined the manual-cum-computational effort required for the model development in terms of 2-time factors: time-to-live(time until 1st code modification) and time-for-modification(time between 2 codebase changes) taken together in a mathematical model to compute Effort Factor(quantitative measure of determining effort needed to design ML algorithms). The study is novel in terms of how the effort required to create ML models is calculated with a particular focus on the manual effort direction. The results and the findings obtained can be used for determining the total time needed to synthesize ML models & frameworks in terms of code change cycles and implementation strategies marking a 25% performance increase in the current method above the Standard Pipeline.
... The Bayesian inference procedures have been developed generally under squared error loss function DeGroot [10] introduced several types of loss functions and he obtains Bayes estimators under this loss function. An example of a symmetric loss function is the DeGroot loss function defined by 2 L , ...
In this paper, Himanshu distribution is considered for Bayesian analysis. The expressions for Bayes estimators of the parameter have been derived under squared error, precautionary, entropy, K-loss, Al-Bayyati’s loss, DeGroot and minimum expected loss functions by using beta prior.
... Maneerat et al. [6] utilized the conjugate families proposed by DeGroot [38] for a normal random sample to derive the posterior distribution of parameters in the normal-gamma prior. The posterior distributions of δ i 0 , µ i , and σ 2 i are as follows: ...
The coefficient of quartile variation is a valuable measure used to assess data dispersion when it deviates from a normal distribution or displays skewness. In this study, we focus specifically on the delta-lognormal distribution. The lognormal distribution is characterized by its asymmetrical nature and comprises exclusively positive values. However, when these values undergo a logarithmic transformation, they conform to a symmetrical (or normal) distribution. Consequently, this research aims to establish confidence intervals for the difference between coefficients of quartile variation within lognormal distributions incorporating zero values. We employ the Bayesian, generalized confidence interval, and fiducial generalized confidence interval methods to construct these intervals, involving data simulation using RStudio software. We evaluate the performance of these methods based on coverage probabilities and average lengths. Our findings indicate that the Bayesian method, employing Jeffreys’ prior, performs well in low variability, while the generalized confidence interval method is more suitable for higher variability. Therefore, we recommend using both approaches to construct confidence intervals for the difference between the coefficients of the quartile variation in lognormal distributions that include zero values. Furthermore, we apply these methods to rainfall data in Thailand to illustrate their alignment with actual and simulated data.
... (4) Donde P (M j | y) denota la probabilidad posterior del modelo M j , y Var ( | y) son el valor esperado y la varianza de los parámetros, y 2 K es el número total de todas las combinaciones lineales en el modelo de regresión. El cálculo de la probabilidad posterior del modelo y la estimación de parámetros en el modelo de regresión lineal es un tema bien conocido en la literatura de estadística bayesiana, por lo que aquí solo se describen de manera general, los pasos principales utilizados, especialmente, aquellos relacionados con el marco de premediación del modelo. 2 Para simplificar los cálculos se utiliza un conjugado natural normal-Gamma antes de los parámetros de regresión (DeGroot 1970, Koop 2003 por lo tanto, se asumen los a-priori estándar no informativos para el intercepto α, que son parámetros comunes en todos los modelos de regresión: ...
The existence of heterogeneity in the literature that addresses the sources of economic growth, from an empirical point of view, generates a problem of uncertainty. The objective is to identify robust determinants of economic growth in Mexico by reducing the uncertainty of the model. To do so, the Bayesian Model Averaging (BMA) methodology is proposed, which analyzes many explanatory variables simultaneously. Thus, 28 possible determinants are considered in a sample that includes the 32 federal entities, to include the period2010-2021. The BMA constructs various possible combinations of models to extract the most robust determinants. Similarly, the instrumental variables BMA (IVBMA) is used to consider possible endogeneity problems. The results show a set of significant economic, institutional, and social variables to understand economic growth in Mexico.
... We use the term probabilistic classifier to refer to a classification system that outputs posterior probabilities. The posterior probabilities produced by a good probabilistic classifier can be used to make optimal decisions for a given cost function using Bayes decision theory (DeGroot, 1970;Bernardo and Smith, 1994;Jaynes, 2003), and they can be readily interpreted by an end user or passed on to a downstream system. Evaluating the quality of the posteriors produced by a classification system is not a trivial task since, unlike for the evaluation of categorical decisions for which class labels are used as ground truth, there are no ground-truth posteriors against which to compare the system-generated posteriors. ...
Most machine learning classifiers are designed to output posterior probabilities for the classes given the input sample. These probabilities may be used to make the categorical decision on the class of the sample; provided as input to a downstream system; or provided to a human for interpretation. Evaluating the quality of the posteriors generated by these system is an essential problem which was addressed decades ago with the invention of proper scoring rules (PSRs). Unfortunately, much of the recent machine learning literature uses calibration metrics -- most commonly, the expected calibration error (ECE) -- as a proxy to assess posterior performance. The problem with this approach is that calibration metrics reflect only one aspect of the quality of the posteriors, ignoring the discrimination performance. For this reason, we argue that calibration metrics should play no role in the assessment of posterior quality. Expected PSRs should instead be used for this job, preferably normalized for ease of interpretation. In this work, we first give a brief review of PSRs from a practical perspective, motivating their definition using Bayes decision theory. We discuss why expected PSRs provide a principled measure of the quality of a system's posteriors and why calibration metrics are not the right tool for this job. We argue that calibration metrics, while not useful for performance assessment, may be used as diagnostic tools during system development. With this purpose in mind, we discuss a simple and practical calibration metric, called calibration loss, derived from a decomposition of expected PSRs. We compare this metric with the ECE and with the expected score divergence calibration metric from the PSR literature and argue, using theoretical and empirical evidence, that calibration loss is superior to these two metrics.
... The performance of nine different algorithms in predicting birth weight was evaluated primarily based on MAE, MAPE, RMSE, and ME parameters. The model with the lowest MAE value was considered the best (Table 3) (9). When predicting newborn weight using AC, FL, BPD, maternal age, gestational week according to the last menstrual period, and gender parameters, the elastic net regressor (ENR) algorithm, with the lowest MAE value (284 grams), demonstrated the best prediction performance. ...
fetal weight estimation via artificial intelligence
... 2) and thus informs the experimental design policy about the relative likelihoods of various outcomes. Bayesian decision theory refers to a prescriptive decision-making policy in which the costs and benefits of taking an action in different states of the world are averaged according to the probabilities of those states of the world (DeGroot, 2005;Berger, 2013). ADO's policy of selecting the stimulus that maximizes the global utility is a special case of a Bayesian decision theoretic method. ...
Adaptive design optimization (ADO) is a state-of-the-art technique for experimental design (Cavagnaro et al., 2010). ADO dynamically identifies stimuli that, in expectation, yield the most information about a hypothetical construct of interest (e.g., parameters of a cognitive model). To calculate this expectation, ADO leverages the modeler’s existing knowledge, specified in the form of a prior distribution. Informative priors align with the distribution of the focal construct in the participant population. This alignment is assumed by ADO’s internal assessment of expected information gain. If the prior is instead misinformative , i.e., does not align with the participant population, ADO’s estimates of expected information gain could be inaccurate. In many cases, the true distribution that characterizes the participant population is unknown, and experimenters rely on heuristics in their choice of prior and without an understanding of how this choice affects ADO’s behavior. Our work introduces a mathematical framework that facilitates investigation of the consequences of the choice of prior distribution on the efficiency of experiments designed using ADO. Through theoretical and empirical results, we show that, in the context of prior misinformation , measures of expected information gain are distinct from the correctness of the corresponding inference. Through a series of simulation experiments, we show that, in the case of parameter estimation, ADO nevertheless outperforms other design methods. Conversely, in the case of model selection, misinformative priors can lead inference to favor the wrong model, and rather than mitigating this pitfall, ADO exacerbates it.
... where the notation N k denotes the neighbors of agent k. ■ As demonstrated in [12], under Assumptions 1-3, repeated application of the updates in (10) and (11) allows agents to almost surely learn the true current state s n : ...
This study proposes the use of a social learning method to estimate a global state within a multi-agent off-policy actor-critic algorithm for reinforcement learning (RL) operating in a partially observable environment. We assume that the network of agents operates in a fully-decentralized manner, possessing the capability to exchange variables with their immediate neighbors. The proposed design methodology is supported by an analysis demonstrating that the difference between final outcomes, obtained when the global state is fully observed versus estimated through the social learning method, is -bounded when an appropriate number of iterations of social learning updates are implemented. Unlike many existing dec-POMDP-based RL approaches, the proposed algorithm is suitable for model-free multi-agent reinforcement learning as it does not require knowledge of a transition model. Furthermore, experimental results illustrate the efficacy of the algorithm and demonstrate its superiority over the current state-of-the-art methods.
... Decision-making under uncertainty has long been studied in statistical decision theory [108,32,13,99]. Let u(y, a) be a utility function that summarises the utility for the decision-maker. The optimal decision is then a ⋆ = arg max a∈A y∈Y u(a, y)p(y|a) . ...
We study the legal challenges in automated decision-making by analysing conventional algorithmic fairness approaches and their alignment with antidiscrimination law in the United Kingdom and other jurisdictions based on English common law. By translating principles of anti-discrimination law into a decision-theoretic framework, we formalise discrimination and propose a new, legally informed approach to developing systems for automated decision-making. Our investigation reveals that while algorithmic fairness approaches have adapted concepts from legal theory, they can conflict with legal standards, highlighting the importance of bridging the gap between automated decisions, fairness, and anti-discrimination doctrine.
... In a Bayesian setting, one is provided with the additional information of the prior distribution over the hypothesis space, opening the door to a much wider array of possibilities. Maximum a posteriori (MAP) (DeGroot, 2005), for example, maximises the posterior probability (or probability density) of the chosen hypothesis, whereas Bayes estimation (Berger, 2013) minimises an expected loss function. ...
Point estimation is a fundamental statistical task. Given the wide selection of available point estimators, it is unclear, however, what, if any, would be universally-agreed theoretical reasons to generally prefer one such estimator over another. In this paper, we define a class of estimation scenarios which includes commonly-encountered problem situations such as both ``high stakes'' estimation and scientific inference, and introduce a new class of estimators, Error Intolerance Candidates (EIC) estimators, which we prove is optimal for it. EIC estimators are parameterised by an externally-given loss function. We prove, however, that even without such a loss function if one accepts a small number of incontrovertible-seeming assumptions regarding what constitutes a reasonable loss function, the optimal EIC estimator can be characterised uniquely. The optimal estimator derived in this second case is a previously-studied combination of maximum a posteriori (MAP) estimation and Wallace-Freeman (WF) estimation which has long been advocated among Minimum Message Length (MML) researchers, where it is derived as an approximation to the information-theoretic Strict MML estimator. Our results provide a novel justification for it that is purely Bayesian and requires neither approximations nor coding, placing both MAP and WF as special cases in the larger class of EIC estimators.
... By minimizing the DLF, the optimal decision rules or estimators can be selected that can lead to improved performance and minimized expected loss. The DLF, Lðy;ŷÞ ¼ ðyÀŷÞ 2 y 2 ; was introduced by [45] and was later utilized by [46] to obtain BEs. The estimatorŷ, which minimizes the PR, is used for parameter θ. ...
Bayesian Control charts are emerging as the most efficient statistical tools for monitoring manufacturing processes and providing effective control over process variability. The Bayesian approach is particularly suitable for addressing parametric uncertainty in the manufacturing industry. In this study, we determine the monitoring threshold for the shape parameter of the Inverse Gaussian distribution (IGD) and design different exponentially-weighted-moving-average (EWMA) control charts based on different loss functions (LFs). The impact of hyperparameters is investigated on Bayes estimates (BEs) and posterior risks (PRs). The performance measures such as average run length (ARL), standard deviation of run length (SDRL), and median of run length (MRL) are employed to evaluate the suggested approach. The designed Bayesian charts are evaluated for different settings of smoothing constant of the EWMA chart, different sample sizes, and pre-specified false alarm rates. The simulative study demonstrates the effectiveness of the suggested Bayesian method-based EWMA charts as compared to the conventional classical setup-based EWMA charts. The proposed techniques of EWMA charts are highly efficient in detecting shifts in the shape parameter and outperform their classical counterpart in detecting faults quickly. The proposed technique is also applied to real-data case studies from the aerospace manufacturing industry. The quality characteristic of interest was selected as the monthly industrial production index of aircraft from January 1980 to December 2022. The real-data-based findings also validate the conclusions based on the simulative results.
... , m k , we then receive a gain which is a function of the variables observed. This problem is a subset of a general class of other optimal stopping problems that all aim to find a sequential procedure to maximise the expected reward (see section 13.4 of [2] for a more extensive discussion of this class of problem). The secretary problem is arguably the most well known (see [3,4]), and it has a wide range of variations (see [5,6]), but there is also a rich literature of other examples (see [7]). ...
We study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an optimal strategy to maximise the expected reward. In our previous work, we presented a methodology, borrowing techniques from applied mathematics, for obtaining asymptotic expressions for the expectation duration of the optimal stopping time where one stop is permitted. In this study, we generalise further to the case where more than one stop is permitted, with an updated objective function of maximising the expected sum of the variables chosen. We formulate a complete generalisation for an exponential family as well as the uniform distribution by utilising an inductive approach in the formulation of the stopping rule. Explicit examples are shown for common probability functions as well as simulations to verify the asymptotic calculations.
... The goal of spatial prediction is to find the optimal predictor Y opt (s 0 ) of the true process at an unobserved location s 0 , as a function of z. In decision theory, Y opt (s 0 ) is the minimizer of an expected loss function or risk function (DeGroot, 2005). That is, ...
... Therefore, after the training of the Total FL algorithm, there is a comparison between the Inference results (Predicted green lines in the figures) and the Real expected results (Actual Test orange lines). We adopt the Mean Absolute Error (MAE) [38], Mean Squared Error or Loss [39] (MSE), Root Mean Squared Error (RMSE) [40], Mean Absolute Percentage Error (MAPE) [41], and Symmetric Mean Absolute Percentage Error (SMAPE) [42] to give an overview of the accuracy achieved for each data type case (n = number of nodes,y i is the pragmatic value (orange lines in the following figures),ŷ p is the predicted value (green lines)): ...
Cloud computing and relevant emerging technologies have presented ordinary methods for processing edge-produced data in a centralized manner. Presently, there is a tendency to offload processing tasks as close to the edge as possible to reduce the costs and network bandwidth used. In this direction, we find efforts that materialize this paradigm by introducing distributed deep learning methods and the so-called Federated Learning (FL). Such distributed architectures are valuable assets in terms of efficiently managing resources and eliciting predictions that can be used for proactive adaptation of distributed applications. In this work, we focus on deep learning local loss functions in multi-cloud environments. We introduce the MulticloudFL system that enhances the forecasting accuracy, in dynamic settings, by applying two new methods that enhance the prediction accuracy in applications and resources monitoring metrics. The proposed algorithm’s performance is evaluated via various experiments that confirm the quality and benefits of the MulticloudFL system, as it improves the prediction accuracy on time-series data while reducing the bandwidth requirements and privacy risks during the training process.
... Satellite QPE products are generally designed and optimized to have the lowest possible mean squared error (MSE) at the pixel resolution. It is however well known that, when the residual errors cannot be reduced to a negligible quantity, MSE-optimal estimation algorithms tend to favor smooth solutions and to compress the dynamical range of the retrieved variable [8,9]; the preservation of statistical extremes is generally "sacrificed" to the uncertainty. For satellite QPEs, the compressing effect of MSE minimization is scale-dependent: the variability at the finer spatial and temporal scales, which is generally associated larger uncertainty, is smoothed to a higher degree. ...
... I do so for both the sequential design problem and the nonsequential optimal design problem. The analyses are based on the theory of optimal design in Bayesian statistical decision theory (e.g., Bandemer et al., 1977;Bandemer & Nather, 1980;Chernoff, 1972;DeGroot, 1970;Fedorov, 1972;Keener, 1984;Lindley, 1972;Strasser, 1985;Wald, 1950). ...
M. Oaksford and N. Chater (1994, 1996) presented a rational analysis of Wason's selection task in which human performance was argued to be optimal when contrasted with the normative yardstick of Bayesian statistics rather than formal logic. In the present article, it is shown that selecting data according to expected information gain, as proposed by Oaksford and Chater, leads to suboptimal performance in Bayesian hypothesis testing. Procedures are presented that are better justified normatively, their psychological implications are explored, and a number of novel predictions are derived under the sequential as well as the more adequate nonsequential interpretation of the task.
... According to this idea, choosing between two uncertain courses of action is conducted in much the same way as a test of two hypotheses on the basis of a sequence of experimental observations (cf. DeGroot, 1970). ...
Decision field theory provides for a mathematical foundation leading to a dynamic, stochastic theory of decision behavior in an uncertain environment. This theory is used to explain (a) violations of stochastic dominance, (b) violations of strong stochastic transitivity, (c) violations of independence between alternatives, (d) serial position effects on preference, (e) speed–accuracy tradeoff effects in decision making, (f) the inverse relation between choice probability and decision time, (g) changes in the direction of preference under time pressure, (h) slower decision times for avoidance as compared with approach conflicts, and (i) preference reversals between choice and selling price measures of preference. The proposed theory is compared with 4 other theories of decision making under uncertainty.
... Chi-square distribution with p degrees of freedom (DeGroot, 2005). Then the distribution function of Y is ...
This work explores the possibilities of using discrete geometric modeling tools as a means of optimization design of building envelopes with the aim of further minimizing energy and resource costs at various stages of its life cycle. In particular, scientific methods and approaches are analyzed that allow to increase the energy efficiency of thermal envelopes of buildings by reducing heat loss due to reducing the surface area of the simulated envelope and minimizing the use of structural materials for its construction. The symbiotic combination of the mentioned design approaches not only allows to reduce the consumption of energy resources at the stage of operation of the building, but also indirectly allows to reduce the costs of energy carriers at the stage of direct production of building structures and products. That leads to a reduction in emissions of greenhouse gases and other pollutants into the atmospheric air, which makes the proposed approach attractive for modern ecological construction and one that fully complies with the principles of green construction and contributes to reducing the carbon footprint of architectural objects, providing at the same time gradual achievement of at least 9 sustainable development goals. From the mathematical point of view, solving the problem of simultaneous minimization of costs of construction materials and maximum reduction of heat losses is carried out by constructing an objective function represented by the sum of the volume of relevant building materials and finding its extreme values with the imposition of special functional conditions. These conditions are discrete functional analogues of the differential properties (namely, the value of the average curvature of the surface) of minimal surfaces having the smallest possible area on a given reference contour. In the modeling process, discretely represented shells are considered as analogues of spatial rod momentless structures or mesh structures, the elements of which work only in compression or tension. It can be seen that the work of all elements of the corresponding structures takes place within the limits of elastic deformations without loss of stability.
Bayesian decision theory is a mathematical framework that models reasoning and decision-making under uncertain conditions. The Bayesian paradigm originated as a theory of how people should operate, not a theory of how they actually operate. Nevertheless, cognitive scientists increasingly use it to describe the actual workings of the human mind. Over the past few decades, cognitive science has produced impressive Bayesian models of mental activity. The models postulate that certain mental processes conform, or approximately conform, to Bayesian norms. Bayesian models offered within cognitive science have illuminated numerous mental phenomena, such as perception, motor control, and navigation. This Element provides a self-contained introduction to the foundations of Bayesian cognitive science. It then explores what we can learn about the mind from Bayesian models offered by cognitive scientists.
As alternatives to the normal distributions, t distributions are widely applied in robust analysis for data with outliers or heavy tails. The properties of the multivariate t distribution are well documented in Kotz and Nadarajah's book, which, however, states a wrong conclusion about the conditional distribution of the multivariate t distribution. Previous literature has recognized that the conditional distribution of the multivariate t distribution also follows the multivariate t distribution. We provide an intuitive proof without directly manipulating the complicated density function of the multivariate t distribution.
The model of rational decision-making in most of economics and statistics is expected utility theory (EU) axiomatised by von Neumann and Morgenstern, Savage and others. This is less the case, however, in financial economics and mathematical finance, where investment decisions are commonly based on the methods of mean-variance (MV) introduced in the 1950s by Markowitz. Under the MV framework, each available investment opportunity ("asset") or portfolio is represented in just two dimensions by the ex ante mean and standard deviation of the financial return anticipated from that investment. Utility adherents consider that in general MV methods are logically incoherent. Most famously, Norwegian insurance theorist Borch presented a proof suggesting that two-dimensional MV indifference curves cannot represent the preferences of a rational investor (he claimed that MV indifference curves "do not exist"). This is known as Borch's paradox and gave rise to an important but generally little-known philosophical literature relating MV to EU. We examine the main early contributions to this literature, focussing on Borch's logic and the arguments by which it has been set aside.
Consider a real-valued function that can only be observed with stochastic noise at a finite set of design points within a Euclidean space. We wish to determine whether there exists a convex function that goes through the true function values at the design points. We develop an asymptotically consistent Bayesian sequential sampling procedure that estimates the posterior probability of this being true. In each iteration, the posterior probability is estimated using Monte Carlo simulation. We offer three variance reduction methods -- change of measure, acceptance-rejection, and conditional Monte Carlo. Numerical experiments suggest that the conditional Monte Carlo method should be preferred.
We present an objective Bayes method for covariance selection in Gaussian multivariate regression models whose error term has a covariance structure which is Markov with respect to a Directed Acyclic Graph (DAG). The scope is covariate-adjusted sparse graphical model selection, a topic of growing importance especially in the area of genetical genomics (eQTL analysis). Specifically, we provide a closed-form expression for the marginal likelihood of any DAG (with small parent sets) whose computation virtually requires no subjective elicitation by the user and involves only conjugate matrix normal Wishart distributions. This is made possible by a specific form of prior assignment, whereby only one prior under the complete DAG model need be specified, based on the notion of fractional Bayes factor. All priors under the other DAG models are derived using prior modularity, and global parameter independence, in the terminology of Geiger & Heckerman (2002). Since the marginal likelihood we obtain is constant within each class of Markov equivalent DAGs, our method naturally specializes to covariate-adjusted decomposable graphical models.
This chapter presents a Bayesian analysis of semiparametric models in which the vector of parameters of interest is characterized by a moment equation. Powerful representations of the Dirichlet processes are used and provide an efficient numerical strategy to deal with such models. The so-called noninformative prior specification gives a Bayesian interpretation of the bootstrap method, but some pathologies of this prior measure are pointed out. Several numerical applications illustrate the presentation.
Cuando los parámetros p , en la distribución de Bernoulli y λ, en la de Poisson, son muy pequeños, su estimación es un problema díficil porque, a menudo, en muestras aleatorias no se presenta el caso de interés. En estudios epidemiológicos sobre enfermedades poco comunes es frecuente encontrar este tipo de resultados. Se presentan algunas soluciones a este problema.
This paper addresses decision-aiding problems that involve multiple objectives and uncertain states of the world. Inspired by the capability approach, we focus on cases where a policy maker chooses an act that, combined with a state of the world, leads to a set of choices for citizens. While no preferential information is available to construct importance parameters for the criteria, we can obtain likelihoods for the different states. To effectively support decision-aiding in this context, we propose two procedures that merge the potential set of choices for each state of the world taking into account their respective likelihoods. Our procedures satisfy several fundamental and desirable properties that characterize the outcomes.
This paper investigates the incidence of limited attention in a high-stakes business setting: a bar owner may be unable to purge transitory shocks from noisy profit signals when deciding whether to exit. Combining a 24-year monthly panel on the alcohol revenues from every bar in Texas with weather data, we find suggestive evidence that inexperienced, distantly located owners may overreact to the transitory component of revenue relative to the persistent component. This apparent asymmetric response is muted under higher revenue fluctuations. We formulate and estimate a structural model to endogenize attention allocation by owners with different thinking cost. Under the assumptions of the model, we find that 3.9% bars make incorrect exit decisions due to limited attention. As exits are irreversible, permanent decisions, small mistakes at the margin interpreting profit signals can lead to large welfare losses for entrepreneurs.
We use randomized treatments that provide different types of information about the first and/or second moments of future economic growth to generate exogenous changes in the perceived macroeconomic uncertainty of treated households. The effects on their spending decisions relative to an untreated control group are measured in follow-up surveys. Our results indicate that, after taking into account first moments, higher macroeconomic uncertainty induces households to significantly and persistently reduce their total monthly spending in subsequent months. Changes in spending are broad based across spending categories and apply to larger durable good purchases as well. (JEL D12, D81, D84, E21, E23, G51)
In this article, the maximum likelihood estimation approach is considered for estimating the unknown parameters of some members of the elliptical distributions using the depth-based multivariate record values, a new innovative concept of records recently defined by the authors. The problem is discussed under some scenarios in which the Mahalanobis and projection depth functions are employed. For all the scenarios, the likelihood equations are formulated and the maximum likelihood estimates are obtained through numerical methods. The simulation studies were implemented to evaluate the behavior of these maximum likelihood estimations and demonstrate the good performance of MLEs by the multivariate depth-based records. Also, we have applied our proposed depth-based maximum likelihood estimation of a real data set related to the drought of Kermanshah city, Iran, which shows very satisfactory compared to the maximum likelihood estimation with complete data.
MSC codes: 62H12, 62G32.
ResearchGate has not been able to resolve any references for this publication.