Within contemporary science, it is common practice to compare data points to the average, i.e., to the statistical mean. Because this practice is so familiar, it might at first appear not to be the sort of thing that requires explanation. But recent research in cognitive science and in the history of science gives us reason to adopt the opposite perspective. Cognitive science research on the ways people ordinarily make sense of the world suggests that, instead of using a purely statistical (...) notion of the average, people tend to use a value-laden notion of the normal. Similarly, historical research on the scientific practices used in other time periods indicates that, prior to the nineteenth century, scientists tended to make use of a notion of the normal that was overtly value-laden. These findings give us reason to rethink certain familiar facts about scientific practice. In particular, they suggest that the fact that scientists so often make use of the statistical average should be seen as a highly surprising fact, the sort of thing that calls out for explanation. (shrink)

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.

ABSTRACT Average utilitarianism and several related axiologies, when paired with the standard expectational theory of decision-making under risk and with reasonable empirical credences, can find their practical prescriptions overwhelmingly determined by the minuscule probability that the agent assigns to solipsism—that is, to the hypothesis that there is only one welfare subject in the world, namely, herself. This either (i) constitutes a reductio of these axiologies, (ii) suggests that they require bespoke decision theories, or (iii) furnishes an unexpected argument for ethical (...) egoism. (shrink)

When faced with uncertainty, consequentialists often advocate choosing the option with the largest expected utility, as calculated using the arithmetic average. I provide some arguments to suggest that instead, one should consider choosing the option with the largest geometric average of utility. I explore the difference between these two approaches in a variety of ethical dilemmas and argue that geometric averaging has some appealing properties as a normative decision-making tool.

This paper compares two well-known arguments in the units of selection literature, one due to , the other due to . Both arguments concern the legitimacy of averaging fitness values across contexts and making inferences about the level of selection on that basis. The first three sections of the paper shows that the two arguments are incompatible if taken at face value, their apparent similarity notwithstanding. If we accept Sober and Lewontin's criterion for when averaging genic fitnesses across diploid genotypes (...) is illegitmate, we cannot accept Sober and Wilson's criterion for when averaging individual fitnesses across groups is illegitimate, and vice versa. The final section suggests a possible way of reconciling the two arguments, by invoking an ambiguity in the concept of genic selection. (shrink)

For many philosophers working in the area of Population Ethics, it seems that either they have to confront the Repugnant Conclusion , or they have to confront the Non-Identity Problem . To them it seems there is no escape, they either have to face one problem or the other. However, there is a way around this, allowing us to escape the Repugnant Conclusion, by using what I will call Negative Average Preference Utilitarianism – which though similar to anti-frustrationism, has some (...) important differences in practice. Current “positive” forms of utilitarianism have struggled to deal with the Repugnant Conclusion, as their theory actually entails this conclusion; however, it seems that a form of Negative Average Preference Utilitarianism easily escapes this dilemma. (shrink)

This article investigates the semantics of sentences that express numerical averages, focusing initially on cases such as 'The average American has 2.3 children'. Such sentences have been used both by linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and (...) Stephen Yablo have used them to provide evidence that apparent singular reference need not be taken as ontologically committing. We develop a fully general and independently justified compositional semantics in which such constructions are assigned truth conditions that are not ontologically problematic, and show that our analysis is superior to all extant rivals. Our analysis provides evidence that a good semantics yields a sensible ontology. It also reveals that natural language contains genuine singular terms that refer to numbers. (shrink)

With the advent of the era of artificial intelligence, deep reinforcement learning has achieved unprecedented success in high-dimensional and large-scale artificial intelligence tasks. However, the insecurity and instability of the DRL algorithm have an important impact on its performance. The Soft Actor-Critic algorithm uses advanced functions to update the policy and value network to alleviate some of these problems. However, SAC still has some problems. In order to reduce the error caused by the overestimation of SAC, we propose a new (...) SAC algorithm called Averaged-SAC. By averaging the previously learned action-state estimates, it reduces the overestimation problem of soft Q-learning, thereby contributing to a more stable training process and improving performance. We evaluate the performance of Averaged-SAC through some games in the MuJoCo environment. The experimental results show that the Averaged-SAC algorithm effectively improves the performance of the SAC algorithm and the stability of the training process. (shrink)

Conventionally, multilevel analysis of covariance (ML-ANCOVA) has been the recommended approach for analyzing treatment effects in quasi-experimental multilevel designs with treatment application at the cluster-level. In this paper, we introduce the generalized ML-ANCOVA with linear effect functions that identifies average and conditional treatment effects in the presence of treatment-covariate interactions. We show how the generalized ML-ANCOVA model can be estimated with multigroup multilevel structural equation models that offer considerable advantages compared to traditional ML-ANCOVA. The proposed model takes into account measurement (...) error in the covariates, sampling error in contextual covariates, treatment-covariate interactions, and stochastic predictors. We illustrate the implementation of ML-ANCOVA with an example from educational effectiveness research where we estimate average and conditional effects of early transition to secondary schooling on reading comprehension. (shrink)

Jeffrey conditioning tells an agent how to update her priors so as to grant a given probability to a particular event. Weighted averaging tells an agent how to update her priors on the basis of testimonial evidence, by changing to a weighted arithmetic mean of her priors and another agent’s priors. We show that, in their respective settings, these two seemingly so different updating rules are axiomatized by essentially the same invariance condition. As a by-product, this sheds new light on (...) the question how weighted averaging should be extended to deal with cases when the other agent reveals only parts of her probability distribution. The combination of weighted averaging and Jeffrey conditioning is a comprehensive updating rule to deal with such cases, which is again axiomatized by invariance under embedding. We conclude that, even though one may dislike Jeffrey conditioning or weighted averaging, the two make a natural pair when a policy for partial testimonial evidence is needed. (shrink)

Formal and social epistemologists have devoted significant attention to the question of how to aggregate the credences of a group of agents who disagree about the probabilities of events. Moss and Pettigrew argue that group credences can be a linear mean of the credences of each individual in the group. By contrast, I argue that if the epistemic value of a credence function is determined solely by its accuracy, then we should, where possible, aggregate the underlying statistical models that individuals (...) use to generate their credence functions, using “stacking” techniques from statistics and machine learning first developed by Wolpert. (shrink)

Average-NPs, such as the one in the title of this paper, have been claimed to be ‘linguistically identical’ to any other definite-NPs but at the same time to be ‘semantically inconsistent’ with these other definite-NPs. To some this is an ironclad proof of the irrelevance of semantics to linguistics. We argue that both of the initial claims are wrong: average-NPs are not ‘linguistically identical’ to other definite-NPs but instead show a number of interesting divergences, and we provide a plausible semantic (...) account for them that is not ‘semantically inconsistent’ with the account afforded other definite-NPs but in fact blends quite nicely with one standard account of the semantics for NPs. (shrink)

The origin of the genetic code has been attributed in part to an accidental assignment of codons to amino acids. Although several lines of evidence indicate the subsequent expansion and improvement of the genetic code, the hypothesis of Francis Crick concerning a frozen accident occurring at the early stage of genetic code evolution is still widely accepted. Considering Crick’s hypothesis, mathematical descriptions of hypothetical scenarios involving a huge number of possible coexisting random genetic codes could be very important to explain (...) the origin and evolution of a selected genetic code. This work aims to contribute in this regard, that is, it provides a theoretical framework in which statistical parameters of error functions are calculated. Given a genetic code and an amino acid property, the functional code robustness is estimated by means of a known error function. In this work, using analytical calculations, general expressions for the average and standard deviation of the error function distributions of completely random codes with standard stop codons were obtained. As a possible biological application of these results, any set of amino acids and any pure or mixed amino acid properties can be used in the calculations, such that, in case of having to select a set of amino acids to create a genetic code, possible advantages of natural selection of the genetic codes could be discussed. (shrink)

Within contemporary science, it is common practice to compare data points to the _average_, i.e., to the statistical mean. Because this practice is so familiar, it might at first appear not to be the sort of thing that requires explanation. But recent research in cognitive science gives us reason to adopt the opposite perspective. Research on the cognitive processes involved in people’s ordinary efforts to make sense of the world suggests that, instead of using a purely statistical notion of the (...) average, people tend to use a value-laden notion of the _normal_. This finding about ordinary cognition gives us reason to rethink certain familiar facts about scientific practice. In particular, it suggests that the fact that scientists so often make use of the statistical average should be seen as a highly surprising fact, the sort of thing that calls out for explanation. To understand it, we turn to work in the history of science, and especially to work on the ways in which the practice of science changed over the course of the 19th century. (shrink)

There has been increasing interest in recent years in the development of approaches to estimate causal effects when the number of potential confounders is prohibitively large. This growth in interest has led to a number of potential estimators one could use in this setting. Each of these estimators has different operating characteristics, and it is unlikely that one estimator will outperform all others across all possible scenarios. Coupling this with the fact that an analyst can never know which approach is (...) best for their particular data, we propose a synthetic estimator that averages over a set of candidate estimators. Averaging is widely used in statistics for problems such as prediction, where there are many possible models, and averaging can improve performance and increase robustness to using incorrect models. We show that these ideas carry over into the estimation of causal effects in high-dimensional scenarios. We show theoretically that averaging provides robustness against choosing a bad model, and show empirically via simulation that the averaging estimator performs quite well, and in most cases nearly as well as the best among all possible candidate estimators. Finally, we illustrate these ideas in an environmental wide association study and see that averaging provides the largest benefit in the more difficult scenarios that have large numbers of confounders. (shrink)

In this paper, we propose the model averaging estimation method for multiplicative error model and construct the corresponding weight choosing criterion based on the Kullback–Leibler divergence with a hyperparameter to avoid the problem of overfitting. The resulting model average estimator is proved to be asymptotically optimal. It is shown that the Kullback–Leibler model averaging estimator asymptotically minimizes the in-sample Kullback–Leibler divergence and improves the forecast accuracy of out-of-sample even under different loss functions. In simulations, we show that the KLMA estimator (...) compares favorably with smooth-AIC estimator, smooth-BIC estimator, and Mallows model averaging estimator, especially when some nonlinear noise is added to the data generation process. The empirical applications in the daily range of S&P500 and price duration of IBM show that the out-of-sample forecasting capacity of the KLMA estimator is better than that of other methods. (shrink)

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, (...) we show that, for any nonexpansive linear operator T on a separable Hilbert space and any element f , it is possible to compute a bound on the rate of convergence of Anf from T , f , and the norm f ∗ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2 (X ), then there is a computable bound on the rate of convergence of the sequence Anf. (shrink)

Good scientific explanations sometimes appear to make use of averages. Using concrete examples from current economic theory, I argue that some confusions about how averages might work in explanations lead to both philosophical and economic problems about the interpretation of the theory. I formulate general conditions on potentially proper uses of averages to refine a notion of average explanation. I then try to show how this notion provides a means for resolving longstanding philosophical problems in economics and other quantitative social (...) sciences. (shrink)

Trapezoidal cubic fuzzy numbers (TzCFNs) are an extraordinary cubic fuzzy set on a real number set. TzCFNs are useful for dealing with well-known quantities in decision data and decision making problems themselves. This paper is about multi-attribute group decision making problems in which the attribute values are stated with TzCFNs, which are solved by developing a new decision method based on power average operators of TzCFNs. The new operation laws for TzCFNs are given. Hereby, the power average operator of real (...) numbers is extended to four kinds of power average operators of TzCFNs, involving the power average operator of TzCFNs, the weighted power average operator of TzCFNs, the power ordered weighted average operator of TzCFNs, and the power hybrid average operator of TzCFNs. In the proposed group decision method, the individual overall evaluation values of alternatives are generated by using the power average operator of TzCFNs. Applying the hybrid average operator of TzCFNs, the specific general evaluation standards of alternatives are then combined into the collective ones, which are used to rank the alternatives. The example analysis shows the practicality and effectiveness of the proposed method. (shrink)

It has sometimes been claimed (usually without evidence) that the harm caused by an individual's participation in a greenhouse-gas-intensive economy is negligible. Using data from several contemporary sources, this paper attempts to estimate the harm done by an average American. This estimate is crude, and further refinements are surely needed. But the upshot is that the average American is responsible, through his/her greenhouse gas emissions, for the suffering and/or deaths of one or two future people.

Heterogeneous treatment effects represent a major issue for medicine as they undermine reliable inference and clinical decision-making. To overcome the issue, the current vision of precision and personalized medicine acknowledges the need to control individual variability in response to treatment. In this paper, we argue that gene-treatment-environment interactions (G × T × E) undermine inferences about individual treatment effects from the results of both genomics-based methodologies—such as genome-wide association studies (GWAS) and genome-wide interaction studies (GWIS)—and randomized controlled trials (RCTs). Then, (...) we argue that N-of-1 trials can be a solution to overcome difficulties in handling individual variability in treatment response. Although this type of trial has been suggested as a promising strategy to assess individual treatment effects, it nonetheless has limitations that limit its use in everyday clinical practice. We analyze the existing variability within the designs of N-of-1 trials in terms of a continuum where each design prioritizes epistemic and pragmatic considerations. We then support wider use of the designs located at the pragmatic end of the explanatory-pragmatic continuum. (shrink)

We treat the classical dynamics of the hydrogen atom in perpendicular electric and magnetic fields as a celestial mechanics problem. By expressing the Hamiltonian in appropriate action–angle variables, we separate the different time scales of the motion. The method of averaging then allows us to reduce the system to two degrees of freedom, and to classify the most important periodic orbits.

Contemporary population ethics is dominated by views that aggregate by summing, whether of well-being or of some construct based on well-being. In contrast, average well-being is generally considered axiologically irrelevant. To many of us, however, the number of future people does not seem important, as long as it is sufficient to enable rich and varied life experiences, and as long as the population continues throughout time. It therefore seems relatively plausible to aggregate future well-being by averaging. In particular, it seems (...) plausible to value high average well-being at any particular time, and to do so for all future times. I present a time-sensitive version of the Average View that underpins such axiological intuitions. I also address a series of issues and objections that confront such a view. (shrink)

We propose an "explanation scheme" for why the Gibbs phase average technique in classical equilibrium statistical mechanics works. Our account emphasizes the importance of the Khinchin-Lanford dispersion theorems. We suggest that ergodicity does play a role, but not the one usually assigned to it.

The problem of missingness in observational data is ubiquitous. When the confounders are missing at random, multiple imputation is commonly used; however, the method requires congeniality conditions for valid inferences, which may not be satisfied when estimating average causal treatment effects. Alternatively, fractional imputation, proposed by Kim 2011, has been implemented to handling missing values in regression context. In this article, we develop fractional imputation methods for estimating the average treatment effects with confounders missing at random. We show that the (...) fractional imputation estimator of the average treatment effect is asymptotically normal, which permits a consistent variance estimate. Via simulation study, we compare fractional imputation’s accuracy and precision with that of multiple imputation. (shrink)

Seemingly every theory of population ethics is confronted with unpalatable implications. While various approaches to the subject have been taken, including non-consequentialist approaches, this area has been dominated by utilitarian thought. The two main approaches to population ethics have been total utilitarianism () and average utilitarianism (). According to TU, we should seek to bring about the state of affairs that maximizes the total amount of happiness. According to AU, we should seek to bring about the state of affairs that (...) maximizes average per capita happiness. Both theories have been afflicted by seemingly strong objections, and as a result, numerous variations and hybrids have been introduced. Despite the widespread disagreement in the field, though, a near consensus has developed in rejecting AU as an absurd view. In this article, however, I will go against the grain and argue that AU is the theory of population ethics that we should endorse. (shrink)

This paper examines the creation of what I have called “The Other Average Man” of nineteenth-century science. This other average man was created during the course of the career of the Belgian polymath Adolphe Quetelet and his development of physique sociale and the controversial l’homme moyen. Rather than discuss Quetelet’s abstract average man, which has received considerable attention, the paper focuses on the real average men that Quetelet imagined would do the work of providing data for physique sociale. To do (...) so, it investigates three moments in Quetelet’s administrative career in Belgium: his early life in Ghent and Brussels during the brief polity of the United Kingdom of the Netherlands, his remarkable transformation of the Académie Royale de Belgique in the 1820s, and an 1846 debate at the Academy on the possible determinism entailed by Quetelet’s statistical account of human behavior. These stories not only point to the creation of another average man in Quetelet’s social physics, but also demonstrate the importance of Belgium’s unique historical context for understanding one of nineteenth-century Europe’s most enduring contributions to the social sciences. (shrink)