Probabilities,causation, and logic programming in conditional reasoning: reply to Stenning and Van Lambalgen (2016)

Oaksford and Chater (2014 Oaksford, M., &; Chater, N. (2014). Probabilistic single function dual process theory and logic programming as approaches to non-monotonicity in human vs. artificial reasoning. Thinking and Reasoning, 20, 269–295. doi:10.1080/13546783.2013.877401[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Thinking and Reasoning, 20, 269–295) critiqued the logic programming (LP) approach to nonmonotonicity and proposed that a Bayesian probabilistic approach to conditional reasoning provided a more empirically adequate theory. The current paper is a reply to Stenning and van Lambalgen's rejoinder to this earlier paper entitled ‘Logic programming, probability, and two-system accounts of reasoning: a rejoinder to Oaksford and Chater’ (2016) in Thinking and Reasoning. It is argued that causation is basic in human cognition and that explaining how abnormality lists are created in LP requires causal models. Each specific rejoinder to the original critique is then addressed. While many areas of agreement are identified, with respect to the key differences, it is concluded the current evidence favours the Bayesian approach, at least for the moment.     

Inferring hidden causal structure

We used a new method to assess how people can infer unobserved causal structure from patterns of observed events. Participants were taught to draw causal graphs, and then shown a pattern of associations and interventions on a novel causal system. Given minimal training and no feedback, participants in Experiment 1 used causal graph notation to spontaneously draw structures containing one observed cause, one unobserved common cause, and two unobserved independent causes, depending on the pattern of associations and interventions they saw. We replicated these findings with less-informative training (Experiments 2 and 3) and a new apparatus (Experiment 3) to show that the pattern of data leads to hidden causal inferences across a range of prior constraints on causal knowledge.     

贝叶斯因子及其在JASP中的实现

统计推断在科学研究中起到关键作用, 然而当前科研中最常用的经典统计方法——零假设检验(Null hypothesis significance test, NHST)却因难以理解而被部分研究者误用或滥用。有研究者提出使用贝叶斯因子(Bayes factor)作为一种替代和(或)补充的统计方法。贝叶斯因子是贝叶斯统计中用来进行模型比较和假设检验的重要方法, 其可以解读为对零假设H₀或者备择假设H₁的支持程度。其与NHST相比有如下优势：同时考虑H₀和H₁并可以用来支持H₀、不“严重”地倾向于反对H₀、可以监控证据强度的变化以及不受抽样计划的影响。目前, 贝叶斯因子能够很便捷地通过开放的统计软件JASP实现, 本文以贝叶斯t检验进行示范。贝叶斯因子的使用对心理学研究者来说具有重要的意义, 但使用时需要注意先验分布选择的合理性以及保持数据分析过程的透明与公开。     

Reasoning about the future: Doom and Beauty

According to the Doomsday Argument we have to rethink the probabilities we assign to a soon or not so soon extinction of mankind when we realize that we are living now, rather early in the history of mankind. Sleeping Beauty finds herself in a similar predicament: on learning the date of her first awakening, she is asked to re-evaluate the probabilities of her two possible future scenarios. In connection with Doom, I argue that it is wrong to assume that our ordinary probability judgements do not already reflect our place in history: we justify the predictive use we make of the probabilities yielded by science (or other sources of information) by our knowledge of the fact that we live now, a certain time before the possible occurrence of the events the probabilities refer to. Our degrees of belief should change drastically when we forget the date—importantly, this follows without invoking the “Self Indication Assumption”. Subsequent conditionalization on information about which year it is cancels this probability shift again. The Doomsday Argument is about such probability shifts, but tells us nothing about the concrete values of the probabilities—for these, experience provides the only basis. Essentially the same analysis applies to the Sleeping Beauty problem. I argue that Sleeping Beauty “thirders” should be committed to thinking that the Doomsday Argument is ineffective; whereas “halfers” should agree that doom is imminent—but they are wrong.     

Extensions of Rasch's multiplicative poisson model

Consideration will be given to a model developed by Rasch that assumes scores observed on some types of attainment tests can be regarded as realizations of a Poisson process. The parameter of the Poisson distribution is assumed to be a product of two other parameters, one pertaining to the ability of the subject and a second pertaining to the difficulty of the test. Rasch's model is expanded by assuming a prior distribution, with fixed but unknown parameters, for the subject parameters. The test parameters are considered fixed. Secondly, it will be shown how additional between- and within-subjects factors can be incorporated. Methods for testing the fit and estimating the parameters of the model will be discussed, and illustrated by empirical examples.     

Statistical inference based on latent ability estimates

The quality of approximations to first and second order moments (e.g., statistics like means, variances, regression coefficients) based on latent ability estimates is being discussed. The ability estimates are obtained using either the Rasch, or the two-parameter logistic model. Straightforward use of such statistics to make inferences with respect to true latent ability is not recommended, unless we account for the fact that the basic quantities are estimates. In this paper true score theory is used to account for the latter; the counterpart of observed/true score being estimated/true latent ability. It is shown that statistics based on the true score theory are virtually unbiased if the number of items presented to each examinee is larger than fifteen. Three types of estimators are compared: maximum likelihood, weighted maximum likelihood, and Bayes modal. Furthermore, the (dis)advantages of the true score method and direct modeling of latent ability is discussed.     

The relationship between Fisher's exact test and Pearson's chi-square test: A bayesian perspective

It is demonstrated in this paper that two major tests for 2 × 2 talbes are highly related from a Bayesian perspective. Although it is well-known that Fisher's exact and Pearson's chi-square tests are asymptotically equivalent, the present analysis shows that a formal similarity also exists in small samples. The key assumption that leads to the resemblance is the presence of a continuous parameter measuring association. In particular, it is shown that Pearson's probability can be obtained by integrating a two-moment approximation to the posterior distribution of the log-odds ratio. Furthermore, Pearson's chi-square test gave an excellent approximation to the actual Bayes probability in all 2×2 tables examined, except for those with extremely disproportionate marginal frequencies.     

How surprising is a simple pattern? Quantifying "Eureka!"

Simple patterns are compelling. When all the observed facts fit into a simple theory or "story," we are intuitively convinced that the pattern must be real rather than random. But how surprising is a simple pattern, really? That is, given a pattern of featural data, such as the properties of a set of objects, how unlikely would the pattern be if they were actually generated at random? In conventional statistics dealing with patterns of numbers, this type of question would be answered by reference to a null distribution such as the t distribution. This paper gives the analogous answer in the realm of concept learning, that is, the formation of generalizations from patterns of featural data. Using a formal but psychologically valid definition of complexity, I derive and exhibit the distribution of subjective complexity under the hypothesis of no pattern. This leads directly to a number of applications, including a statistical test indicating whether an observed pattern is sufficiently simple that it is not likely to have been an accident: literally, the "significance of simplicity."     

Empirical bayes estimates of domain scores under binomial and hypergeometric distributions for test scores

We introduce two simple empirical approximate Bayes estimators (EABEs)— and —for estimating domain scores under binomial and hypergeometric distributions, respectively. Both EABEs (derived from corresponding marginal distributions of observed test scorex without relying on knowledge of prior domain score distributions) have been proven to hold -asymptotic optimality in Robbins' sense of convergence in mean. We found that, where and are the monotonized versions of and under Van Houwelingen's monotonization method, respectively, the convergence rate of the overall expected loss of Bayes risk in either or depends on test length, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, and outperform their maximum likelihood counterparts over the middle range of domain scales. In terms of mean-squared error, we also found that: (a) given a unimodal prior distribution of domain scores, performs better than both and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterogeneity; (b) performs as well as when prior distribution is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimodal prior distribution exists.The authors are indebted to both anonymous reviewers, especially Reviewer 2, and the Editor for their invaluable comments and suggestions. Thanks are also due to Yuan-Chin Chang and Chin-Fu Hsiao for their help with our simulation and programming work.     

An investigation of hierarchical Bayes procedures in item response theory

Hierarchical Bayes procedures for the two-parameter logistic item response model were compared for estimating item and ability parameters. Simulated data sets were analyzed via two joint and two marginal Bayesian estimation procedures. The marginal Bayesian estimation procedures yielded consistently smaller root mean square differences than the joint Bayesian estimation procedures for item and ability estimates. As the sample size and test length increased, the four Bayes procedures yielded essentially the same result.The authors wish to thank the Editor and anonymous reviewers for their insightful comments and suggestions.