### PAPER Rich analysis and rational models: inferring individual behavior from infant looking data

"... Studies of infant looking times over the past 50 years have provided profound insights about cognitive development, but their dependent measures and analytic techniques are quite limited. In the context of infants ’ attention to discrete sequential events, we show how a Bayesian data analysis approa ..."

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Studies of infant looking times over the past 50 years have provided profound insights about cognitive development, but their dependent measures and analytic techniques are quite limited. In the context of infants ’ attention to discrete sequential events, we show how a Bayesian data analysis approach can be combined with a rational cognitive model to create a rich data analysis framework for infant looking times. We formalize (i) a statistical learning model, (ii) a parametric linking between the learning model’s beliefs and infants ’ looking behavior, and (iii) a data analysis approach and model that infers parameters of the cognitive model and linking function for groups and individuals. Using this approach, we show that recent findings from Kidd, Piantadosi and Aslin (2012) of a U-shaped relationship between look-away probability and stimulus complexity even holds within infants and is not due to averaging subjects with different types of behavior. Our results indicate that individual infants prefer stimuli of intermediate complexity, reserving attention for events that are moderately predictable given their probabilistic expectations about the world.

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"... © 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to ..."

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© 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Entropic Priors for Short-Term Stochastic Process Classification

### Posterior Normality and Reference Priors for . . .

, 2009

"... In this article, we study asymptotic normality of the posterior distribution of the natural parameter in an exponential family based on independent and identically distributed (i.i.d.) data, that is, in terms of expected Kullback-Leibler divergence, when the number of parameters p is increasing wit ..."

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In this article, we study asymptotic normality of the posterior distribution of the natural parameter in an exponential family based on independent and identically distributed (i.i.d.) data, that is, in terms of expected Kullback-Leibler divergence, when the number of parameters p is increasing with the sample size n. We use this to generate an asymptotic expansion of the Shannon mutual information, p is allowed to increase with n. The leading term, dependent on the prior in the asymptotic expansion, can be optimized to give Jeffreys’ prior as the reference prior in the absence of nuisance parameters. In two examples, we determine the rates at which p = pn can be allowed to increase while still retaining the reference prior.

### Examining the Role of a Non-informative Prior Function Through Weakly Informative Prior Densities

"... A non-informative prior function is discussed from the standpoint of a weakly informative prior density, though it has been pursued in relation only to the sampling densities. We note that a weakly informative prior density is useful for examining a no-informative prior function. Some disadvantages ..."

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A non-informative prior function is discussed from the standpoint of a weakly informative prior density, though it has been pursued in relation only to the sampling densities. We note that a weakly informative prior density is useful for examining a no-informative prior function. Some disadvantages of a non-informative prior function are pointed out. On the other hand, a non-informative prior function is compared favorably with a degenerated prior density on an unknown point. The role of the posterior density compared with the marginal likelihood is discussed.

### Natural Induction: An Objective Bayesian Approach

"... Abstract. The statistical analysis of a sample taken from a finite population is a classic problem for which no generally accepted objective Bayesian results seem to exist. Bayesian solutions to this problem may be very sensitive to the choice of the prior, and there is no consensus as to the approp ..."

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Abstract. The statistical analysis of a sample taken from a finite population is a classic problem for which no generally accepted objective Bayesian results seem to exist. Bayesian solutions to this problem may be very sensitive to the choice of the prior, and there is no consensus as to the appropriate prior to use. This paper uses new developments in reference prior theory to justify and generalize Perks (1947) ([15]) ‘rule of succession ’ — determining the probability that a new element from a population will have a property, given that all n previous elements from a random sample possessed the property — and to propose a new objective Bayesian solution to the ‘law of natural induction ’ problem — determining the probability that all elements in a finite population have the property, given that all previous elements had the property. The prior used for the first problem is the reference prior for an underlying hypergeometric probability model, a prior first suggested by Jeffreys (1946) ([10]) and recently justified on the basis of an exchange-ability argument in Berger, Bernardo and Sun (2009) ([4]). The reference prior in the second problem arises as a modification to this prior that results from declaring the quantity of interest to be whether or not all the elements in the finite population have the property under scrutiny. Inducción en las Ciencias de la Naturaleza:

### Is there a two-envelope paradox?

"... We address the two-envelope paradox, studied over a number of years. A sta-tistical analysis is provided based on a classical inference point of view as well as from a Bayesian perspective. In the classical analysis it is stressed that there is no paradox but rather an erroneous use of an expected v ..."

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We address the two-envelope paradox, studied over a number of years. A sta-tistical analysis is provided based on a classical inference point of view as well as from a Bayesian perspective. In the classical analysis it is stressed that there is no paradox but rather an erroneous use of an expected value, something that has been said before by some authors. A different, related problem is discussed briefly where the issues of “utility ” and “personal preferences ” arise naturaly. From a Bayesian point of view, the analysis centers on the degeneracy caused by the implied (though correct), improper prior which arises from the explicit argument used when stating the paradoxical result. The implied prior coincides with that in many articles that claim that underneath the paradox is the mis-taken interpretation of infinite quantities. The classical analysis of this note is based on some ideas presented in O’Reilly (2006) and the Bayesian analysis on Quintana and O’Reilly (2008). The aim of this working document is to have both analyses in the same manuscript.

### Principles of Statistical Inference

"... Statistical theory aims to provide a foundation for studying the collection and interpretation of data that does not depend on the particular details of the substantive field in which the data is being considered. This gives a systematic way to approach new problems, and a common language for summar ..."

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Statistical theory aims to provide a foundation for studying the collection and interpretation of data that does not depend on the particular details of the substantive field in which the data is being considered. This gives a systematic way to approach new problems, and a common language for summarizing results; ideally the foundations and common language ensure that statistical aspects of one study, or of several studies on closely related phenomena, can in broad terms be understood by the non-specialist. We discuss some principles of statistical inference, to outline how these are, or could be, used to inform the interpretation of results, and to provide a greater degree of coherence for the foundations of statistics. 1