We study a model of social learning on rooted regular trees. An agent is stationed at each vertex of $\mathbb{T}_{m}$, the rooted tree in which each vertex has precisely m children, and at any time step $t \in \mathbb{N}_{0}$, the agent is allowed to select one of two available technologies: B and R. Let the technology chosen by the agent at vertex v of $\mathbb{T}_{m}$, at time step t, be $C_{t}(v)$. We begin with the independent and identically distributed (i.i.d.) collection $\{C_{0}(v)\,:\, v \in \mathbb{T}_{m}\}$, where $C_{0}(v)=B$ with probability $\pi_{0}$. During the epoch t, the agent at vertex v performs an experiment that results in success with probability $p_{B}$ if $C_{t}(v)=B$, and with probability $p_{R}$ if $C_{t}(v)=R$. If the children of v are denoted $v_{1}, \ldots, v_{m}$, the agent at v updates their technology to $C_{t+1}(v)=B$ if the number of successes among all $v_{i}$ (where $i \in \{1,2,\ldots,m\}$) with $C_{t}(v_{i})=B$ exceeds, strictly, the number of successes among all $v_{j}$ (where $j \in \{1,2,\ldots,m\}$) with $C_{t}(v_{j})=R$. If these two numbers are equal then the agent at v sets $C_{t+1}(v)=B$ with probability $1/2$. In all other cases, $C_{t+1}(v)=R$. We show that $\{C_{t}(v)\,:\, v \in \mathbb{T}_{m}\}$ is i.i.d. as well, with $C_{t}(v)=B$ with probability $\pi_{t}$, where the sequence $\{\pi_{t}\}_{t \in \mathbb{N}_{0}}$ converges to a fixed point $\pi$, in [0, 1], of a function $g_{m}$. We show that for $m \geqslant 3$, there exists a $p(m) \in (0,1)$ such that $g_{m}$ has the unique fixed point $1/2$ when $p \leqslant p(m)$, and three distinct fixed points, of the form $\alpha$, $1/2$, and $1-\alpha$, for some $\alpha \in [0,1/2)$ when $p > p(m)$. When $m=3$, $p_{B}=1$, and $p_{R} \in [0,1)$, we show that the function $g_{3}$ (i) has a unique fixed point, 1, when $p_{R} < \sqrt{3}-1$, (ii) has two distinct fixed points, one of which is 1, when $p_{R} = \sqrt{3}-1$, and (iii) has three distinct fixed points, one of which is 1, when $p_{R} > \sqrt{3}-1$. When $g_{m}$ has multiple fixed points, we also specify which of these fixed points $\pi$ equals, depending on $\pi_{0}$. Finally, for $m=2$, we describe the behaviour of $g_{2}$ for all values of $p_{B}$ and $p_{R}$.

The present paper introduces a new explanatory item response model to account for the learning that takes place during a psychometric test due to the repeated use of the operations involved in the items. The proposed model is an extension of the operation-specific learning model (Fischer and Formann in Appl Psychol Meas 6:397–416, 1982; Scheiblechner in Z für Exp Angew Psychol 19:476–506, 1972; Spada in Spada and Kempf (eds.) Structural models of thinking and learning, Huber, Bern, Germany, pp 227–262, 1977). The paper discusses special cases of the model, which, together with the general formulation, differ in the type of response in which the model states that learning occurs: (1) correct and incorrect responses equally (non-contingent learning); (2) correct responses only (contingent learning); and (3) correct and incorrect responses to a different extent (differential contingent learning). A Bayesian framework is adopted for model estimation and evaluation. A simulation study is conducted to examine the performance of the estimation and evaluation methods in recovering the true parameters and selecting the true model. Finally, an empirical study is presented to illustrate the applicability of the model to detect learning effects using real data.

In the last decade, reading research has seen a paradigmatic shift. A new wave of computational models of orthographic processing that offer various forms of noisy position or context-sensitive coding have revolutionized the field of visual word recognition. The influx of such models stems mainly from consistent findings, coming mostly from European languages, regarding an apparent insensitivity of skilled readers to letter order. Underlying the current revolution is the theoretical assumption that the insensitivity of readers to letter order reflects the special way in which the human brain encodes the position of letters in printed words. The present article discusses the theoretical shortcomings and misconceptions of this approach to visual word recognition. A systematic review of data obtained from a variety of languages demonstrates that letter-order insensitivity is neither a general property of the cognitive system nor a property of the brain in encoding letters. Rather, it is a variant and idiosyncratic characteristic of some languages, mostly European, reflecting a strategy of optimizing encoding resources, given the specific structure of words. Since the main goal of reading research is to develop theories that describe the fundamental and invariant phenomena of reading across orthographies, an alternative approach to model visual word recognition is offered. The dimensions of a possible universal model of reading, which outlines the common cognitive operations involved in orthographic processing in all writing systems, are discussed.

One general and three specialized models of the Bush–Mosteller type are presented to describe the kind of learning experiment where the response of the learner is always reinforced. Inhomogeneity is admitted. The random sequences of response probabilities and of responses associated with the different models are considered. Information about the existence and the distribution of asymptotic response probabilities is provided. The stress is on sufficient and necessary conditions for convergence (a.s. or with positive probability) of the response sequence, which is what ‘learning' means.