needed reduction and spine strategies for the lambda calculus abstract a redex r in a lambda term m is called needed if in every reduction of m to normal form some residual of r is contracted among others the following results are proved 1 r is needed in m iff r is contracted in the leftmost reduction path of m 2 let r m0 m1 m2 reduce redexes ri mi mi 1 and have the property that i j i rj is needed in mj then r is normalising i e if m0 has a normal form then r is finite and terminates at that normal form 3 neededness is an undecidable property but has several efficiently decidable approximations various versions of the so called spine redexes

infinitary lambda calculus in a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems in this paper we perform the same task for the lambda calculus from the viewpoint of infinitary rewriting the bohm model of the lambda calculus can be seen as an infinitary term model in contrast to term rewriting there are several different possible notions of infinite term which give rise to different bohm like models which embody different notions of lazy or eager computation

active inference curiosity and insight this article offers a formal account of curiosity and insight in terms of active bayesian inference it deals with the dual problem of inferring states of the world and learning its statistical structure in contrast to current trends in machine learning e g deep learning we focus on how people attain insight and understanding using just a handful of observations which are solicited through curious behavior we use simulations of abstract rule learning and approximate bayesian inference to show that minimizing expected variational free energy leads to active sampling of novel contingencies this epistemic behavior closes explanatory gaps in generative models of the world thereby reducing uncertainty and satisfying curiosity we then move from epistemic learning to model selection or structure learning to show how abductive processes emerge when agents test plausible hypotheses about symmetries i e invariances or rules in their generative models the ensuing bayesian model reduction evinces

an approach to producing space time covariance functions on spheres for space time processes on global or large scales it is critical to use models that respect the earth s spherical shape the covariance functions of such processes should be not only positive definite on sphere time but also capable of capturing the dynamics of the processes well we develop space time covariance functions on sphere time that are flexible in producing space time interactions especially space time asymmetries our idea is to consider a sum of independent processes in which each process is obtained by applying a first order differential operator to a fully symmetric process on sphere time the resulting covariance functions can produce various types of space time interactions and give different covariance structures along different latitudes our approach yields explicit expressions for the covariance functions which has great advantages in computation moreover it applies equally well to generating asymmetric space time covariance functions on flat or other spatial domains we

computing bayes factors using a generalization of the savage dickey density ratio abstract we present a simple method for computing bayes factors the method derives from observing that in general a bayes factor can be written as the product of a quantity called the savage dickey density ratio and a correction factor both terms are easily estimated from posterior simulation in some cases it is possible to do these computations without ever evaluating the likelihood

highlights in infinitary rewriting and lambda calculus we present some highlights from the emerging theory of infinitary rewriting both for first order term rewriting systems and l calculus in the first section we introduce the framework of infinitary rewriting for first order rewrite systems so without bound variables we present a recent observation concerning the continuity of infinitary rewriting in the second section we present an excursion to the infinitary l calculus after the main definitions we mention a recent observation about infinite looping l terms that is terms that reduce in one step to themselves next we describe the fundamental trichotomy in the semantics of l calculus bohm trees levy longo trees and berarducci trees we conclude with a short description of a new refinement of bohm tree semantics called clocked semantics

classes of nonseparable spatio temporal stationary covariance functions abstract suppose that a random process z s t indexed in space and time has spatio temporal stationary covariance c h u where h d d 1 is a spatial lag and u is a temporal lag separable spatio temporal covariances have the property that they can be written as a product of a purely spatial covariance and a purely temporal covariance their ease of definition is counterbalanced by the rather limited class of random processes to which they correspond in this article we derive a new approach that allows one to obtain many classes of nonseparable spatio temporal stationary covariance functions and fit several such classes to spatio temporal data on wind speed over a region in the tropical western pacific ocean

levels of undecidability in rewriting undecidability of various properties of first order term rewriting systems is well known an undecidable property can be classified by the complexity of the formula defining it this classification gives rise to a hierarchy of distinct levels of undecidability starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy where quantification over function variables is allowed in this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies we consider properties related to normalization strong normalization weak normalization and dependency problems and properties related to confluence confluence local confluence and the unique normal form property for all of these we distinguish between the single term version and the uniform version where appropriate we also distinguish between ground and open terms most uniform properties are p 2 0 complete the particular problem of local confluence turns out to be p 2 0 complete for ground terms but only p 2 0 complete and thereby recursively enumerable for open terms the most surprising result concerns dependency pair problems without minimality flag we prove this problem to be p 1 1 complete hence not in the arithmetical hierarchy but properly in the analytic hierarchy some of our results are new or have appeared in our earlier publications others are based on folklore constructions and are included for completeness as their precise classifications have hardly been noticed previously

nonseparable stationary covariance functions for space time data geostatistical approaches to spatiotemporal prediction in environmental science climatology meteorology and related fields rely on appropriate covariance models this article proposes general classes of nonseparable stationary covariance functions for spatiotemporal random processes the constructions are directly in the space time domain and do not depend on closed form fourier inversions the model parameters can be associated with the data s spatial and temporal structures respectively and a covariance model with a readily interpretable space time interaction parameter is fitted to wind data from ireland

the dimple in gneiting s spatial temporal covariance model gneiting 2002 proposed a nonseparable covariance model for spatial temporal data in present paper we show that in certain circumstances his model possesses a counterintuitive dimple which detracts from its modelling appeal

statistical methods for regular monitoring data summary meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics particularly when there are little or no missing data this work investigates methods for modelling such data and ways of approximating the associated likelihood functions models for processes on the sphere crossed with time are emphasized especially models that are not fully symmetric in space time two approaches to obtaining such models are described the first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function the second is based on a representation of space time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time various models are applied to a data set of daily winds at 11 sites in ireland over 18 years spectral and space time domain diagnostic procedures are used to assess the quality of the fits the spectral in time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space time dependences of the data about as well

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

infinitary lambda calculus and discrimination of berarducci trees we propose an extension of lambda calculus for which the berarducci trees equality coincides with observational equivalence when we observe rootstable or rootactive behavior of terms in one direction the proof is an adaptation of the classical bohm out technique in the other direction the proof is based on confluence for strongly converging reductions in this extension

approximating likelihoods for large spatial data sets summary likelihood methods are often difficult to use with large irregularly sited spatial data sets owing to the computational burden even for gaussian models exact calculations of the likelihood for n observations require o n3 operations since any joint density can be written as a product of conditional densities based on some ordering of the observations one way to lessen the computations is to condition on only some of the past observations when computing the conditional densities we show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating through theoretical numerical and practical examples we show that there can often be considerable benefit in conditioning on some distant observations as well

infinitary combinatory reduction systems we define infinitary combinatory reduction systems icrss thus providing the first notion of infinitary higher order rewriting the systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary l calculus are special cases furthermore we generalise a number of known results from first order infinitary rewriting and infinitary l calculus to icrss in particular for fully extended left linear icrss we prove the well known compression property and for orthogonal icrss we prove that 1 if a set of redexes u has a complete development then all complete developments of u end in the same term and that 2 any tiling diagram involving strongly convergent reductions s and t can be completed iff at least one of s t and t s is strongly convergent we also prove an ancillary result of independent interest a set of redexes in an orthogonal icrs has a complete development iff the set has the so called finite jumps property

transfinite reductions in orthogonal term rewriting systems strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems otrss for these we prove the transfinite parallel moves lemma and the compressing lemma strongness is necessary as shown by counterexamples normal forms which we allow to be infinite are unique in contrast to normal forms strongly converging fair reductions result in normal forms

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

estimation and model identification for continuous spatial processes summary formal parameter estimation and model identification procedures for continuous domain spatial processes are introduced the processes are assumed to be adequately described by a linear model with residuals that follow a second order stationary gaussian random field and data are assumed to consist of noisy observations of the process at arbitrary sampling locations a general class of two dimensional rational spectral density functions with elliptic contours is used to model the spatial covariance function an iterative estimation procedure alleviates many of the computational difficulties of conventional maximum likelihood estimation for non lattice data the procedure is applied to several generated data sets and to an actual ground water data set

adapting innocent game models for the bohm tree λ theory we present a game model of the untyped calculus with equational theory equal to the bohm tree theory b which is universal i e every element of the model is definable by some term this answers a question of di gianantonio franco and honsell we build on our earlier work which uses the methods of innocent game semantics to develop a universal model inducing the maximal consistent sensible theory h to our knowledge these are the first syntax independent universal models of the untyped calculus

estimating space and space time covariance functions for large data sets a weighted composite likelihood approach in this article we propose two methods for estimating space and space time covariance functions from a gaussian random field based on the composite likelihood idea the first method relies on the maximization of a weighted version of the composite likelihood function while the second one is based on the solution of a weighted composite score equation this last scheme is quite general and could be applied to any kind of composite likelihood an information criterion for model selection based on the first estimation method is also introduced the methods are useful for practitioners looking for a good balance between computational complexity and statistical efficiency the effectiveness of the methods is illustrated through examples simulation experiments and by analyzing a dataset on ozone measurements

lazy productivity via termination abstract we present a procedure for transforming strongly sequential constructor based term rewriting systems trss into context sensitive trss in such a way that productivity of the input system is equivalent to termination of the output system thereby automated termination provers become available for proving productivity a trs is called productive if all its finite ground terms are constructor normalizing and all inductive constructor paths through the resulting possibly non wellfounded constructor normal form are finite to our knowledge this is the first complete transformation from productivity to termination the transformation proceeds in two steps i the strongly sequential trs is converted into a shallow trs where patterns do not have nested constructors ii the shallow trs is transformed into a context sensitive trs where rewriting below constructors and in arguments not consumed from is disallowed furthermore we show how lazy evaluation can be encoded by strong sequentiality thus extending our transformation to e g haskell programs finally we present a simple but fruitful extension of matrix interpretations to make them applicable for proving termination of context sensitive trss

cross covariance functions for multivariate random fields based on latent dimensions summary the problem of constructing valid parametric cross covariance functions is challenging we propose a simple methodology based on latent dimensions and existing covariance models for univariate random fields to develop flexible interpretable and computationally feasible classes of cross covariance functions in closed form we focus on spatio temporal cross covariance functions that can be nonseparable asymmetric and can have different covariance structures for instance different smoothness parameters in each component we discuss estimation of these models and perform a small simulation study to demonstrate our approach we illustrate our methodology on a trivariate spatio temporal pollution dataset from california and demonstrate that our cross covariance performs better than other competing models

blur generated non separable space time models statistical space time modelling has traditionally been concerned with separable covariance functions meaning that the covariance function is a product of a purely temporal function and a purely spatial function we draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant we show that this model has a non separable covariance function the model is well suited to a wide range of realistic problems which will be poorly fitted by separable models the model operates successively in time the spatial field at time t 1 is obtained by blurring the field at time t and adding a spatial random field the model is first introduced at discrete time steps and the limit is taken as the length of the time steps goes to 0 this gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals under certain conditions the blurring must be a gaussian smoothing kernel we also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously

space time covariance functions this work considers a number of properties of space time covariance functions and how these relate to the spatial temporal interactions of the process first it examines how the smoothness away from the origin of a space time covariance function affects for example temporal correlations of spatial differences models that are not smoother away from the origin than they are at the origin such as separable models have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments these results are used to obtain a parametric class of spectral densities whose corresponding space time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time second this work considers models that

non existent statman s double fixed point combinator does not exist indeed we give a negative answer to a problem of r statman concerning the existence in the lambda calculus of a fixed point combinatorysuch thaty si yholds

approximate bayesian inference in conditionally independent hierarchical models parametric empirical bayes models abstract we consider two stage models of the kind used in parametric empirical bayes peb methodology calling them conditionally independent hierarchical models we suppose that there are k units which may be experimental subjects cities study centers etcetera at the first stage the observation vectors yi for units i 1 k are independently distributed with densities p yi i or more generally p yi i at the second stage the unit specific parameter vectors i are iid with densities p i the peb approach proceeds by regarding the second stage distribution as a prior and noting that if were known inference about could be based on its posterior since is not known the simplest peb methods estimate the parameter by maximum likelihood or some variant and then treat as if it were known to be equal to this estimate although this procedure is sometimes satisfactory a well known defect is that it neglects the uncertainty due to the estimation of in this article w

bayesian model reduction and empirical bayes for group dcm studies this technical note describes some bayesian procedures for the analysis of group studies that use nonlinear models at the first within subject level e g dynamic causal models and linear models at subsequent between subject levels its focus is on using bayesian model reduction to finesse the inversion of multiple models of a single dataset or a single hierarchical or empirical bayes model of multiple datasets these applications of bayesian model reduction allow one to consider parametric random effects and make inferences about group effects very efficiently in a few seconds we provide the relatively straightforward theoretical background to these procedures and illustrate their application using a worked example this example uses a simulated mismatch negativity study of schizophrenia we illustrate the robustness of bayesian model reduction to violations of the commonly used laplace assumption in dynamic causal modelling and show how its recursive application can facilitate both classical and bayesian inference about group differences finally we consider the application of these empirical bayesian procedures to classification and prediction

post hoc bayesian model selection this note describes a bayesian model selection or optimization procedure for post hoc inferences about reduced versions of a full model the scheme provides the evidence marginal likelihood for any reduced model as a function of the posterior density over the parameters of the full model it rests upon specifying models through priors on their parameters under the assumption that the likelihood remains the same for all models considered this provides a quick and efficient scheme for scoring arbitrarily large numbers of models after inverting a single full model in turn this enables the selection among discrete models that are distinguished by the presence or absence of free parameters where free parameters are effectively removed from the model using very precise shrinkage priors an alternative application of this post hoc model selection considers continuous model spaces defined in terms of hyperparameters sufficient statistics of the prior density over model parameters in this instance the prior model can be optimized with respect to its evidence the expressions for model evidence become remarkably simple under the laplace gaussian approximation to the posterior density special cases of this scheme include savage dickey density ratio tests for reduced models and automatic relevance determination in model optimization we illustrate the approach using general linear models and a more complicated nonlinear state space model

network discovery with dcm this paper is about inferring or discovering the functional architecture of distributed systems using dynamic causal modelling dcm we describe a scheme that recovers the dynamic bayesian dependency graph connections in a network using observed network activity this network discovery uses bayesian model selection to identify the sparsity structure absence of edges or connections in a graph that best explains observed time series the implicit adjacency matrix specifies the form of the network e g cyclic or acyclic and its graph theoretical attributes e g degree distribution the scheme is illustrated using functional magnetic resonance imaging fmri time series to discover functional brain networks crucially it can be applied to experimentally evoked responses activation studies or endogenous activity in task free resting state fmri studies unlike conventional approaches to network discovery dcm permits the analysis of directed and cyclic graphs furthermore it eschews implausible markovian assumptions about the serial independence of random fluctuations the scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability relative to other networks the networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed and reciprocal effective connectivity between connected nodes or regions we envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting state studies

estimation and model selection for an ide based spatio temporal model a state space model of the stochastic spatio temporal integro difference equation ide is derived based on multidimensional sampling theory the dimensions of the state space and parameter space of the model are identified from the spatial bandwidth of the system and the support of the redistribution kernel of the ide when both the bandwidth and the kernel support are unknown a method to propose a number of state space and parameter space dimensions is presented these chosen dimensions result in a number of candidate model structures bayesian model selection making use of bayes factor the data augmentation algorithm and importance sampling is then used to identify the model best suited to represent the data in a maximum a posteriori sense

productivity of stream definitions we give an algorithm for deciding productivity of a large and natural class of recursive stream definitions a stream definition is called productive if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit whereas productivity is undecidable for stream definitions in general we show that it can be decided for pure stream definitions for every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements called pebbles in a finite pebbleflow net work and the production of a pebbleflow net associated with a pure stream definition that is the amount of pebbles the net is able to produce at its output port can be calculated by reducing nets to trivial nets