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2018


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Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective

Tronarp, F., Kersting, H., Särkkä, S., Hennig, P.

ArXiv preprint 2018, arXiv:1810.03440 [stat.ME], October 2018 (article)

Abstract
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consists of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a Bayesian state estimation problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers, which were formulated in terms of generating synthetic measurements of the vector field, come out as specific approximations. We derive novel solvers, both Gaussian and non-Gaussian, from the Bayesian state estimation problem posed in this paper and compare them with other probabilistic solvers in illustrative experiments.

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link (url) Project Page [BibTex]

2018



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Convergence Rates of Gaussian ODE Filters

Kersting, H., Sullivan, T. J., Hennig, P.

arXiv preprint 2018, arXiv:1807.09737 [math.NA], July 2018 (article)

Abstract
A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives a priori as a Gauss--Markov process $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. We prove worst-case local convergence rates of order $h^{q+1}$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $h^q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyse how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we present explicit formulas for the steady states and show that the posterior confidence intervals are well calibrated in all considered cases that exhibit global convergence---in the sense that they globally contract at the same rate as the truncation error.

pn

link (url) Project Page [BibTex]

link (url) Project Page [BibTex]


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Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

Kanagawa, M., Hennig, P., Sejdinovic, D., Sriperumbudur, B. K.

Arxiv e-prints, arXiv:1805.08845v1 [stat.ML], 2018 (article)

Abstract
This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.

pn

arXiv [BibTex]

arXiv [BibTex]


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Schema-related cognitive load influences performance, speech, and physiology in a dual-task setting: A continuous multi-measure approach

Wirzberger, M., Herms, R., Esmaeili Bijarsari, S., Eibl, M., Rey, G. D.

Cognitive Research: Principles and Implications, 3:46, Springer Nature, 2018 (article)

Abstract
Schema acquisition processes comprise an essential source of cognitive demands in learning situations. To shed light on related mechanisms and influencing factors, this study applied a continuous multi-measure approach for cognitive load assessment. In a dual-task setting, a sample of 123 student participants learned visually presented symbol combinations with one of two levels of complexity while memorizing auditorily presented number sequences. Learners’ cognitive load during the learning task was addressed by secondary task performance, prosodic speech parameters (pauses, articulation rate), and physiological markers (heart rate, skin conductance response). While results revealed increasing primary and secondary task performance over the trials, decreases in speech and physiological parameters indicated a reduction in the overall level of cognitive load with task progression. In addition, the robustness of the acquired schemata was confirmed by a transfer task that required participants to apply the obtained symbol combinations. Taken together, the observed pattern of evidence supports the idea of a logarithmically decreasing progression of cognitive load with increasing schema acquisition, and further hints on robust and stable transfer performance, even under enhanced transfer demands. Finally, theoretical and practical consequences consider evidence on desirable difficulties in learning as well as the potential of multimodal cognitive load detection in learning applications.

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DOI [BibTex]

DOI [BibTex]


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Counterfactual Mean Embedding: A Kernel Method for Nonparametric Causal Inference

Muandet, K., Kanagawa, M., Saengkyongam, S., Marukata, S.

Arxiv e-prints, arXiv:1805.08845v1 [stat.ML], 2018 (article)

Abstract
This paper introduces a novel Hilbert space representation of a counterfactual distribution---called counterfactual mean embedding (CME)---with applications in nonparametric causal inference. Counterfactual prediction has become an ubiquitous tool in machine learning applications, such as online advertisement, recommendation systems, and medical diagnosis, whose performance relies on certain interventions. To infer the outcomes of such interventions, we propose to embed the associated counterfactual distribution into a reproducing kernel Hilbert space (RKHS) endowed with a positive definite kernel. Under appropriate assumptions, the CME allows us to perform causal inference over the entire landscape of the counterfactual distribution. The CME can be estimated consistently from observational data without requiring any parametric assumption about the underlying distributions. We also derive a rate of convergence which depends on the smoothness of the conditional mean and the Radon-Nikodym derivative of the underlying marginal distributions. Our framework can deal with not only real-valued outcome, but potentially also more complex and structured outcomes such as images, sequences, and graphs. Lastly, our experimental results on off-policy evaluation tasks demonstrate the advantages of the proposed estimator.

ei pn

arXiv [BibTex]

arXiv [BibTex]


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Model-based Kernel Sum Rule: Kernel Bayesian Inference with Probabilistic Models

Nishiyama, Y., Kanagawa, M., Gretton, A., Fukumizu, K.

Arxiv e-prints, arXiv:1409.5178v2 [stat.ML], 2018 (article)

Abstract
Kernel Bayesian inference is a powerful nonparametric approach to performing Bayesian inference in reproducing kernel Hilbert spaces or feature spaces. In this approach, kernel means are estimated instead of probability distributions, and these estimates can be used for subsequent probabilistic operations (as for inference in graphical models) or in computing the expectations of smooth functions, for instance. Various algorithms for kernel Bayesian inference have been obtained by combining basic rules such as the kernel sum rule (KSR), kernel chain rule, kernel product rule and kernel Bayes' rule. However, the current framework only deals with fully nonparametric inference (i.e., all conditional relations are learned nonparametrically), and it does not allow for flexible combinations of nonparametric and parametric inference, which are practically important. Our contribution is in providing a novel technique to realize such combinations. We introduce a new KSR referred to as the model-based KSR (Mb-KSR), which employs the sum rule in feature spaces under a parametric setting. Incorporating the Mb-KSR into existing kernel Bayesian framework provides a richer framework for hybrid (nonparametric and parametric) kernel Bayesian inference. As a practical application, we propose a novel filtering algorithm for state space models based on the Mb-KSR, which combines the nonparametric learning of an observation process using kernel mean embedding and the additive Gaussian noise model for a state transition process. While we focus on additive Gaussian noise models in this study, the idea can be extended to other noise models, such as the Cauchy and alpha-stable noise models.

pn

arXiv [BibTex]

arXiv [BibTex]


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Attention please! Enhanced attention control abilities compensate for instructional impairments in multimedia learning

Wirzberger, M., Rey, G. D.

Journal of Computers in Education, 5(2):243-257, Springer Nature, 2018 (article)

Abstract
Learners exposed to multimedia learning contexts have to deal with a variety of visual stimuli, demanding a conducive design of learning material to maintain limitations in attentional resources. Within the current study, effects and constraints arising from two selected impairing features are investigated in more detail within a computer-based learning task on factor analysis. A sample of 53 students received a combination of textual and pictorial elements that explained the topic, while impaired attention was systematically induced in a 2 × 2 factorial between-subjects design by interrupting system-notifications (with vs. without) and seductive text passages (with vs. without). Learners’ ability for controlled attention was assessed with a standardized psychological attention inventory. Approaching the results, learners receiving seductive text passages spent significantly more time on the learning material. In addition, a moderation effect of attention control abilities on the relationship between interruptions and retention performance resulted. Explanations for the obtained findings are discussed referring to mechanisms of compensation, load, and activation.

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DOI [BibTex]

DOI [BibTex]


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A probabilistic model for the numerical solution of initial value problems

Schober, M., Särkkä, S., Philipp Hennig,

Statistics and Computing, Springer US, 2018 (article)

Abstract
We study connections between ordinary differential equation (ODE) solvers and probabilistic regression methods in statistics. We provide a new view of probabilistic ODE solvers as active inference agents operating on stochastic differential equation models that estimate the unknown initial value problem (IVP) solution from approximate observations of the solution derivative, as provided by the ODE dynamics. Adding to this picture, we show that several multistep methods of Nordsieck form can be recast as Kalman filtering on q-times integrated Wiener processes. Doing so provides a family of IVP solvers that return a Gaussian posterior measure, rather than a point estimate. We show that some such methods have low computational overhead, nontrivial convergence order, and that the posterior has a calibrated concentration rate. Additionally, we suggest a step size adaptation algorithm which completes the proposed method to a practically useful implementation, which we experimentally evaluate using a representative set of standard codes in the DETEST benchmark set.

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PDF Code DOI Project Page [BibTex]


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The Computational Challenges of Pursuing Multiple Goals: Network Structure of Goal Systems Predicts Human Performance

Reichman, D., Lieder, F., Bourgin, D. D., Talmon, N., Griffiths, T. L.

PsyArXiv, 2018 (article)

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DOI [BibTex]

DOI [BibTex]


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The moderating role of arousal on the seductive detail effect in a multimedia learning setting

Schneider, S., Wirzberger, M., Rey, G. D.

Applied Cognitive Psychology, Wiley, 2018 (article)

Abstract
Arousal has been found to increase learners' attentional resources. In contrast, seductive details (interesting but learning‐irrelevant information) are considered to distract attention away from relevant information and, thus, hinder learning. However, a possibly moderating role of arousal on the seductive detail effect has not been examined yet. In this study, arousal variations were induced via audio files of false heartbeats. In consequence, 100 participants were randomly assigned to a 2 (with or without seductive details) × 2 (lower vs. higher false heart rates) between‐subjects design. Data on learning performance, cognitive load, motivation, heartbeat frequency, and electro‐dermal activity were collected. Results show learning‐inhibiting effects for seductive details and learning‐enhancing effects for higher false heart rates. Cognitive processes mediate both effects. However, the detrimental effect of seductive details was not present when heart rate was higher. Results indicate that the seductive detail effect is moderated by a learner's state of arousal.

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DOI [BibTex]

DOI [BibTex]


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Rational metareasoning and the plasticity of cognitive control

Lieder, F., Shenhav, A., Musslick, S., Griffiths, T. L.

{PLoS Computational Biology}, 14(4):e1006043, Public Library of Science, 2018 (article)

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Project Page [BibTex]

Project Page [BibTex]


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Over-representation of extreme events in decision making reflects rational use of cognitive resources

Lieder, F., Griffiths, T. L., Hsu, M.

Psychological Review, 125(1):1-32, 2018 (article)

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[BibTex]

[BibTex]

2015


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Probabilistic Interpretation of Linear Solvers

Hennig, P.

SIAM Journal on Optimization, 25(1):234-260, 2015 (article)

ei pn

Web PDF link (url) DOI [BibTex]

2015


Web PDF link (url) DOI [BibTex]


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Modeling interruption and resumption in a smartphone task: An ACT-R approach

Wirzberger, M., Russwinkel, N.

i-com, 14(2), Walter de Gruyter GmbH, 2015 (article)

Abstract
This research aims to inspect human cognition when being interrupted while performing a smartphone task with varying levels of mental demand. Due to its benefits especially in the early stages of interface development, a cognitive modeling approach is used. It applies the cognitive architecture ACT-R to shed light on task-related cognitive processing. The inspected task setting involves a shopping scenario, manipulating interruption via product advertisements and mental demands by the respective number of people shopping is done for. Model predictions are validated through a corresponding experimental setting with 62 human participants. Comparing model and human data in a defined set of performance-related parameters displays mixed results that indicate an acceptable fit – at least in some cases. Potential explanations for the observed differences are discussed at the end.

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DOI [BibTex]

DOI [BibTex]


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Probabilistic numerics and uncertainty in computations

Hennig, P., Osborne, M. A., Girolami, M.

Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2179), 2015 (article)

Abstract
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data have led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimizers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

ei pn

PDF DOI [BibTex]

PDF DOI [BibTex]


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The optimism bias may support rational action

Lieder, F., Goel, S., Kwan, R., Griffiths, T. L.

NIPS 2015 Workshop on Bounded Optimality and Rational Metareasoning, 2015 (article)

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[BibTex]

[BibTex]


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Rational use of cognitive resources: Levels of analysis between the computational and the algorithmic

Griffiths, T. L., Lieder, F., Goodman, N. D.

Topics in Cognitive Science, 7(2):217-229, Wiley, 2015 (article)

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[BibTex]

[BibTex]


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Model-based strategy selection learning

Lieder, F., Griffiths, T. L.

The 2nd Multidisciplinary Conference on Reinforcement Learning and Decision Making, 2015 (article)

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Project Page [BibTex]

Project Page [BibTex]