Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

Promoting Harmonious Relations and Equitable Well-Being: Peace Psychology and “Intractable” Conflicts

The chapter explores Bar-Tal’s legacy in relation to key concepts, perspectives, and findings that comprise the growing field of peace psychology, specifically the promotion of sustainable peace through the indivisible constructs of harmonious relations and equitable wellbeing. Analyzed through a peace psychology lens, Bar-Tal’s work highlights both the barriers to and bridges for achieving sustainable peace. Central concepts from his work, such as fear, insecurity, and an ethos of conflict, demonstrate key obstacles to fostering harmonious intergroup relations based on social justice. Bar-Tal’s work also identifies processes that can overcome these barriers, which is consistent with peace psychology’s emphasis on the development of constructive responses to violence and conflict. For example, the chapter outlines how confidence-building mechanisms, mutually respectful identities, and reconciliation processes, may help foster an ethos of peace that can be embedded in the structure of societies through peace education. The chapter concludes with implications and suggestions for future research, with a focus on the role of young people in settings of prolonged intergroup division and generational approaches to peacebuilding, as conceptualized through a peace psychology lens.

An evaluation of portion size estimation aids precision, ease of use and likelihood of future use

OBJECTIVE: The present study aimed to evaluate the precision, ease of use and likelihood of future use of portion size estimation aids (PSEA).

DESIGN: A range of PSEA were used to estimate the serving sizes of a range of commonly eaten foods and rated for ease of use and likelihood of future usage.

SETTING: For each food, participants selected their preferred PSEA from a range of options including: quantities and measures; reference objects; measuring; and indicators on food packets. These PSEA were used to serve out various foods (e.g. liquid, amorphous, and composite dishes). Ease of use and likelihood of future use were noted. The foods were weighed to determine the precision of each PSEA.

SUBJECTS: Males and females aged 18-64 years (n 120).

RESULTS: The quantities and measures were the most precise PSEA (lowest range of weights for estimated portion sizes). However, participants preferred household measures (e.g. 200 ml disposable cup) - deemed easy to use (median rating of 5), likely to use again in future (all scored either 4 or 5 on a scale from 1='not very likely' to 5='very likely to use again') and precise (narrow range of weights for estimated portion sizes). The majority indicated they would most likely use the PSEA preparing a meal (94 %), particularly dinner (86 %) in the home (89 %; all P<0·001) for amorphous grain foods.

CONCLUSIONS: Household measures may be precise, easy to use and acceptable aids for estimating the appropriate portion size of amorphous grain foods.