A Bayes interpretation of stacking for M-complete and M-open settings

In M-open problems where no true model can be conceptualized, it is common to back off from modeling and merely seek good prediction. Even in M-complete problems, taking a predictive approach can be very useful. Stacking is a model averaging procedure that gives a composite predictor by combining individual predictors from a list of models using weights that optimize a cross-validation criterion. We show that the stacking weights also asymptotically minimize a posterior expected loss. Hence we formally provide a Bayesian justification for cross-validation. Often the weights are constrained to be positive and sum to one. For greater generality, we omit the positivity constraint and relax the `sum to one' constraint. A key question is `What predictors should be in the average?' We first verify that the stacking error depends only on the span of the models. Then we propose using bootstrap samples from the data to generate empirical basis elements that can be used to form models. We use this in two computed examples to give stacking predictors that are (i) data driven, (ii) optimal with respect to the number of component predictors, and (iii) optimal with respect to the weight each predictor gets.Comment: 37 pages, 2 figure

Weakly Submodular Functions

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the "standard greedy algorithm" achieves a constant approximation ratio where the constant (experimentally) converges to 5.95 as the cardinality constraint increases. For a general matroid constraint, a simple local search algorithm achieves a constant approximation ratio where the constant (analytically) converges to 10.22 as the rank of the matroid increases

Dementia Education: What are the Needs of Post-Secondary Students in London, Ontario?

Dementia is a chronic and progressive syndrome characterized by the disturbance of multiple brain functions. As of 2008, an estimated 500,000 Canadians will have a dementia diagnosis and is predicted to rise to 1.1 million Canadians in 2038. A lack of dementia awareness has been identified by McCormick Dementia Services. This study examines the current dementia knowledge of a small cross-section of post-secondary students in London, Ontario. A sample size of twenty-eight participants took an online survey in which students identified that they were able to recognize and had sufficient knowledge of dementia. The survey revealed that although adequate knowledge of dementia was present, the participants were unaware of various resources that could be found in their community to further educate themselves outside of the Alzheimer Society. 100% of students think it would be valuable to learn more about dementia. The participants’ expressed that if a youth and dementia education one-day symposium were offered, they were willing to attend on their own accord to further educate themselves. The survey indicated a demand for more opportunities to be made accessible for students to get involved and to gain further understanding of dementia. A more comprehensive study is recommended to revaluate post-secondary student’s interest in a youth and dementia education one-day symposium

THE AR- PROPERTY AND THE FIXED POINT PROPERTY FOR COMPACT MAPS OF A SOME CONVEX SUBSET IN THE SPACE LP(0 < p < 1)

Joint Research on Environmental Science and Technology for the Eart

Characteristics of the participants in mass sporting events in Vietnam

This study aimed to examine the characteristics of athletes and spectators at mass sports competitions, focusing on aspects such as quantity, age, gender, educational level, occupation, and income to provide a comprehensive demographic overview. The study employed a sociological survey method, sampling 10 localities out of 63 provinces and cities across Vietnam. The survey included 520 spectators (Group 1) and 220 athletes (Group 2), resulting in a total sample size of 740 participants. The results are illustrated using graphical representations. The majority of spectators at mass sport events are aged 23 to under 60, whereas athletes show a more even age distribution, indicating lower student participation in spectating. Men significantly outnumber women. Most participants hold high school or university degrees, with the largest group being unskilled workers. Students, freelancers, and retirees are more evenly distributed. Professional field participants make up 35.5% of the total. Income-wise, most participants are middle to high-income earners, with nearly half earning between 5 and 10 million VND per month, and 42.3% earning over 10 million VND. Low-income and very high-income individuals are mostly spectators, while medium to high-income individuals are more likely to be athletes

Using the Bayesian Shtarkov solution for predictions

AbstractThe Bayes Shtarkov predictor can be defined and used for a variety of data sets that are exceedingly hard if not impossible to model in any detailed fashion. Indeed, this is the setting in which the derivation of the Shtarkov solution is most compelling. The computations show that anytime the numerical approximation to the Shtarkov solution is ‘reasonable’, it is better in terms of predictive error than a variety of other general predictive procedures. These include two forms of additive model as well as bagging or stacking with support vector machines, Nadaraya–Watson estimators, or draws from a Gaussian Process Prior

The formal definition of reference priors under a general class of divergence

"May 2014."Dissertation Supervisor: Dr. Dongchu Sun.Includes vita.Bayesian analysis is widely used recently in both theory and application of statistics. The choice of priors plays a key role in any Bayesian analysis. There are two types of priors: subjective priors and objective priors. In practice, however, the difficulties of subjective elicitation and time restrictions frequently limit us to use the objective priors constructed by some formal rules. In this dissertation, our methodology is using reference analysis to derive objective priors. Objective Bayesian inference makes inference depending only on the assumed model and the available data. The prior distribution used to make an inference is least informative in a certain information-theoretic sense. Recently, Berger, Bernardo and Sun (2009) derived reference priors rigorously in the contexts under Kullback-Leibler divergence. In special cases with common support and other regularity conditions, Ghosh, Mergel and Liu (2011) derived a general f-divergence criterion for prior selection. We generalize Ghosh, Mergel and Liu's (2011) results to the case without common support and show how an explicit expression for the reference prior can be obtained under posterior consistency. The explicit expression can be used to derive new reference priors both analytically and numerically.Includes bibliographical references (pages 126-127)

A Bayes Interpretation of Stacking for M-Complete and M-Open Settings

In M-open problems where no true model can be conceptualized, it is common to back off from modeling and merely seek good prediction. Even in M-complete problems, taking a predictive approach can be very useful. Stacking is a model averaging procedure that gives a composite predictor by combining individual predictors from a list of models using weights that optimize a cross validation criterion. We show that the stacking weights also asymptotically minimize a posterior expected loss. Hence we formally provide a Bayesian justification for cross-validation. Often the weights are constrained to be positive and sum to one. For greater generality, we omit the positivity constraint and relax the ‘sum to one’ constraint

Metric compatibility and determination in complete metric spaces

It was established in [8] that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if $f$ is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of [8] and [18]