scholarly journals Learning Coefficient of Vandermonde Matrix-Type Singularities in Model Selection

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 561
Author(s):  
Miki Aoyagi
Keyword(s):  
Model Selection ◽  
Information Criteria ◽  
Learning Systems ◽  
Vandermonde Matrix ◽  
Selection Methods ◽  
Learning Models ◽  
Matrix Type ◽  
Log Canonical Threshold ◽  
Log Canonical ◽  
Blowing Up

In recent years, selecting appropriate learning models has become more important with the increased need to analyze learning systems, and many model selection methods have been developed. The learning coefficient in Bayesian estimation, which serves to measure the learning efficiency in singular learning models, has an important role in several information criteria. The learning coefficient in regular models is known as the dimension of the parameter space over two, while that in singular models is smaller and varies in learning models. The learning coefficient is known mathematically as the log canonical threshold. In this paper, we provide a new rational blowing-up method for obtaining these coefficients. In the application to Vandermonde matrix-type singularities, we show the efficiency of such methods.

Learning Coefficient of Generalization Error in Bayesian Estimation and Vandermonde Matrix-Type Singularity

2012 ◽  
Vol 24 (6) ◽  
pp. 1569-1610 ◽  
Author(s):  
Miki Aoyagi ◽  
Kenji Nagata
Keyword(s):  
Bayesian Estimation ◽  
Probabilistic Models ◽  
Vandermonde Matrix ◽  
Generalization Error ◽  
Normal Mixture ◽  
New Approach ◽  
Matrix Type ◽  
Hierarchical Learning ◽  
Log Canonical ◽  
Normal Mixture Models

The term algebraic statistics arises from the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry (Sturmfels, 2009 ). The purpose of our study is to consider the generalization error and stochastic complexity in learning theory by using the log-canonical threshold in algebraic geometry. Such thresholds correspond to the main term of the generalization error in Bayesian estimation, which is called a learning coefficient (Watanabe, 2001a , 2001b ). The learning coefficient serves to measure the learning efficiencies in hierarchical learning models. In this letter, we consider learning coefficients for Vandermonde matrix-type singularities, by using a new approach: focusing on the generators of the ideal, which defines singularities. We give tight new bound values of learning coefficients for the Vandermonde matrix-type singularities and the explicit values with certain conditions. By applying our results, we can show the learning coefficients of three-layered neural networks and normal mixture models.

scholarly journals A Comparative Study of the Lasso-type and Heuristic Model Selection Methods

2013 ◽  
Vol 233 (4) ◽  
pp. 526-549 ◽  
Author(s):  
Ivan Savin
Keyword(s):  
Model Selection ◽  
Search Space ◽  
Information Criteria ◽  
Adaptive Lasso ◽  
Heuristic Model ◽  
Selection Methods ◽  
Monte Carlo Simulation Study ◽  
Discrete Search ◽  
Highly Correlated ◽  
Correlated Predictors

Summary This study presents a first comparative analysis of Lasso-type (Lasso, adaptive Lasso, elastic net) and heuristic subset selection methods. Although the Lasso has shown success in many situations, it has some limitations. In particular, inconsistent results are obtained for pairwise highly correlated predictors. An alternative to the Lasso is constituted by model selection based on information criteria (IC), which remain consistent in the situation mentioned. However, these criteria are hard to optimize due to a discrete search space. To overcome this problem, an optimization heuristic (Genetic Algorithm) is applied. To this end, results of a Monte-Carlo simulation study together with an application to an actual empirical problem are reported to illustrate the performance of the methods.

scholarly journals Learning Coefficient in Bayesian Estimation of Restricted Boltzmann Machine

2013 ◽  
Vol 4 (1) ◽  
Author(s):  
Miki Aoyagi
Keyword(s):  
Bayesian Estimation ◽  
Restricted Boltzmann Machine ◽  
Boltzmann Machine ◽  
Generalization Error ◽  
Learning Models ◽  
Log Canonical Threshold ◽  
The Real ◽  
Hierarchical Learning ◽  
Log Canonical ◽  
The Ideal

We consider the real log canonical threshold for the learning model in Bayesian estimation.This threshold corresponds to a learning coefficient of generalization error in Bayesianestimation, which serves to measure learning efficiency in hierarchical learning models [30, 31, 33].In this paper, we clarify the ideal which gives the log canonical threshold of the restricted Boltzmannmachine and consider the learning coefficients of this model.

scholarly journals Explainable AI: A Review of Machine Learning Interpretability Methods

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 18
Author(s):  
Pantelis Linardatos ◽  
Vasilis Papastefanopoulos ◽  
Sotiris Kotsiantis
Keyword(s):  
Artificial Intelligence ◽  
Machine Learning ◽  
Black Box ◽  
Learning Systems ◽  
Model Complexity ◽  
Learning Models ◽  
New Methods ◽  
Industrial Adoption ◽  
Machine Learning Models ◽  
The Way

Recent advances in artificial intelligence (AI) have led to its widespread industrial adoption, with machine learning systems demonstrating superhuman performance in a significant number of tasks. However, this surge in performance, has often been achieved through increased model complexity, turning such systems into “black box” approaches and causing uncertainty regarding the way they operate and, ultimately, the way that they come to decisions. This ambiguity has made it problematic for machine learning systems to be adopted in sensitive yet critical domains, where their value could be immense, such as healthcare. As a result, scientific interest in the field of Explainable Artificial Intelligence (XAI), a field that is concerned with the development of new methods that explain and interpret machine learning models, has been tremendously reignited over recent years. This study focuses on machine learning interpretability methods; more specifically, a literature review and taxonomy of these methods are presented, as well as links to their programming implementations, in the hope that this survey would serve as a reference point for both theorists and practitioners.

scholarly journals Model Selection Procedures in Bounds Test of Cointegration: Theoretical Comparison and Empirical Evidence

Economies ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 49 ◽  
Author(s):  
Waqar Badshah ◽  
Mehmet Bulut
Keyword(s):  
Model Selection ◽  
Akaike Information Criterion ◽  
Bayesian Information Criterion ◽  
Selection Process ◽  
Information Criterion ◽  
Small Sample ◽  
Information Criteria ◽  
Path Model ◽  
Sample Sizes ◽  
Bounds Test

Only unstructured single-path model selection techniques, i.e., Information Criteria, are used by Bounds test of cointegration for model selection. The aim of this paper was twofold; one was to evaluate the performance of these five routinely used information criteria {Akaike Information Criterion (AIC), Akaike Information Criterion Corrected (AICC), Schwarz/Bayesian Information Criterion (SIC/BIC), Schwarz/Bayesian Information Criterion Corrected (SICC/BICC), and Hannan and Quinn Information Criterion (HQC)} and three structured approaches (Forward Selection, Backward Elimination, and Stepwise) by assessing their size and power properties at different sample sizes based on Monte Carlo simulations, and second was the assessment of the same based on real economic data. The second aim was achieved by the evaluation of the long-run relationship between three pairs of macroeconomic variables, i.e., Energy Consumption and GDP, Oil Price and GDP, and Broad Money and GDP for BRICS (Brazil, Russia, India, China and South Africa) countries using Bounds cointegration test. It was found that information criteria and structured procedures have the same powers for a sample size of 50 or greater. However, BICC and Stepwise are better at small sample sizes. In the light of simulation and real data results, a modified Bounds test with Stepwise model selection procedure may be used as it is strongly theoretically supported and avoids noise in the model selection process.

scholarly journals Using Model Selection Criteria to Choose the Number of Principal Components

2021 ◽  
Vol 20 (3) ◽  
pp. 450-461
Author(s):  
Stanley L. Sclove
Keyword(s):  
Model Selection ◽  
Principal Components ◽  
Bayesian Information Criterion ◽  
Selection Criteria ◽  
Information Criterion ◽  
Information Criteria ◽  
Akaike's Information Criterion ◽  
Model Selection Criteria ◽  
Adequate Number ◽  
Number Of Principal Components

AbstractThe use of information criteria, especially AIC (Akaike’s information criterion) and BIC (Bayesian information criterion), for choosing an adequate number of principal components is illustrated.

scholarly journals Bayesian Model Averaging to Account for Model Uncertainty in Estimates of a Vaccine's Effectiveness

2021 ◽  
Author(s):  
Carlos R Oliveira ◽  
Eugene D Shapiro ◽  
Daniel M Weinberger
Keyword(s):  
Model Selection ◽  
Model Uncertainty ◽  
Bayesian Model ◽  
Bayesian Model Averaging ◽  
Model Averaging ◽  
Selection Methods ◽  
Final Model ◽  
Negative Case ◽  
Confounder Selection ◽  
Control Study

Vaccine effectiveness (VE) studies are often conducted after the introduction of new vaccines to ensure they provide protection in real-world settings. Although susceptible to confounding, the test-negative case-control study design is the most efficient method to assess VE post-licensure. Control of confounding is often needed during the analyses, which is most efficiently done through multivariable modeling. When a large number of potential confounders are being considered, it can be challenging to know which variables need to be included in the final model. This paper highlights the importance of considering model uncertainty by re-analyzing a Lyme VE study using several confounder selection methods. We propose an intuitive Bayesian Model Averaging (BMA) framework for this task and compare the performance of BMA to that of traditional single-best-model-selection methods. We demonstrate how BMA can be advantageous in situations when there is uncertainty about model selection by systematically considering alternative models and increasing transparency.

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