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Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...

Uncertainty: Overview

Uncertainty: Overview

In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.

Propagation of Uncertainty from Random Error

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An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...

Propagation of Uncertainty from Systematic Error

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The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...

Uncertainty in Measurement: Accuracy and Precision

Uncertainty in Measurement: Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or the accepted value. Precise values agree with each other; accurate values agree with a true value.

Interpretation of Confidence Intervals

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A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
Confidence intervals have confidence coefficients that are crucial for their interpretation. The most common confidence coefficients are 0.90, 0.95, and 0.99, which can be written as percentages–90%, 95%, and 99%, respectively.
Suppose a person calculates a confidence interval with a confidence coefficient of 0.95. In that case, they can...

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Bayesian Dynamic Borrowing of Historical Information with Applications to the Analysis of Large-Scale Assessments.

Psychometrika·2022

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On the Quantification of Model Uncertainty: A Bayesian Perspective.

¹University of Wisconsin-Madison, Madison, USA. dkaplan@education.wisc.edu.

|March 15, 2021

Summary

This summary is machine-generated.

Bayesian model averaging (BMA) addresses statistical model selection uncertainty by averaging over possible models, unlike methods yielding a single final model. This approach accounts for inherent uncertainty in statistical analysis.

Keywords:

Bayesian model averaging Bayesian stacking prediction

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Area of Science:

Statistics
Computational Statistics

Background:

Model selection is a long-standing challenge in statistical literature.
Traditional methods like stepwise regression, ridge regression, lasso, and elastic net yield a single model, ignoring selection uncertainty.
Bayesian model averaging (BMA) offers an alternative by averaging predictions across a model space.

Purpose of the Study:

To provide a comprehensive review of Bayesian model averaging (BMA).
To focus on BMA's foundations in Bayesian decision theory and predictive modeling.
To discuss practical aspects including prior selection, prediction evaluation, and extensions like Bayesian stacking.

Main Methods:

Review of Bayesian model averaging (BMA) principles.
Exploration of Bayesian decision theory and predictive modeling frameworks.
Discussion of parameter and model prior selection, prediction evaluation, and Bayesian stacking.

Main Results:

BMA directly incorporates model uncertainty by averaging, unlike single-model selection methods.
The review details theoretical underpinnings and practical applications of BMA.
Extensions like Bayesian stacking are presented to address BMA assumptions.

Conclusions:

BMA provides a robust framework for handling model uncertainty in statistical analysis.
The paper offers guidance on implementing BMA, including prior specification and prediction assessment.
Future research directions relevant to psychometrics are highlighted.