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Propagation of Uncertainty from Random Error

Propagation of Uncertainty from Random Error

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...

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...

Propagation of Uncertainty from Systematic Error

Propagation of Uncertainty from Systematic Error

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: 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.

Random Error

Random Error

Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...

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.

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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

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Asymptotics of Bayesian Uncertainty Estimation in Random Features Regression.

Youngsoo Baek¹, Samuel I Berchuck², Sayan Mukherjee³

¹Department of Statistical Science, Duke University, Durham, NC 27705.

Advances in Neural Information Processing Systems

|December 15, 2025

Summary

This study compares the posterior predictive distribution and maximum a posteriori (MAP) estimator in overparameterized random features regression. Asymptotic agreement between these Bayesian and frequentist approaches depends on the signal-to-noise ratio and data dimensions.

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Area of Science:

Statistics
Machine Learning
Computational Science

Background:

The behavior of Bayesian and frequentist estimators in overparameterized models is a key area of research.
Understanding the relationship between posterior predictive distributions and maximum a posteriori (MAP) estimators is crucial for model evaluation.

Purpose of the Study:

To compare the posterior predictive distribution (Bayesian model average) with the risk of the maximum a posteriori (MAP) estimator in random features regression.
To analyze their asymptotic behavior in the overparameterized regime, focusing on the role of signal-to-noise ratio and dimensionality.

Main Methods:

Theoretical analysis of the variance of the posterior predictive distribution and the risk of the MAP estimator.
Asymptotic analysis in two regimes: model dimensions growing faster than samples, and samples growing faster than dimensions.
Numerical simulations to investigate finite-dimensional properties and distributional characteristics.

Main Results:

Asymptotic agreement between the posterior predictive distribution and MAP estimator risk is governed by a phase transition in the signal-to-noise ratio when dimensions grow faster than samples.
These quantities also asymptotically agree when the number of samples grows faster than model dimensions.
Numerical simulations reveal finer distributional properties for finite dimensions.

Conclusions:

The study establishes conditions for asymptotic agreement between Bayesian and frequentist estimators in overparameterized random features regression.
A conjecture is proposed regarding Gaussian fluctuations and similarities to findings in Gaussian sequence models, suggesting broader theoretical implications.