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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...

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: 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.

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.

Prediction Intervals

Prediction Intervals

The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y.

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Related Experiment Video

Updated: Jun 22, 2025

Deep Neural Networks for Image-Based Dietary Assessment

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

Density regression and uncertainty quantification with Bayesian deep noise neural networks.

Daiwei Zhang¹, Tianci Liu², Jian Kang³

¹Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, Pennsylvania, 19104, USA.

Stat

|July 3, 2024

Summary

We introduce Bayesian deep noise neural networks (B-DeepNoise) to accurately quantify uncertainty in deep learning predictions. This novel approach improves density estimation and uncertainty quantification for continuous outcomes.

Keywords:

Bayesian methods machine learning neural networks

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Deep Neural Networks for Image-Based Dietary Assessment

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Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks

Published on: March 2, 2015

Area of Science:

Artificial Intelligence
Machine Learning
Computational Statistics

Background:

Deep neural networks (DNNs) offer high predictive accuracy but struggle with quantifying prediction uncertainty, particularly for continuous outcomes.
Accurate uncertainty quantification is crucial for reliable decision-making in various applications.

Purpose of the Study:

To propose a novel Bayesian deep neural network, B-DeepNoise, that effectively quantifies uncertainty in predictions.
To extend Bayesian DNNs by incorporating random noise variables across all hidden layers.

Main Methods:

Developed the Bayesian deep noise neural network (B-DeepNoise) model.
Implemented a closed-form Gibbs sampling algorithm for posterior computation, avoiding complex tuning.
Established a recursive representation for predictive density and analyzed predictive variance.

Main Results:

B-DeepNoise demonstrated superior performance in density estimation and uncertainty quantification compared to existing methods.
The model effectively approximates complex predictive density functions and learns random variations in outcomes.
Validated the model's utility through experiments and a neuroimaging application.

Conclusions:

B-DeepNoise offers a robust solution for uncertainty quantification in deep learning.
The proposed Gibbs sampling method simplifies posterior computation.
The model shows significant promise for scientific studies requiring accurate uncertainty estimates.