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Related Concept Videos

The Uncertainty Principle04:08

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Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...
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Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such measurements result in exact numbers. By counting the eggs in a carton, for instance, one can determine exactly how many eggs are there in the carton. Similarly, the numbers of defined quantities are also exact. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms. Quantities...
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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.
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All the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a measured value; for example, if a scale that shows weight to the nearest pound reads “140,” then the 1 (hundreds), 4 (tens), and 0 (ones) are all significant (measured) values.
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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...
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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 Comprehensive Framework for Uncertainty Quantification of Voxel-Wise Supervised Deep Learning Models in IVIM MRI.

Nicola Casali1,2, Alessandro Brusaferri1, Giuseppe Baselli1,2

  • 1Istituto di Sistemi e Tecnologie Industriali Intelligenti per il Manifatturiero Avanzato, Consiglio Nazionale delle Ricerche, Milan, Italy.

NMR in Biomedicine
|January 20, 2026
PubMed
Summary

This study introduces a deep learning framework using deep ensembles of mixture density networks for more accurate intravoxel incoherent motion (IVIM) parameter estimation in MRI. The method quantifies uncertainty, improving reliability in diffusion MRI analysis.

Keywords:
aleatoric uncertaintycalibrationdeep ensembleepistemic uncertaintyintravoxel incoherent motionmagnetic resonance imagingmixture density networks

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Area of Science:

  • Medical Imaging and Physics
  • Machine Learning in Medical Diagnostics
  • Quantitative MRI Techniques

Background:

  • Accurate estimation of intravoxel incoherent motion (IVIM) parameters from diffusion-weighted MRI is challenging due to ill-posed inverse problems and noise sensitivity.
  • Existing methods struggle with noise, especially in the perfusion compartment, limiting the reliability of IVIM parameter quantification.
  • Uncertainty quantification is crucial for interpreting the reliability of estimated parameters in diffusion MRI.

Purpose of the Study:

  • To develop and evaluate a probabilistic deep learning framework for robust IVIM parameter estimation.
  • To enable the quantification and decomposition of predictive uncertainty into aleatoric (AU) and epistemic (EU) components.
  • To benchmark the proposed framework against existing non-probabilistic and probabilistic methods.

Main Methods:

  • A probabilistic deep learning framework utilizing deep ensembles (DEs) of mixture density networks (MDNs) was developed.
  • Supervised training was performed on synthetic data, with evaluation on simulated and in vivo mouse brain MRI datasets.
  • Uncertainty quantification reliability was assessed using calibration curves, predictive distribution sharpness, and CRPS.

Main Results:

  • MDNs demonstrated more calibrated and sharper predictive distributions for diffusion coefficient (D) and perfusion fraction (f) compared to other methods.
  • Slight overconfidence was noted for the pseudodiffusion coefficient (D*), but MDNs yielded smoother in vivo D* estimates.
  • Elevated epistemic uncertainty (EU) in vivo indicated a potential mismatch with real acquisition conditions, highlighting the value of DEs.

Conclusions:

  • The proposed deep ensemble MDN framework provides comprehensive uncertainty quantification for IVIM fitting, identifying unreliable estimates.
  • This approach enhances the reliability and interpretability of diffusion MRI parameter estimation.
  • The framework is adaptable for fitting other physical models with appropriate adjustments.