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Related Concept Videos

Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
The Van der Waals Equation01:26

The Van der Waals Equation

The ideal gas law is based on two simplifying assumptions: first, that there are no intermolecular attractions between gas molecules, and second, that the volume occupied by the molecules themselves is negligible compared with the volume of the container. However, these assumptions don't hold up under all conditions - specifically, at high pressures and low temperatures, as gas tends to deviate from ideal gas behavior.The van der Waals equation is an enhanced version of the ideal gas law,...
Gibbs Free Energy02:39

Gibbs Free Energy

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs. This new property is called the Gibbs free energy (G) (or simply the free...
Maxwell's Thermodynamic Relations01:23

Maxwell's Thermodynamic Relations

Maxwell's thermodynamic relations are very useful in solving problems in thermodynamics. Each of Maxwell's relations relates a partial differential between quantities that can be hard to measure experimentally to a partial differential between quantities that can be easily measured. These relations are a set of equations derivable from the symmetry of the second derivatives and the thermodynamic potentials.
All thermodynamic potentials are exact differentials. Therefore, their second-order...
Calculation of First-Law Quantities II01:24

Calculation of First-Law Quantities II

The first law of thermodynamics establishes that the change in internal energy of a system is given by ΔU = q + w, where q is the heat exchanged, and w is the work performed. For a perfect gas, both internal energy (U) and enthalpy (H) depend solely on temperature. Consequently, for any change of state, whether reversible or irreversible, the internal energy change is determined by integrating the heat capacity at constant volume, and the enthalpy change by integrating the heat capacity at...

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Related Experiment Video

Updated: Jul 19, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Variational free energy and the Laplace approximation.

Karl Friston1, Jérémie Mattout, Nelson Trujillo-Barreto

  • 1The Wellcome Department of Imaging Neuroscience, Institute of Neurology, UCL, 12 Queen Square, London, WC1N 3BG, UK. k.friston@fil.ion.ucl.ac.uk

Neuroimage
|October 24, 2006
PubMed
Summary

This study shows how restricted maximum likelihood (ReML) can approximate Bayesian log-evidence for model selection. This method aids in comparing models with varying covariance components and selecting priors in hierarchical models.

Related Experiment Videos

Last Updated: Jul 19, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Area of Science:

  • Statistical modeling
  • Machine learning
  • Bayesian inference

Background:

  • Variational free energy is used to approximate log-evidence in Bayesian model selection.
  • Restricted Maximum Likelihood (ReML) is a method for estimating model parameters, particularly in mixed models.
  • Bayesian model averaging and selection require accurate approximations of the log-evidence.

Purpose of the Study:

  • To derive variational free energy under Laplace approximation, considering model complexity from additional parameters.
  • To demonstrate how ReML can be adjusted to approximate log-evidence for model selection.
  • To explore the relationships between Variational Bayes, Expectation Maximization (EM), and ReML.

Main Methods:

  • Derivation of variational free energy using Laplace approximation.
  • Contextualizing ReML within variational learning and EM frameworks.
  • Analyzing the objective function of ReML for log-evidence approximation.

Main Results:

  • ReML objective function can be adjusted for log-evidence approximation, enabling model selection for models with different covariance components.
  • Established relationships between Variational Bayes, EM, and ReML from variational principles.
  • Showed EM is equivalent to full variational treatment when precisions are linear in hyperparameters.

Conclusions:

  • ReML can be utilized for principled model selection, especially in hierarchical models with automatic relevance determination (ARD).
  • The study clarifies the formal connections between key statistical learning methods.
  • Insights into dynamic models inform regularization of free energy ascent schemes like EM and ReML.