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Related Concept Videos

Entropy02:39

Entropy

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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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Entropy01:18

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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
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Entropy and the Second Law of Thermodynamics01:20

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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation  between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
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Entropy and the Second Law of Thermodynamics01:26

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Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
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Third Law of Thermodynamics02:38

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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Related Experiment Video

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Modeling the Size Spectrum for Macroinvertebrates and Fishes in Stream Ecosystems
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Learning maximum entropy models from finite-size data sets: A fast data-driven algorithm allows sampling from the

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  • 1Sorbonne Universités, UPMC Univ Paris 06, INSERM, CNRS, Institut de la Vision, 17 rue Moreau, 75012 Paris, France.

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Summary

We developed a new algorithm for training maximum entropy models that improves learning speed and accuracy. This data-driven approach, by rectifying parameter space, avoids overfitting and underfitting, outperforming standard methods on neural data.

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Area of Science:

  • Computational neuroscience
  • Statistical modeling
  • Machine learning

Background:

  • Maximum entropy models are crucial for inferring probability distributions from data.
  • Standard training methods like steepest descent face challenges with complex parameter spaces.
  • Understanding learning dynamics is key to optimizing model performance.

Purpose of the Study:

  • To characterize and improve the learning dynamics of maximum entropy models for large, finite datasets.
  • To develop a computationally efficient method for parameter space rectification.
  • To introduce a novel algorithm that samples from the posterior distribution to prevent overfitting.

Main Methods:

  • Analysis of steepest descent dynamics and its limitations due to parameter space curvature.
  • Development of a dataset-property-based rectification method for the parameter space.
  • Solving the long-time limit of stochastic dynamics using Gibbs sampling.
  • Application to learning pairwise Ising models from retina neuron recordings.

Main Results:

  • Steepest descent dynamics are suboptimal due to inhomogeneous parameter space curvature.
  • The proposed rectification method improves learning efficiency without significant computational cost.
  • Stochastic dynamics converge to a stationary distribution, with rectified dynamics matching the posterior distribution.
  • The 'rectified' algorithm demonstrates superior performance over steepest descent on neural data.

Conclusions:

  • A novel, fast, data-driven algorithm for training maximum entropy models has been developed.
  • The algorithm effectively samples from the parameters' posterior distribution, mitigating under- and overfitting.
  • This approach offers significant advantages for analyzing complex datasets, as shown in neural recordings.