Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Cartesian Vector Notation01:28

Cartesian Vector Notation

1.9K
Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
1.9K
Dimensional Analysis01:23

Dimensional Analysis

2.6K
Dimensional analysis is a powerful tool that is used in physics and engineering to understand and predict the behavior of physical systems. The basic idea behind dimensional analysis is to express physical quantities in terms of fundamental dimensions such as the mass, length, and time. Derived dimensions like the velocity, acceleration, and force are derived from the combinations of these fundamental dimensions.
Dimensional analysis allows us to analyze and compare physical quantities on a...
2.6K
Dimensional Analysis02:19

Dimensional Analysis

19.4K
The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that mathematical equations must meet the following two rules. The first rule is that, in an equation, the expressions on each side of the equal sign must have the same dimensions. This is fairly intuitive since we can only add or subtract quantities of the same type (dimension). The second rule states that, in an equation, the arguments of any of the...
19.4K
Dimensional Analysis03:40

Dimensional Analysis

51.4K
Dimensional analysis, also known as the factor label method, is a versatile approach for mathematical operations. The main principle behind this approach is: the units of quantities must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex and multi-step calculations involving several different quantities and their units.
Conversion Factors and Dimensional Analysis
The unit...
51.4K
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

15.5K
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
15.5K
Fisher's Exact Test01:08

Fisher's Exact Test

1.4K
Fisher's exact test is a statistical significance test widely used to analyze 2x2 contingency tables, particularly in situations where sample sizes are small. Unlike the chi-squared test, which approximates P-values and assumes minimum expected frequencies of at least five in each cell, Fisher's exact test calculates the exact probability (P-value) of observing the data or more extreme results under the null hypothesis. This feature makes it especially valuable when the assumptions of...
1.4K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

No evidence for persistent natural plague reservoirs in historical and modern Europe.

Proceedings of the National Academy of Sciences of the United States of America·2022
Same author

Estimation and tests for power-transformed and threshold GARCH models.

Journal of econometrics·2020
Same author

Jackknife approach to the estimation of mutual information.

Proceedings of the National Academy of Sciences of the United States of America·2018
Same author

Cumulative effects of air pollution on public health.

Statistics in medicine·2005
Same author

Smoothing for spatiotemporal models and its application to modeling muskrat-mink interaction.

Biometrics·2004

Related Experiment Video

Updated: May 5, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

2.1K

An Approach to Fisher-Rao Metric for Infinite Dimensional Non-Parametric Information Geometry.

Bing Cheng1, Howell Tong2,3,4

  • 1Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

Entropy (Basel, Switzerland)
|May 4, 2026
PubMed
Summary

This study overcomes computational barriers in non-parametric information geometry by introducing a structural decomposition of the tangent space. This enables the Covariate Fisher Information Matrix (cFIM) for tractable statistical inference and provides a geometric basis for the Manifold Hypothesis.

Keywords:
Cramer–Rao Lower BoundFishe-r-Rao metricG-entropyKullback–Leibler divergenceStein scoreinformation geometrymanifold hypothesisnon-parametricorthogonal decompositiontangent space

More Related Videos

Development of New Methods for Quantifying Fish Density Using Underwater Stereo-video Tools
09:32

Development of New Methods for Quantifying Fish Density Using Underwater Stereo-video Tools

Published on: November 20, 2017

8.8K
Quantification of Information Encoded by Gene Expression Levels During Lifespan Modulation Under Broad-range Dietary Restriction in C. elegans
09:23

Quantification of Information Encoded by Gene Expression Levels During Lifespan Modulation Under Broad-range Dietary Restriction in C. elegans

Published on: August 16, 2017

9.4K

Related Experiment Videos

Last Updated: May 5, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

2.1K
Development of New Methods for Quantifying Fish Density Using Underwater Stereo-video Tools
09:32

Development of New Methods for Quantifying Fish Density Using Underwater Stereo-video Tools

Published on: November 20, 2017

8.8K
Quantification of Information Encoded by Gene Expression Levels During Lifespan Modulation Under Broad-range Dietary Restriction in C. elegans
09:23

Quantification of Information Encoded by Gene Expression Levels During Lifespan Modulation Under Broad-range Dietary Restriction in C. elegans

Published on: August 16, 2017

9.4K

Area of Science:

  • * Information Geometry
  • * Non-parametric Statistics
  • * Manifold Learning

Background:

  • * The
  • intractability barrier
  • in infinite-dimensional non-parametric information geometry hinders classical optimization and estimation.
  • * The Fisher-Rao metric's unbounded inverse in infinite dimensions poses computational challenges.
  • * Orlicz spaces offer a solution by providing exponential integrability for score functions and Fréchet differentiability for Kullback-Leibler divergence.

Purpose of the Study:

  • * To resolve the intractability barrier in non-parametric information geometry.
  • * To introduce a novel framework for statistical inference in infinite-dimensional spaces.
  • * To provide a formal mathematical justification for the Manifold Hypothesis.

Main Methods:

  • * Construction of statistical manifolds on Orlicz spaces (L0Φ(Pf)).
  • * Introduction of Structural Decomposition of the Tangent Space (TfM=S⊕S⟂) into covariate (S) and orthogonal (S⟂) subspaces.
  • * Derivation of the Covariate Fisher Information Matrix (cFIM) as a computable Hilbertian slice of the metric functional.

Main Results:

  • * Proof of the Trace Theorem (HG(f)=Tr(Gf)), identifying G-entropy as a geometric invariant.
  • * Demonstration of the geometric invariance of the cFIM as a covariant tensor under reparameterization.
  • * Establishment of the cFIM as the local Hessian of the KL-divergence and characterization of the Efficiency Standard via a generalized Cramer-Rao Lower Bound.
  • * Formal justification of the Manifold Hypothesis through the identification of low-dimensional information-concentrated subspaces.

Conclusions:

  • * The proposed framework resolves computational intractability in non-parametric information geometry.
  • * The Covariate Fisher Information Matrix (cFIM) enables efficient semi-parametric inference.
  • * Statistical information is characterized as a geometric interaction between data, system, and mechanism, validating the Manifold Hypothesis.