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Related Concept Videos

Implicit Differentiation with Partial Derivatives01:27

Implicit Differentiation with Partial Derivatives

Implicit differentiation with partial derivatives is used when a relationship between two variables is defined implicitly rather than explicitly. Instead of solving one variable in terms of the other, the variables remain connected through a single equation. In this setting, one variable is treated as depending on the other, and differentiation is applied directly to the entire relation.To differentiate an implicit relation, the chain rule is applied to every term in the equation. Because one...
Implicit Differentiation: Problem Solving01:29

Implicit Differentiation: Problem Solving

Curves defined implicitly, where variables cannot be separated algebraically, require specialized techniques for analysis. The conchoid of Nicomedes exemplifies such a case. Its equation links x and y in a way that prevents isolation of one variable, making implicit differentiation essential to determine the slope and behavior at any point on the curve.The implicit form of the conchoid can be expressed as:To differentiate this equation, y is treated as a function of x, and the chain rule is...
Implicit Differentiation01:25

Implicit Differentiation

In classical mechanics, motion is often described through relationships between spatial coordinates and time. A car moving along a straight highway with constant acceleration serves as a simple case where velocity is an explicit function of time. This scenario results in a linear equation, enabling straightforward analysis using basic differentiation techniques.In contrast, a satellite in circular orbit follows a path defined by an implicit function. The position of the satellite is constrained...
Applications of Integration to Probability Density Functions01:27

Applications of Integration to Probability Density Functions

Continuous probability distributions are used to model random variables that can take on any real value within a specified range. These variables do not take on isolated or countable values but rather exist on a continuum. For example, the height of an individual can be measured with increasing precision—such as 163.5 or 165.25 centimeters—demonstrating that height is a continuous random variable.The behavior of such variables is described using a probability density function (PDF), which...
Statically Indeterminate Problem Solving01:16

Statically Indeterminate Problem Solving

Statically indeterminate problems are those where statics alone can not determine the internal forces or reactions. Consider a structure comprising two cylindrical rods made of steel and brass. These rods are joined at point B and restrained by rigid supports at points A and C. Now, the reactions at points A and C and the deflection at point B are to be determined. This rod structure is classified as statically indeterminate as the structure has more supports than are necessary for maintaining...
Second Derivatives of Implicit Functions01:29

Second Derivatives of Implicit Functions

Elliptical arches are fundamental in architectural and structural engineering, offering aesthetic appeal and structural efficiency. The shape of an elliptical arch follows a constrained geometric relationship where the height and horizontal position are implicitly related. This means that the height y cannot be explicitly expressed as a function of the horizontal position x, necessitating implicit differentiation for slope and curvature analysis.The equation of an ellipse centered at the origin...

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Updated: Jun 30, 2026

A Tactile Automated Passive-Finger Stimulator (TAPS)
19:44

A Tactile Automated Passive-Finger Stimulator (TAPS)

Published on: June 3, 2009

Simplifying debiased inference via automatic differentiation and probabilistic programming.

Alex Luedtke1

  • 1Department of Statistics, University of Washington.

Journal of the Royal Statistical Society. Series B, Statistical Methodology
|June 29, 2026
PubMed
Summary
This summary is machine-generated.

We developed Dimple, an algorithm that simplifies creating efficient estimators by automating complex statistical derivations. This tool makes advanced statistical estimation more accessible for researchers and data scientists.

Keywords:
Hadamard differentiabilityasymptotic efficiencyautomatic differentiationdifferentiable programmingefficient influence functionpathwise differentiabilityprobabilistic programming

Related Experiment Videos

Last Updated: Jun 30, 2026

A Tactile Automated Passive-Finger Stimulator (TAPS)
19:44

A Tactile Automated Passive-Finger Stimulator (TAPS)

Published on: June 3, 2009

Area of Science:

  • Statistics
  • Computer Science
  • Machine Learning

Background:

  • Constructing efficient estimators typically requires complex mathematical derivations, such as calculating the efficient influence function.
  • This complexity limits the accessibility of advanced statistical estimation techniques to a broader audience.

Purpose of the Study:

  • To introduce Dimple, an algorithm designed to simplify the construction of efficient estimators.
  • To make efficient estimation more accessible by automating the derivation of necessary components.

Main Methods:

  • Dimple takes computer code defining a parameter of interest and outputs an efficient estimator.
  • It utilizes automatic differentiation applied to the statistical functional, avoiding manual derivation of the efficient influence function.
  • The approach requires expressing the functional as a composition of primitives meeting a novel differentiability condition.

Main Results:

  • Dimple successfully generates efficient estimators without requiring users to compute the efficient influence function.
  • The algorithm identifies necessary nuisance parameters through the functional composition.
  • A Python implementation demonstrates that Dimple enables efficient estimation with minimal code.

Conclusions:

  • Dimple significantly simplifies the process of constructing efficient estimators.
  • The algorithm broadens the accessibility of advanced statistical estimation techniques.
  • Reusable software primitives enhance efficiency and applicability across diverse estimation problems.