What is Fisher information: Definition and 19 Discussions
In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior (according to the Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information is also used in the calculation of the Jeffreys prior, which is used in Bayesian statistics.
The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test.
Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have been shown to obey maximum Fisher information. The level of the maximum depends upon the nature of the system constraints.
Hi,
I have been studying the Fisher matrix to apply in a project. I understand how to compute a fisher matrix when you have a simple model for example which is linear in the model parameters (in that case the derivatives of the model with respect to the parameters are independent of the...
I'm getting interested in quantum-enhanced metrology and have come across the quantum Fisher information (QFI) as a measure of how much a quantum state ##|\Psi(\theta)\rangle## changes with respect to some variable, for example, the phase accumulated during an interferometer, ##\theta##. This is...
In the context of a forecast, I am currently working on Fisher's formalism which is part of a more general theory, that of information. My problem applies to estimating cosmological parameters from input data with the Fisher formalism and recipes to build a Fisher matrix. The context is...
Hello,
I follow the post https://www.physicsforums.com/threads/cross-correlations-what-size-to-select-for-the-matrix.967222/#post-6141227 .
It talks about the constraints on cosmological parameters and forecast on futur Dark energy surveys with Fisher's matrix formalism.
Below a capture of...
Hello,
I am working on Fisher's formalism in order to get constraints on cosmological parameters.
I am trying to do cross-correlation between 2 types of galaxy populations (LRG/ELG) into a total set of 3 types of population (BGS,LRG,ELG).
From the following article...
I am currently studying Fisher's formalism as part of parameter estimation.
From this documentation :
They that Fisher matrix is the inverse matrix of the covariance matrix. Initially, one builds a matrix "full" that takes into account all the parameters.
1) Projection : We can then do...
I'm trying to construct a Fisher Forecast for the upcoming S4 CMB survey. I don't understand
what the C_l is in this formula. It is H(z) and the Angular Distance? Or is it some covariance matrix and if it is a covariance matrix how do I calculate it considering the experiment hasn't been done...
I'm trying to recreate the results of this paper https://arxiv.org/pdf/1607.08016.pdf
to obtain the constraints for the matter density and Hubble constant h.
However every time I try to create there results my Fisher Matrix has elements of order of 10^14 which is far to high. I suspect this is...
I'm doing a Fisher Forecast to satisfy my experimental physics requirement. I never done anything like this before so I'm kind of in the dark of how to proceed. I have done some readings on Fisher Forecasting and would like to outline how I would do it and then ask if you can tell me if I'm...
Homework Statement
Let X1, X2,...Xn be a random sample from pdf,
f(x|θ) = θx-2 where 0 < θ ≤ x < ∞
Find the MLE of θMy attempt:
Likelihood fxn: L(θ|x) = ∏θx-2 = θn∏ θx-2
And to find MLE, I take Log of that function and partial derivative (w.r.t θ, of log L(θ|x) and set that = 0, and get...
Homework Statement
Let X1, X2,...Xn be a random sample from pdf,
f(x|θ) = θx-2 where 0 < θ ≤ x < ∞
Find the MLE of θMy attempt:
Likelihood fxn: L(θ|x) = ∏θx-2 = θn∏ θx-2
And to find MLE, I take Log of that function and partial derivative (w.r.t θ, of log L(θ|x) and set that = 0, and get...
Hello everyone,
I am currently considering a set of random variables, \vec{x} = [x_1,x_2,...x_N] which are know to follow a multivariate normal distribution,
P(\vec{x}) \propto \mathrm{exp}(-\frac{1}{2}(\vec{x}-\vec{\mu})^\mathrm{T}\Sigma^{-1}(\vec{x}-\vec{\mu}))
The covariance matrix Σ and...
Hello,
I'm trying to calculate Fisher information (and eventually the Cramer-Rao lower bound) for this particular pdf with Mathematica:
\text{pte}[t,\Theta ] = \frac{P_{\text{ec}}}{\tau _d-\tau _r}\left[e^{\frac{-(t-\Theta )}{\tau _d}}-e^{\frac{-(t-\Theta )}{\tau _r}}\right]...
I don't understand the following step regarding the (i,j)^{th} element of the Fisher Information Matrix, \textbf{J}:
J_{ij}\triangleq\mathcal{E}\left\{ \frac{\partial}{\partial\theta_{i}}L_{\textbf{x}}(\textbf{θ})\frac{\partial}{\partial\theta_{j}}
L_{\mathbf{x}}(\textbf{θ})\right\}
\\...
Hi everyone.
Homework Statement
We are given N spins 1/2. A rotation is defined as
\rho_\theta=e^{-i\theta J_n}\rho_\theta e^{i\theta J_n}
on an Hilbert Space H, with
J_n=n_xJ_x+n_yJ_y+n_zJ_z\:,\quad n_x^2+n_y^2+n_z^2=1,
and \theta isn't related to any observable.
Given a quantum state...
How to understand "fisher information"?
Hello, I am trying to understand what "fisher information is."
It is defined as V [∂/∂∅(lnf(X,∅)) ]=E[ (∂/∂∅[lnf(X,∅)])^2 ].
From Wikipedia:
Can you please help me understand why this is the case? How can this be explained by looking at the...
Maybe it is a good idea to have mathematical tutorials like in the subforum about Physics. I have no homework anymore :smile: but tend to think that doing mathematics is a skill that always pays back at the end.
First thing I want to know what https://en.wikipedia.org/wiki/Fisher_information...
Have some of you read this book by Roy Friedel?
What are your comment?
I am also interrested in other sources about "information" and "Fisher information" as it could be used in physics.
Thanks
Michel