Questions tagged [fisher-information]

The Fisher information measures the curvature of the log-likelihood and can be used to assess the efficiency of estimators.

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Does the fisher information matrix exist when we can only calculate the quasi-likelihood and not the full-likelihood function?

Does the fisher information matrix exist when we can only calculate the quasi-likelihood and not the full-likelihood function? In GEE, the full-likelihood isn't calculated, but the variances are ...
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expected value of a score function (the gradient of the log-likelihood function)

according to the Wikipedia: https://en.wikipedia.org/wiki/Score_(statistics), expected value of a score function should equals to zero and the proof is following: \begin{equation} \begin{aligned} \...
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Clarifying the definition of Fisher information

I'm studying the Fisher information, which leads me to the wiki page: https://en.wikipedia.org/wiki/Fisher_information It says: Formally, the partial derivative with respect to θ of the natural ...
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What is the geometric relationship between the covariance matrix and the inverse of the covariance matrix?

The covariance matrix represents the dispersion of data points while the inverse of the covariance matrix represents the tightness of data points. How is the dispersion and tightness related ...
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MLEs multivariate normal distribution estimation

I’m a beginner in this field, I hope the problem will be clear… . Under some regularity assumption the MLE estimators of unknown parameters are unbiased and their distributions is a multivariate ...
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Proof Sample Variance is Minimum Variance Unbiased Estimator for Unknown Mean

I am trying to prove that the unbiased sample variance is a minimum variance estimator. In this problem I have a Normal distribution with unknown mean (and the variance is the parameter to estimate so ...
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54 views

Cramér-Rao lower bound with an uncertain observable

Let's say I have an observable $X$ that depends on a parameter $\theta$, and that I can find an expression for the Cramér-Rao lower bound for estimating $\theta$, as a function of $X$: $\sigma_{CRB}(X)...
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Why Fisher Information uses Log Likelihood and not Plain Likelihood [duplicate]

I would like to know that to determine Fisher information from the Likelihood model, why do we take the log of the likelihood first instead of using normal likelihood ?
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Conflicting Definition of Information in Statistics | Fisher Vs Shannon

The notion of information as per Shannon is that if the probability of RV is close to 1, there is little information in that RV because we are more certain about the outcome of the RV so there is ...
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Deriving C-R inequality from H-C-R bound

As mentioned in the title, I want to derive the Cramer-Rao Lower bound from the Hammersly-Chapman-Robbins lower bound for the variance of a statistic $T$. The statement for the H-C-R lower bound is ...
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Why is the Fisher information of a one-parameter family of exponential distributions the inverse of the variance?

Let $$ f(x; \lambda) = h(x) e^{\eta(\lambda) J(x) + \xi(\lambda)} $$ be a one-parameter family of distributions. In my course notes, it is claimed that $$ I(\lambda) := \mathbb{E}_\lambda\left[ \left(...
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Fisher Information | Second Derivative of Likelihood Vs Second Derivative of Log Likelihood

I watched this video on Fisher Information and it is mentioned that in Taylor series expansion of the likelihood function the second derivative is parabola which is not a good approximation and a ...
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How to compute variance of Cox model coefficient estimate using Fisher information?

We have Cox proportional hazards model: $$ \lambda(t,x) = \lambda_0(t)exp(\boldsymbol \beta'\boldsymbol x),$$ where $\boldsymbol \beta$ and $\boldsymbol x$ are vectors. To make it simple, lets say ...
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Inequalities on Fisher Information / expected second derivative?

Under some regularity conditions we can compute fisher information as $ - \mathbb{E}_{\theta_0} [\frac{\partial}{\partial \theta^2} \ln f(x;\theta_0)] $ I was wondering if there are some kind of ...
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Fisher's information for weibull 2 parameter

I want to find Fisher's information $I(\theta)$ with $-E(\frac{d^2(ln(fx))}{d\theta^2})$ but in two parameters, the expectation is too complex for me. Is there something about the Weibull distribution ...
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Connection between the two expressions of fisher information

I've seen two major expressions for the fisher information: $$I_{\mathbf{X}}(\theta) = \mathbb{V}ar(\ell'(\theta|\mathbf{X}))$$ where $\ell(\theta)$ is the log-likelihood function for $\mathbf{X}$...
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Differences between Sampler, MonteCarlo, Metropolis-Hasting method, MCMC method and Fisher formalism

1) I make confusions about what we call a "sampler". From what I understand, a sampler allows to generate a distribution of points that follows a known PDF (probability distribution function), doesn't ...
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MLE for bivariate Normal distribution and Fisher Information

I got very confused trying to understand the meaning of Fisher information, and link it with the information for parameters contained in samples. Suppose $(X_1,X_2) \sim \mathcal{N}(\begin{bmatrix} \...
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What is the difference between standard errors using the inverse of hessian and calculated using the the inverse of hessian and Fisher information?

In one of R packages for advanced survival analysis, the frailtypack, the output contains standard errors calculated in two ways, named: H (using the inverse of Hessian) and HIH (...
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Score asymptotically belongs to the span of Fisher's information matrix

I am reading a paper from Poskitt and Tremayne titled "Testing the specification of a fitted autoregressive-moving average model". The paper is concerned with a solution to the singularity of Fisher'...
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explanation of why an UMVUE doesn't necessarily have to achieve the CRLB?

I'm studying uniformly minimum variance unbiased estimator(UMVUE). I have seen question on this site asking why the UMVUE doesn't achieve the CRLB(Cramer Rao lower bound), and all of the answers have ...
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Information contained in sample about the…estimator?

I have always understood Fisher information to represent the information contained in data regarding some unknown parameter. In my professor's notes, he writes about the information contained in the ...
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Fisher Information for Cox Model

Actually, I'm working on a Statistical Genetics Article (Schaid and al,2010) in a retrospective likelihood context. In the article, authors present some result about conditional likelihood but I can't ...
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Fisher matrix with penalty function

I am fitting a parametric model to human tracking data. Because my data is in part corrupted (see below why), as such I had to introduce a penalty function to my optimization algorithm. The Fisher ...
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Different formulas for the Fisher Information, for use in Cramér-Rao lower bound. Correct? Assumptions?

I am taking a course in Statistical Inference. We have to calculate the Cramér-Rao lower bound to determine if different statistics are efficient or not. I have read that as long as we have an ...
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Fisher Information with respect to the Standard deviation of Normal distribution

Let $X\sim\mathcal{N}(0,\sigma^2)$ be given. I computed the Fisher Information to be $I(\sigma)=\frac{2}{\sigma^2}$. Note that the Fisher Information for the variance is given by $I(\sigma^2)=\frac{1}{...
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Is it true that Fisher information for a statistic and the sample are equal if and only if the statistic is sufficient?

According to https://en.wikipedia.org/wiki/Fisher_information#Sufficient_statistic we have if and only if, but according to https://projecteuclid.org/download/pdfview_1/euclid.imsc/1362751193 we don'...
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How can I find CRLB for an integer valued Maximum Likelihood Estimator?

Reference: How to find Maximum Likelihood estimates of an *integer* parameter? Extending the above question, how to find the Fisher information and hence CRLB for the integer-valued Estimator using ...
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Fisher Information Matrix for regression problem

In DeepMind's paper [Overcoming catastrophic forgetting in neural networks]( https://arxiv.org/abs/1612.00796), elastic weight consolidation with Fisher Information Matrix is used to tackle the ...
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Signal-to-noise-ratio, Fisher information and and “estimability”

Given a parametric statistical model, is it common to study the quantity $$ Q_{\theta} = \theta^2 I_{\theta} \, ,$$ where $I_{\theta}$ is the Fisher information? (I focus on a single parameter for ...
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Fisher Information for general one parameter exponential family (inconsistent with Poisson distribution)

For one of my hw questions, I was asked to derive Fisher Information for one parameter exponential family. Here's my approach: $$L(\theta) = f(x\mid\theta) = e^{\theta T(x) - \eta(\theta)}h(x)$$ $$\...
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Sampling distribution of the Score statistics of a GLM MODEL

In the context of a GLM (with a distribution that belongs to the exponential family), we often compute the score statistics $$ U = \frac{\partial LogLike(\boldsymbol{\beta};\mathbf{y})}{\partial\...
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Fisher Information for Gamma

Question: Find the fisher's information for $\mathcal{G}(\alpha, \beta)$ ,$\beta$ known. Attempt: Since $\mathcal{G}(\alpha, \beta)$ ,$\beta$ known. $$f(x|\alpha) =\dfrac{x^{\alpha-1}e^{x/\beta}}...
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Fisher information for double exponential distribution [duplicate]

I am asked to find the Fisher information function for double exponential distribution with mean $\mu$ and variance 1. Here's my approach: $l(\mu) = -\frac{1}{2}e^{|x-\mu|}$, so $\log l(\mu) = ...
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Compute the information matries related to normal distribution

This is a problem that I have trouble with. Suppose that we have $X_{1}, \ldots, X_{m}$ are iid $N\left(\mu, \sigma^{2}\right), Y_{1}, \ldots, Y_{n}$ are iid $N\left(0, \sigma^{2}\right),$ the $X$...
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Why do we use $S^2$ while estimating the variance?

Sorry the title is a bit silly, but I currently confront a problem related to Fisher's information. Let $X_1, X_2, \cdots, X_n$ be of $N(\mu , \sigma^2 )$ distribution where $\mu$ is known, $U^2 := ...
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548 views

Fisher information from MLE in R?

Reworded the question: I have read "The Fisher information I(p) is this negative second derivative of the log-likelihood function, averaged over all possible X = {h, N–h}, when we assume some value ...
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Inverse Covariance Matrix of a Gaussian Distribution: Relationship of Precision Matrix and Information Matrix

In the book "Probabilistic Robotics" (Thrun et al.), chapter 3.5.1 states that The canonical parameterization of a multivariate Gaussian is given by a matrix $\Omega$ and a vector $\xi$. The ...
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189 views

Calculate the uncertainty of a MLE

I have minimized the negative LL of a Poisson distribution to get an MLE of three parameters using scipy.minimize w/ Nelder-Mead. I want to calculate the uncertainty of the MLE. From reading, I ...
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Does C-efficiency exist? If so, how is it defined?

When using an algorithm to construct an optimal experimental design one has a measurement of optimality to find an optimal design with this optimality-criterion. For any arbitrary chosen design a D- ...
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Asymptotic Mean Squared Error of Maximum Likelihood estimator

I want to show that $n$ times mean squared error for the maximum likelihood estimator converges to the inverse of Fisher information, where $n$ is the number of samples. But The standard proofs of ...
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Statistical comparison of (covariance) matrices

I am trying to test whether the covariance matrix for the maximum likelihood estimates for a gaussian general lienear model approaches the inverse Fisher information matrix (times 1/n , n being the ...
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Does Fisher Information quantify precision? [duplicate]

Looking at perspective from estimating the actual value from a set of data measured by the instrument. Does Fisher information just quantify the precision of the measurement? What does it say about ...
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What statistic to use in testing the variance of maximum likelihood estimators

(A physicist self-studying statistics here) I was previously confused about the meaning of the standard error in a maximum likelihood estimate. Certain stack exchange posts (linked below) have gone ...
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Efficient GPU (batch) implementation of Empirical Fisher information matrix?

I have seen many implementations. It seems to be a limitation of autograd itself that we can compute the gradient of loglikelihood only one sample at a time. The batch version has been used but in a ...
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Action of Fisher information in “Latent Variable Model Selection…”

I'm having trouble understanding the role of the Fisher information matrix in the assumptions of Chandrasekaran et al. 2012. In the paper, the authors define the Fisher information matrix (i.e., ...
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Sufficient Statistic and Unbiased Estimate in Exponential Family

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he deals with sufficient statistics in exponential distributions ...
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Score function of poisson distribution

When counting the score for poisson distribution I get the log likelihood $$S(\mu ) = \frac{\partial \ell(\lambda )}{\partial \lambda } = \sum_1^n \left(\frac{y_i}{\lambda}-1\right)$$ Textbook says ...
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Fisher information matrix and gradients

I'm a math Ph.D. without formal training in statistics. Quite a few papers on normalization methods in deep learning mention the Fisher information matrix and how it's related to the Riemannian metric ...
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How do we deduce this fisher information relation?

Given a RS $X_{1},X_{2},\ldots,X_{n}$ whose distribution is well known (unless its parameters), how do we prove the following Fischer Information relationship \begin{align*} I_{F}(\theta) =\textbf{E}\...

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