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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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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Fisher Scoring for Constrained optimization
Suppose I am estimating a parameter vector using the usual Fisher Scoring updates;
\begin{equation}
\theta_{s+1} = \theta_{s} + \lambda\mathcal{I}(\theta_{s})^{-1}\frac{dl}{d\theta}\bigg|_{\theta=\...
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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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1answer
28 views
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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35 views
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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39 views
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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1answer
169 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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1answer
74 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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1answer
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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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28 views
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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1answer
31 views
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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2answers
268 views
Score function of poisson distribution
I have a stupid question I haven't figured out.
So when counting the score for poisson distribution I get the log likelihood
$$S(\mu ) = \frac{\partial \ell(\lambda )}{\partial \lambda } = \sum_1^n \...
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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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1answer
39 views
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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39 views
The role of Fisher information matrix in Fisher kernel
I read the original paper proposed the Fisher kernel. The Fisher kernel is defined as $K(X_i,X_j) \propto U_{X_i}I^{-1}U_{X_j}$, where $U_X$ is the Fisher schore and $I$ is the Fisher information ...
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1answer
134 views
If $X\sim\text{Beta}(\theta,1)$, obtain the confidence interval of $100(1-\alpha)\%$ based on the asymptotic distribution of the score function
Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\text{Beta}(\theta,1)$. Obtain the approximate confidence interval of $100(1-\alpha)\%$ based on the asymptotic ...
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1answer
70 views
Given $X\sim\mathcal{N}(0,\sigma^{2})$, obtain the Fischer information of $\sigma$ and $\sigma^{2}$
Suppose the random variable $X\sim\mathcal{N}(0,\sigma^{2})$, where we do not know the value of the standard deviation $\sigma$. Then obtain the Fisher information $I_{F}(\sigma)$ through $X$. Suppose ...
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Fisher Information for a Gaussian Process
Suppose I fit a Gaussian process to data such that the posterior distribution over any output is also a Gaussian process, $\mathcal{G}\mathcal{P}(\mu(x),\sigma^2(x))$ where $x$ is some valid input. ...
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Fisher information in GLM where $Y_1, Y_2,… ,Y_n$ are independent but not identically distributed
When making inference about regression coefficients, our estimate of asymptotic covariance matrix of $\mathbf{\hat {\beta}}$ is
$$\hat{cov}(\hat{\beta}) = I^{-1}(\hat{\beta})$$
where $I$ is Fisher ...
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Weighted D optimality
I am working with D optimality for nonlinear models. My model has 8 parameters and some are more important than others. Is there a way to give weights to each of the parameters?
I am thinking of ...
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1answer
169 views
Computing Empirical Fisher Information matrix for natural gradient
I would like to implement the natural gradient for reinforcement learning as described in the following paper: https://arxiv.org/pdf/1703.02660.pdf
However, I do not know how to compute the empirical ...
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1answer
117 views
Observed information matrix with multivariate normal distribution
$$
\DeclareMathOperator\tr{tr}
\DeclareMathOperator\vecOP{vec}
\newcommand\di{\mathrm{d}}
\newcommand\D{\mathrm{D}}
\newcommand\Hess{\mathrm{H}}
$$
I do not have much experience with matrix ...
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Finding Fisher information [duplicate]
Let $X$ distribution belongs for the family $\mathcal{P}\{P_{\theta}, \theta \in \Theta \}$. We need to find Fisher information $I(\theta)$ according $n$ simple sample, when $P_{\theta}$ is $N(\mu,\...
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1answer
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Cramér-Rao Lower Bound & Fisher information - error in textbook?
I'm currently reading the textbook "Statistics for Mathematician" from Victor Panaretos. On page 65, the author presents the following equation for the Cramér-Rao Lower Bound (Note: I set the ...
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218 views
How to compute variance/confidence intervals from Fisher information matrix. Mistake in this document?
https://www.stat.umn.edu/geyer/5931/mle/mle.pdf
In this document (by the great Geyer nonetheless!)
it calculates confidence intervals using the Fisher matrix:
But the standard deviation is not the ...
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123 views
Singular Fisher Information and nuisance parameters
I am interested in a problem where the Fisher Information matrix is singular and I want to treat some of the parameters as nuisance parameters.
If the FI matrix is singular, that means that some ...
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1answer
201 views
Confidence interval for the 95th percentile of the normal distribution
Let $X_1, .., X_n \sim Normal(\mu, \sigma^2)$.
Let $\tau$ be the 95th percentile of this distribution. Thus,
$P(X_i < \tau) = 0.95$.
What is the $1 - \alpha$ confidence interval for $\tau$?
I ...
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Alternatives to calculating the rank of the information matrix in determining if the model is identifiable
I have a known non-linear model $h \in \mathbf{R}^n$:
$$
y = h(\theta) + \epsilon,
$$
where $\theta\in \mathbf{R}^m$ is a parameter vector, and $\epsilon$ is a normal random variable with zero mean ...
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1answer
147 views
Different version of Wald test statistic formula
I came across two formulas for the Wald test statistic in a maximum likelihood framework:
One has $(R\hat{\theta}-r)'(RI_n^{-1}R')^{-1}(R\hat{\theta}-r)$, where $I_n^{-1}$ is the inverse of the ...
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1answer
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Statistics : Why is the Cramer-Rao Lower Bound (CRLB) inverse of the Fisher Information I(θ) ?
Why is the Cramer-Rao Lower Bound (CRLB) inverse of the Fisher Information I(θ) ? Could someone provide an intuitive explanation? I am having trouble understanding the concept.
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Inference for Maximum Likelihood Estimator Using Particle Filter
How does one compute standard errors for the MLE when using a particle filter approximation to the likelihood? I know that the estimator is asymptotically normal and that the variance-covariance ...
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1answer
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understanding natural policy gradient
I'm reading this paper on Natural Policy Gradient https://papers.nips.cc/paper/2073-a-natural-policy-gradient.pdf and have some questions regarding how it works.
I'm coming at this from an ML ...
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1answer
372 views
Fisher information for MLE with constraint
Supposing I have a probability distribution $f(x|\vec\theta)$, where $x$ is a random variable and $\vec\theta$ is a vector of distribution parameters. I also know that parameters $\vec\theta$ should ...
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1answer
101 views
Why does Fisher use covariance when only variance is needed?
With reference to the following image from here: (can not inline it due to unsupported format)
https://wikimedia.org/api/rest_v1/media/math/render/svg/9af8aa035642689bb2004047416b069a15406447
If we ...
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70 views
Fisher information for models that are not one-to-one
When a parameterized data model and corresponding pdf are known, the Cramér-Rao lower bound provides an lower bound for the variance of an estimator of one of the parameters. That is, given the data ...
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139 views
Observed Fisher Information & Cramer-Rao bound
The Cramer-Rao bound is usually derived for the "expected" or "total" fisher-information. Is a similar result possible for the observed-fisher information?
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Is there a Fisher Information equivalent in MAP Empirical Bayes estimation?
Background
The Fisher information for a linear Gaussian model is
$\mathcal{I}_{\theta} = \frac{X X^T}{\sigma^2} $. This is used in optimal experiment design techniques, for example, maximisation of $|...
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1answer
205 views
Does a quadratic log-likehood mean the MLE is (approximately) normally distributed?
So, in the usual case, one can prove from the asymptotic normality of a maximum likelihood estimator that the corresponding log-likehood surface is quadratic near the MLE (e.g. in the proof of the ...
