The Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ upon which the probability of X depends Let f(X; θ) be the probability density function (or probability mass function) for X conditioned on the value of θ
What does the Fisher information tell us?
What is Fisher Information? Fisher information tells us how much information about an unknown parameter we can get from a sample In other words, it tells us how well we can measure a parameter, given a certain amount of data
How is Fisher information calculated?
Given a random variable y that is assumed to follow a probability distribution f(y;θ), where θ is the parameter (or parameter vector) of the distribution, the Fisher Information is calculated as the Variance of the partial derivative wrt θ of the Log-likelihood function ℓ(θ | y)
Is Fisher information a matrix?
Fisher Information Matrix is defined as the covariance of score function It is a curvature matrix and has interpretation as the negative expected Hessian of log likelihood function
What does the likelihood function tell you?
Likelihood function is a fundamental concept in statistical inference It indicates how likely a particular population is to produce an observed sample Let P(X; T) be the distribution of a random vector X, where T is the vector of parameters of the distribution
What is efficient estimator in statistics?
A measure of efficiency is the ratio of the theoretically minimal variance to the actual variance of the estimator This measure falls between 0 and 1 An estimator with efficiency 10 is said to be an “efficient estimator” The efficiency of a given estimator depends on the population
Can Fisher information be negative?
In statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the “log-likelihood” (the logarithm of the likelihood function)
Can the Fisher information be zero?
The right answer is to allocate bits according the Fisher information (Rissanen wrote about this) If the Fisher information of a parameter is zero, that parameter doesn’t matter We call it “information” because the Fisher information measures how much this parameter tells us about the data
Is the Fisher information always positive?
The Fisher information is the variance of the score, given as I(θ)=E[(∂∂θlnf(x∣θ))2], which is nonnegative
How is Cramer Rao lower bound calculated?
= p(1 − p) m Alternatively, we can compute the Cramer-Rao lower bound as follows: ∂2 ∂p2 log f(x;p) = ∂ ∂p ( ∂ ∂p log f(x;p)) = ∂ ∂p (x p − m − x 1 − p ) = −x p2 − (m − x) (1 − p)2
What is regularity condition?
The regularity condition defined in equation 629 is a restriction imposed on the likelihood function to guarantee that the order of expectation operation and differentiation is interchangeable
What is asymptotic variance?
Though there are many definitions, asymptotic variance can be defined as the variance, or how far the set of numbers is spread out, of the limit distribution of the estimator
How do you show asymptotic normality?
Proof of asymptotic normality Ln(θ)=1nlogfX(x;θ)L′n(θ)=∂∂θ(1nlogfX(x;θ))L′′n(θ)=∂2∂θ2(1nlogfX(x;θ))
Is likelihood a probability density?
Therefore one should not expect the likelihood function to behave like a probability density Okay but the likelihood function is the joint probability density for the observed data given the parameter θ As such it can be normalized to form a probability density function
What are the features of probability density function?
The probability density function is non-negative for all the possible values, ie f(x)≥ 0, for all x Due to the property of continuous random variable, the density function curve is continuous for all over the given range which defines itself over a range of continuous values or the domain of the variable
What is the difference between likelihood and possibility?
As nouns the difference between likelihood and possibility is that likelihood is the probability of a specified outcome; the chance of something happening; probability; the state of being probable while possibility is the quality of being possible
How do you determine if the estimator of the data is relatively efficient?
We can compare the quality of two estimators by looking at the ratio of their MSE If the two estimators are unbiased this is equivalent to the ratio of the variances which is defined as the relative efficiency
What would it mean if an estimator is inefficient compared to another estimator?
Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance An efficient estimator is characterized by a small variance or mean square error, indicating that there is a small deviance between the estimated value and the “true” value
Can an estimator be biased and efficient?
The fact that any efficient estimator is unbiased implies that the equality in (77) cannot be attained for any biased estimator However, in all cases where an efficient estimator exists there exist biased estimators that are more accurate than the efficient one, possessing a smaller mean square error
What is the negative log likelihood?
Negative Log-Likelihood (NLL) We can interpret the loss as the “unhappiness” of the network with respect to its parameters The negative log-likelihood becomes unhappy at smaller values, where it can reach infinite unhappiness (that’s too sad), and becomes less unhappy at larger values
Is a normal distribution asymptotic?
Perhaps the most common distribution to arise as an asymptotic distribution is the normal distribution In particular, the central limit theorem provides an example where the asymptotic distribution is the normal distribution
What is the Cramer Rao lower bound of the variance of an unbiased estimator of theta?
The function 1/I(θ) is often referred to as the Cramér-Rao bound (CRB) on the variance of an unbiased estimator of θ I(θ) = −Ep(x;θ) { ∂2 ∂θ2 logp(X;θ) } and, by Corollary 1, X is a minimum variance unbiased (MVU) estimator of λ
What is the use of Cramer Rao inequality?
The Cramér-Rao Inequality provides a lower bound for the variance of an unbiased estimator of a parameter It allows us to conclude that an unbiased estimator is a minimum variance unbiased estimator for a parameter
Why we use Cramer Rao inequality?
The Cramér–Rao inequality is important because it states what the best attainable variance is for unbiased estimators Estimators that actually attain this lower bound are called efficient It can be shown that maximum likelihood estimators asymptotically reach this lower bound, hence are asymptotically efficient