Derivation of the Cramér-Rao Lower Bound (CRLB)

Derivation of the Cramér-Rao Lower Bound (CRLB)

The Clamell-Lauber lower bound provides a lower bound for measuring how much uncertainty an estimator has in statistics, using the Fisher Information Matrix described in “Overview of the Fisher Information Matrix and Related Algorithms and Examples of Implementations. The procedure for deriving the CRLB is described below.

Suppose that the probability distribution under consideration is \(f(x;\theta)\) with parameter \(\theta\), and that \(X_1, X_2, \ldots, X_n\) are random variables that follow this distribution in independent equidistribution. In this case, the likelihood function (Likelihood Function) is expressed as follows.

\[ L(\theta) = f(x_1;\theta) \cdot f(x_2;\theta) \cdot \ldots \cdot f(x_n;\theta) \]

By taking the log-likelihood function and taking the expected value, we can obtain the Fisher Information Matrix. The diagonal component of the Fisher Information Matrix \(I(\theta)\) represents the inverse of the accuracy of the individual parameters.

The CRLB becomes an inequality that gives a lower bound on the variance of the estimator; the CRLB for one parameter \(\theta_i\) is expressed as follows.

\[ \text{Var}(\hat{\theta}_i) \geq \frac{1}{n \cdot I_{ii}(\theta)} \]

where \(\hat{\theta}_i\) is the estimator of the parameter \(\theta_i\).

If the CRLB for all parameters is expressed as a matrix, the Covariance Matrix can be obtained by taking the inverse matrix. This gives a lower bound on the covariance matrix for the multidimensional parameter vector.

The CRLB indicates that no estimator can have a variance less than this lower bound. However, this lower bound is achieved under very ideal conditions, and the goal is to get close enough to this lower bound in real problems.

Algorithm used to derive the Kramer-Lauber lower bound

The Cramer-Rauber Lower Bound is derived using the Fisher Information Matrix (Fisher Information Matrix). Below is a brief description of the main steps in the CRLB derivation procedure.

1. obtaining the log-likelihood function:

If the probability distribution is \(f(x;\theta)\) and the parameter is \(\theta\), the likelihood function \(L(\theta)\) is \(L(\theta) = f(x_1;\theta) \cdot f(x_2;\theta) \cdot \ldots \cdot f(x_n;\theta)\), which is log-transformed to obtain the log likelihood function \(l(\theta)\).

2. Obtaining the expected value:

Take the expected value of the log-likelihood function \(l(\theta)\). The expected value is the average over the entire data set.

3 Calculate the partial derivatives for the parameters:

Differentiate the expectation by the parameter \(\theta\). This yields a vector that represents how much the variation with respect to the parameters affects the log-likelihood function.

4. computation of the Fisher information matrix:

Partial differentiation is used to compute the Fisher information matrix \(I(\theta)\). The Fisher information matrix is a matrix of parameter variances and covariances, with the diagonal components representing the inverse of the accuracy for the individual parameters.

5. computation of CRLB:

The CRLB is obtained by taking the inverse of the Fisher information matrix and dividing it by \(n\) (the number of data); the CRLB gives a lower bound on the variance of the estimator for each parameter.

The derivation of the CRLB combines concepts from statistics and information theory, and requires advanced mathematics for rigorous derivation. Although the procedure will vary depending on the specific probability distribution and model, the steps above provide the gist of a general derivation procedure.

Application of the Kramer-Lauber Lower Boundary

The Cramer-Rauber lower bound plays an important role in a variety of statistical estimation problems because it provides a theoretical lower bound on the accuracy of the estimator. The following are examples of applications of the CRLB.

1. evaluation of the minimum variance unbiased estimator (MVUE):

The CRLB represents the minimum variance that the Minimum Variance Unbiased Estimator (MVUE) can achieve. If an estimator has a variance close to the CRLB, it is likely to be the best estimator.

2. radar signal processing:

Radar signal processing requires estimates of target position and velocity, and CRLB helps to theoretically evaluate how accurately these estimates can be obtained.

3. communications engineering:

In communications engineering, CRLB is used to estimate signal parameters (e.g., frequency, phase). If the estimate is close to the CRLB, the performance of the estimation can be judged to be near optimal.

4. ecology:

In ecological studies, CRLB is applied to estimate ecological and biological parameters. For example, it is used to estimate specific characteristics of populations or habitats.

5. medicine:

In medical research and clinical diagnosis, CRLBs are applied to estimate physiological parameters of patients and characteristics of medical devices.

In these instances, CRLB is used to theoretically evaluate the performance of estimators; CRLB provides an indication of the extent to which the most efficient estimation is possible in a given statistical model and is useful in evaluating the performance of estimation methods and techniques in practice.

Example implementation of the Cramér-Rao Lower Bound (CRLB)

The CRLB is a theoretical measure and there is no direct implementation example for a concrete data set or probability distribution; the CRLB is derived by computing the Fisher Information Matrix, which in concrete problems is calculated using the derivative of the probability density function, the likelihood function, and the derivative of the log-likelihood function.

The following is a general Python code as a procedure for calculating CRLB. In the following example, the CRLB is computed for the estimation of the mean parameter in a normal distribution.

import numpy as np

def crlb_normal_distribution(variance, sample_size):
    """
    Function to compute the CRLB of the mean parameter in a normal distribution
    :param variance: Variance of Normal Distribution
    :param sample_size: sample size
    :return: CRLB of average parameter
    """
    fisher_information = 1 / (sample_size * variance)
    crlb = 1 / fisher_information
    return crlb

# Example: Calculate the CRLB of the mean parameter for a normal distribution with a variance of 2.0 and a sample size of 100
variance = 2.0
sample_size = 100
crlb_mean = crlb_normal_distribution(variance, sample_size)

print(f"CRLB of average parameter: {crlb_mean}")

In this example, the variance of the normal distribution is known and the CRLB of the mean parameter is calculated. In actual problems, a procedure is needed to calculate the Fisher information matrix based on the probability density function or likelihood function and to derive the CRLB from the inverse matrix. Depending on the data and model, the specific mathematical equations for doing this will vary.

Challenges in deriving the Cramér-Rao Lower Bound (CRLB) and how to address them

Several challenges exist in the derivation of the Cramér-Rao Lower Bound (CRLB). Addressing these challenges requires a careful mathematical approach and consideration of the specific nature of the problem. These challenges and their remedies are described below.

1. very complex probability distributions:

Challenge: Although the CRLB is presented in a form that has some generality, certain probability distributions and models can make the computation of the Fisher information matrix very complex.

Solution: Because the derivation of the CRLB requires rigorous mathematical methods, especially for complex probability distributions, numerical or approximate methods may be used. In special cases, it may be possible to derive it in a simple form.

2. parameter dependencies:

Challenge: CRLB can depend on the choice of parameters, especially in nonlinear models or when correlations are strong, and the results can change depending on which parameters are estimated.

Solution: While specific parameter choices are made in the derivation of the CRLB, the characteristics of the problem should be taken into account when doing so, and the CRLB should be calculated for the most interesting parameters. If the parameters are highly correlated, extensions or special methods may be used to account for the correlation.

3. unknowns in the true model:

Challenge: Derivation of CRLBs requires a true model, but in real-world problems it is common for the true model to be unknown.

Solution: For unknown true models, predictive or hypothetical models may be used to derive the CRLB instead. Methods that account for model uncertainty have also been studied to deal with model uncertainty.

Reference Information and Reference Books

For more information on optimization in machine learning, see also “Optimization for the First Time Reading Notes” “Sequential Optimization for Machine Learning” “Statistical Learning Theory” “Stochastic Optimization” etc.

Reference books include Optimization for Machine Learning

“Machine Learning, Optimization, and Data Science“

“Linear Algebra and Optimization for Machine Learning: A Textbook“