7.4 Maximum likelihood estimation (MLE)

Maximum likelihood estimation (MLE) is a powerful statistical method for estimating parameters of probability distributions. It's widely used in signal processing to extract information from observed data, making it a crucial tool for engineers and researchers in the field.

MLE works by maximizing the likelihood function, which measures the probability of observing the given data. This approach provides consistent and efficient estimators, making it valuable for various applications in signal detection, estimation, and classification.

Basics of maximum likelihood estimation

• Maximum likelihood estimation (MLE) is a statistical method for estimating the parameters of a probability distribution by maximizing a likelihood function
• MLE is widely used in various fields, including signal processing, to estimate unknown parameters from observed data
• MLE provides a consistent approach to parameter estimation problems where the probability distribution depends on the parameters

Definition of MLE

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• MLE is a method of estimating the parameters of a probability distribution by maximizing the likelihood function
• The likelihood function measures the probability of observing the given data as a function of the unknown parameters
• MLE chooses the parameter values that make the observed data most probable

Principles of MLE

• MLE is based on the principle of maximum likelihood, which states that the best estimate of the parameters is the one that maximizes the likelihood function
• The likelihood function is constructed based on the assumed probability distribution of the data
• MLE involves finding the parameter values that maximize the likelihood function, typically by setting the derivative of the log-likelihood function to zero

Assumptions in MLE

• MLE assumes that the data are independently and identically distributed (i.i.d.) according to the assumed probability distribution
• The probability distribution is assumed to be known, but the parameters are unknown
• MLE assumes that the model is correctly specified and that the data are generated from the assumed probability distribution

MLE for parameter estimation

MLE for single parameter

• MLE can be used to estimate a single unknown parameter of a probability distribution
• The likelihood function is constructed based on the assumed probability distribution and the observed data
• The MLE estimate is obtained by maximizing the likelihood function with respect to the unknown parameter

MLE for multiple parameters

• MLE can also be used to estimate multiple unknown parameters simultaneously
• The likelihood function is constructed based on the joint probability distribution of the data and the unknown parameters
• The MLE estimates are obtained by maximizing the likelihood function with respect to all the unknown parameters

Properties of MLE estimators

• MLE estimators are consistent, meaning that they converge to the true parameter values as the sample size increases
• MLE estimators are asymptotically efficient, meaning that they achieve the lowest possible variance among all consistent estimators as the sample size increases
• MLE estimators are invariant under parameter transformations, meaning that the MLE of a function of the parameters is the function of the MLE of the parameters

Derivation of MLE

Likelihood function

• The likelihood function is a function of the unknown parameters, given the observed data
• It measures the probability of observing the given data as a function of the unknown parameters
• The likelihood function is denoted as $L(\theta; x)$, where $\theta$ represents the unknown parameters and $x$ represents the observed data

Log-likelihood function

• The log-likelihood function is the natural logarithm of the likelihood function
• It is often more convenient to work with the log-likelihood function because the logarithm is a monotonically increasing function
• The log-likelihood function is denoted as $\ell(\theta; x) = \log L(\theta; x)$

Maximizing log-likelihood

• MLE involves finding the parameter values that maximize the log-likelihood function
• The maximum of the log-likelihood function occurs at the same parameter values as the maximum of the likelihood function
• Maximizing the log-likelihood function is often easier than maximizing the likelihood function directly

Solving MLE equations

• To find the MLE estimates, we set the partial derivatives of the log-likelihood function with respect to each parameter to zero
• These equations are called the MLE equations or the score equations
• Solving the MLE equations yields the MLE estimates of the parameters

MLE in signal processing applications

MLE for signal detection

• MLE can be used for signal detection in the presence of noise
• The likelihood function is constructed based on the probability distribution of the received signal under different hypotheses (signal present or absent)
• The MLE detector compares the likelihood ratio to a threshold to make a decision

MLE for signal estimation

• MLE can be used to estimate unknown signal parameters, such as amplitude, phase, or frequency
• The likelihood function is constructed based on the probability distribution of the observed signal, given the unknown parameters
• The MLE estimates are obtained by maximizing the likelihood function with respect to the unknown signal parameters

MLE for signal classification

• MLE can be used for signal classification, where the goal is to assign a signal to one of several predefined classes
• The likelihood function is constructed based on the probability distribution of the signal features under each class
• The MLE classifier assigns the signal to the class that maximizes the likelihood function

Advantages and limitations of MLE

Advantages of MLE

• MLE is a versatile and powerful method for parameter estimation
• MLE is consistent, asymptotically efficient, and invariant under parameter transformations
• MLE provides a unified framework for estimating parameters in a wide range of probability distributions

Limitations of MLE

• MLE relies on the assumption that the model is correctly specified and that the data are generated from the assumed probability distribution
• MLE can be sensitive to outliers or model misspecification
• MLE may not always have a closed-form solution, requiring numerical optimization methods

MLE vs other estimation methods

• MLE is one of several parameter estimation methods, including least squares estimation, method of moments, and Bayesian estimation
• MLE is often preferred when the probability distribution is known and the sample size is large
• Other estimation methods may be more appropriate in certain situations, such as when prior information is available (Bayesian estimation) or when the focus is on minimizing the sum of squared errors (least squares estimation)

Numerical methods for MLE

Newton-Raphson method

• The Newton-Raphson method is an iterative algorithm for finding the roots of a function
• It can be used to solve the MLE equations numerically
• The method updates the parameter estimates iteratively based on the first and second derivatives of the log-likelihood function

Expectation-maximization algorithm

• The expectation-maximization (EM) algorithm is an iterative method for finding the MLE estimates in the presence of missing or latent data
• The EM algorithm alternates between an expectation step (E-step) and a maximization step (M-step) until convergence
• The E-step computes the expected log-likelihood, given the current parameter estimates and the observed data
• The M-step updates the parameter estimates by maximizing the expected log-likelihood

Gradient descent optimization

• Gradient descent is an optimization algorithm that can be used to find the MLE estimates
• It iteratively updates the parameter estimates in the direction of the gradient of the log-likelihood function
• The step size is determined by a learning rate, which controls the speed of convergence
• Variants of gradient descent, such as stochastic gradient descent and mini-batch gradient descent, can be used for large-scale problems

Advanced topics in MLE

MLE with constraints

• In some cases, the parameter space may be subject to constraints, such as non-negativity or sum-to-one constraints
• MLE with constraints involves maximizing the likelihood function subject to the given constraints
• Lagrange multipliers or barrier methods can be used to incorporate the constraints into the optimization problem

MLE with missing data

• MLE can be used to estimate parameters in the presence of missing data
• The EM algorithm is a common approach for handling missing data in MLE
• The E-step computes the expected log-likelihood, given the current parameter estimates and the observed data
• The M-step updates the parameter estimates by maximizing the expected log-likelihood

MLE for non-linear models

• MLE can be applied to non-linear models, where the relationship between the parameters and the observed data is non-linear
• Non-linear optimization techniques, such as the Gauss-Newton method or the Levenberg-Marquardt algorithm, can be used to find the MLE estimates
• Iterative methods are often required to solve the non-linear MLE equations

Bayesian MLE approach

• Bayesian MLE combines the principles of MLE with Bayesian inference
• In Bayesian MLE, prior distributions are assigned to the unknown parameters
• The posterior distribution of the parameters is obtained by combining the prior distribution with the likelihood function using Bayes' theorem
• The Bayesian MLE estimate is the mode of the posterior distribution

Practical considerations in MLE

Choice of initial values

• MLE often requires iterative optimization methods, which depend on the choice of initial values for the parameters
• Poor initial values can lead to slow convergence or convergence to a local maximum instead of the global maximum
• Strategies for choosing initial values include using prior knowledge, using the method of moments estimates, or trying multiple random initializations

Convergence of MLE

• The convergence of MLE algorithms depends on factors such as the choice of initial values, the complexity of the model, and the properties of the log-likelihood function
• Monitoring the change in the log-likelihood function or the parameter estimates across iterations can help assess convergence
• Convergence criteria, such as a tolerance threshold for the change in the log-likelihood or the parameter estimates, can be used to determine when to stop the iterations

Computational complexity of MLE

• The computational complexity of MLE depends on the size of the dataset, the number of parameters, and the complexity of the model
• For large datasets or high-dimensional parameter spaces, MLE can be computationally expensive
• Efficient optimization algorithms and parallel computing techniques can be used to speed up the computations

Robustness of MLE estimators

• The robustness of MLE estimators refers to their sensitivity to model misspecification or the presence of outliers
• MLE estimators can be sensitive to outliers, as they aim to maximize the likelihood of the observed data
• Robust estimation methods, such as M-estimators or trimmed likelihood estimators, can be used to mitigate the impact of outliers
• Model diagnostics and goodness-of-fit tests can be used to assess the appropriateness of the assumed model and detect potential issues