Maximum Likelihood Estimation

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21 Aug, 2024

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Maximum Likelihood Estimation (MLE) is a statistical procedure used to estimate a model's parameters by maximizing the likelihood of an event's occurrence within a given data distribution. MLE offers numerous advantages, making it a widely used method for fitting probability data in software applications.

Unlike other methods, MLE does not require linearity, allowing for the modeling of various equations. It is particularly suitable for datasets with heavy censorship, as it can handle failure and right-censored data effectively. Additionally, MLE tends to converge faster toward population parameters as the amount of data increases.

What is Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) involves selecting a probability distribution with a specific set of parameters to determine the likelihood of generating a probability distribution.
MLE is a commonly used procedure to fit probability data in various applications and does not require linearity assumptions, making it versatile.
MLE provides unbiased outcomes for larger samples but may produce biased outcomes when applied to smaller samples.
The main difference between MLE and least squares is that MLE focuses on the dataset itself, while least squares predict the maximum error associated with the distribution.

Maximum Likelihood Estimation Explained

Maximum Likelihood Estimation gets defined as a statistical technique for the estimation of the parameters of a model. In the MLE model, the parameters get chosen to increase the probability that the presumed model produces the data obtained. MLE functions by calculating the occurrence probability per data point associated with a model having the given set of parameters.

Later on, one sums all these probabilities for entire data points. Furthermore, one uses an optimizer for changing parameters to maximize the total of the probabilities. Moreover, if one has to implement MLE, one must:

One uses data generating process model.
One gets able to derive a likelihood function related to the data.
After the likelihood function gets derived, MLE does not appear to be more than a simple optimization problem.

In other words, it works by estimating the likelihood of every data point and then multiplying all of them with each other to get the likelihood. After this, one would obtain a different likelihood value for the given dataset when the distribution parameters change. Furthermore, if one tries to find those parameters that would produce the greatest likelihood, in turn, it gives one the maximum likelihood estimation. As these parameters maximize the likelihood estimates related to actual population parameters, they get termed maximum likelihood estimators.

The main steps to apply MLE:

Using correct distribution for regression or classification problem
Defining the likelihood
Taking the natural log
Reducing the product functions into a function of the summation
Maximizing or minimizing the negative aspect of the objective function
Verifying those safe assumptions comprises uniform priors

Using the above steps and ways to implement MLE would be helpful in successfully estimating distribution data.

Formula

To derive the formula of MLE, one needs to define MLE mathematically. Therefore, let us assume that X1, X2, X3, ..., XN represents any random sample with a θ parameter. Moreover, let it be true that X₁=x1, X₂=x2, X₃=x3, ..., X_N=xn.

The MLE of θ, denoted as θ̂ ML, is the value that maximizes the likelihood function, represented as:

Likelihood function = L(x₁, x₂, ..., xₙ; θ)

One can also say that the MLE of parameter θ getting shown as a random variable,

ML = ML (X₁, X₂, X₃, ..., X_N )

Its value comes out to be ML when X₁=x1, X₂=x2, X₃=x3, ..., X_N=xn.

Calculation Example

We have a random sample of observations from a binomial distribution with parameters n = 3 and θ. The observed values are (x1, x2, x3, x4) = (1, 3, 2, 2), and we want to find the maximum likelihood estimate (MLE) for the parameter θ.

The likelihood function for the binomial distribution is given by:

L(1, 3, 2, 2; θ) = θ^1 * (1 - θ)^2 * θ^3 * (1 - θ)^0 * θ^2 * (1 - θ)^1 * θ^2 * (1 - θ)^1

Simplifying, we get:

L(1, 3, 2, 2; θ) = θ^8 * (1 - θ)^4

To find the value of θ that maximizes the likelihood function, we take the derivative and set it equal to zero:

dL(1, 3, 2, 2; θ)/dθ = 8θ^7 * (1 - θ)^4 - 4θ^8 * (1 - θ)^3 = 0

The above step outlines the path to calculating the maximum likelihood estimate (MLE) for the parameter θ in a binomial distribution using the observed values. The likelihood function is derived, simplified, and then the derivative is taken to find the critical point where the derivative equals zero. The result of setting the derivative equal to zero is not explicitly provided in the example, as it requires further analysis or computational methods to obtain the exact MLE value.

Advantages And Disadvantages

One might wonder why one should use MLE over other methods like least squares regression. However, using MLE should be based on carefully considering its advantages and disadvantages. Let's discuss the advantages and disadvantages of using MLE compared to other techniques in the table below:

MLE advantages	MLE disadvantages
It becomes the best and the most efficient estimator of a parameter if correct assumptions for the model gets used.	It depends on the assumptions of such a model whose derivation function has never been easy.
It has the ability to provide the user with a consistent and flexible approach making it more reliable than other estimators.	It gets highly sensitive to choosing of initiating values of a model, which poses great problems like other models.
It gets applied in different types of applications where other models' assumptions get violated.	The numerical estimation could become quite expensive computationally based on the complexity of the MLE function.
For larger samples, it produces unbiased outcomes.	For a smaller sample, it produces biased outcomes.

Maximum Likelihood Estimation vs Least Squares

Both have been used as methods of parameter estimation but have certain differences, as mentioned in the below table:

Maximum Likelihood Estimation	Least Squares (LS)
It gets used in predicting the likelihood of an event happening.	It acts as the error function predicting the maximum error in MLE's prediction of an event.
It also determines the best-estimated parameter for working dependence and iterative weighted of the design matrix.	It assumes and gets applied when the distribution of the dependent variable remains related in some ways, like linearly to any explicative variables or factors.
It gets used to estimate parameters related to a statistical model that can get used to fit data.	It could get used as a technique for the approximate determination of unknown parameters stationised within a linear regression model.
It could get used as a technique for the approximate determination of unknown parameters stationed within a linear regression model.	It takes that value of the parameter, which minimizes the residual errors.
After that, it fits the model using trial estimated parameter value to calculate the model's mean.	It considers the sum of the square plus derivate concerning the beta parameter regression coefficient to set it as zero.
It also determines the best-estimated parameter for working dependence and the iterative weighted of the design matrix.	After that, it determines that the parameter value minimizes the square error residual sum.
MLE gets used when the parameters do not have any linear relationship.	OLS gets used when the parameters fulfill the linearity assumption.

Frequently Asked Questions (FAQs)

1. What are the important applications of maximum likelihood estimation?

Maximum likelihood estimation (MLE) finds applications in various fields, such as statistics, econometrics, machine learning, and bioinformatics. It is used to estimate model parameters, perform regression analysis, fit probability distributions, train machine learning models, and make predictions based on likelihoods.

2. What are the assumptions of maximum likelihood estimation?

Maximum likelihood estimation relies on certain assumptions for accurate parameter estimation. The key assumptions include the following:
- Correct specification of the underlying probability distribution or model.
- Independence and identically distributed observations.
- Unbiasedness and consistency of the estimator.
- The sufficiently large sample size for asymptotic properties.
- Absence of measurement errors or significant outliers.

3. What are the limitations of maximum likelihood estimation?

Maximum likelihood estimation has certain limitations that should be considered. It heavily relies on the accuracy of the assumed model and distributional assumptions. MLE can be sensitive to the choice of initial parameter values and may not always yield a unique solution. In some cases, it may require complex numerical optimization techniques. Additionally, MLE assumes independent and identically distributed observations, which may not hold in all real-world scenarios.