Maximum likelihood method

Maximum likelihood is a statistical method used to estimate the parameters of a population distribution by finding values that maximize the likelihood of making the observations that were actually observed. In management, it is a method used to identify the most likely value or set of values for a population parameter based on a sample of data. It is a useful tool for finding the best estimate of a parameter from observed data and evaluating the quality of a model. It is also an important tool for making decisions based on data.

Example of maximum likelihood method

• A company has collected data on the ages of 100 employees. The company wishes to estimate the mean age of the employee population. This can be done by applying the maximum likelihood method, where the mean is estimated as the value that maximizes the probability of observing the sample data.
• A medical researcher wants to estimate the probability of a patient developing a certain disease. The researcher collects data from previous patients, such as their age, gender, and other characteristics. Using the maximum likelihood method, the researcher can estimate the probability of the disease based on the observed data.
• A sales team is trying to estimate the average amount of money customers are likely to spend in their store. They collect data on customers' purchases, and then use the maximum likelihood method to estimate the average purchase amount.
• A market researcher is trying to estimate the proportion of people who would be interested in a new product. They collect data on people's attitudes towards the product, and then use the maximum likelihood method to estimate the proportion of people who would be interested.

Formula of maximum likelihood method

The maximum likelihood method is used to estimate the parameters of a population distribution by finding the values that maximize the likelihood of the observed data. The formula for the likelihood of a set of data is given by:

$$L(\theta \mid x) = \prod_{i=1}^n \; f(x_i \mid \theta)$$

where $$\theta$$ is the parameter of the population distribution, $$x$$ is the data, and $$f$$ is the probability density function (PDF) of the population distribution.

The likelihood of the data is a measure of the probability of the observed data given the parameter $$\theta$$. The maximum likelihood estimate (MLE) of the parameter $$\theta$$ is the value that maximizes the likelihood of the data. This can be found by taking the derivative of the likelihood function with respect to $$\theta$$ and setting it to 0:

$$\frac{\partial L(\theta \mid x)}{\partial \theta} = 0$$

The MLE can be used to evaluate the quality of a model, as well as to make decisions based on data.

When to use maximum likelihood method

Maximum likelihood is a useful method for estimating parameters from observational data, and is applicable in a wide range of fields. It can be used to:

• Estimate the parameters of a population distribution, such as the mean, variance, and correlation coefficients, from a sample of data.
• Identify the most probable value or set of values for a parameter based on observed data.
• Assess the quality of a model by calculating the likelihood of a set of observations.
• Make decisions based on data by finding the most likely outcome.
• Compare models and data to find the best fit.
• Test hypotheses about the population parameters.

Types of maximum likelihood method

• Maximum likelihood estimation (MLE): This is a method for finding the most likely value for a parameter by maximizing the likelihood of the observed data. It is a powerful tool for finding the best estimate of a parameter from observed data and evaluating the quality of a model.
• Bayesian maximum likelihood estimation (BMLE): This is a method that uses Bayesian probability theory to estimate the parameters of a population distribution. It is a powerful tool for making decisions based on data.
• Conditional maximum likelihood estimation (CMLE): This is a method that uses the information from observed data to make decisions about the most likely value for a parameter. It is a useful tool for quantifying the uncertainty in parameter estimates.
• Maximum likelihood bootstrap (MLB): This is a method for resampling data to estimate the uncertainty in parameter estimates. It is a useful tool for making decisions based on data.
• Maximum penalized likelihood estimation (MPLE): This is a method that combines the maximum likelihood estimation with a penalty function to reduce the potential for overfitting. It is a useful tool for finding the best estimate of a parameter from observed data.

Maximum likelihood is a powerful and widely used statistical technique for estimating population parameters. It has a number of advantages, including:

• It is relatively easy to understand and implement, making it well suited for use by both professionals and non-professionals.
• Maximum likelihood estimates are usually more accurate than other methods of parameter estimation, such as least squares.
• The estimates produced by maximum likelihood are consistent, meaning that as the sample size increases, the estimates become more accurate.
• Maximum likelihood can be used to identify and estimate parameters for complex models, such as non-linear models and models with multiple parameters.
• Maximum likelihood is a fully automatic method, meaning that it does not require manual adjustments or tuning.
• Maximum likelihood can be used to test the validity of a model, allowing the researcher to assess the quality of the model and make the necessary adjustments.

Limitations of maximum likelihood method

The maximum likelihood method is a powerful tool used in statistics, but it is subject to certain limitations. These limitations include:

• Over-reliance on the assumptions of the model: The maximum likelihood method relies on the assumptions of the model being used, and these assumptions may not be realistic or accurate.
• Limited explanatory power: The maximum likelihood method does not provide much insight into the underlying causes of the data.
• Sensitivity to outliers: Outliers can have a disproportionate influence on the results of the maximum likelihood method.
• Biased estimates: Maximum likelihood estimates can be biased when the data used to fit the model is not representative of the population.
• Computationally intensive: The maximum likelihood method can be computationally intensive and require significant computing resources.

Other approaches related to maximum likelihood method

• Bayesian Estimation: Bayesian estimation is a statistical technique which uses prior information, such as beliefs and prior data, to determine the most likely values of parameters.
• Maximum A Posteriori Estimation (MAP): MAP estimation is a method that combines prior information with observed data to estimate parameters. It is a generalization of maximum likelihood estimation.
• Least Squares Estimation: Least squares estimation is a method of estimating parameters from a given dataset by minimizing the sum of squares of the errors between the observed data and the estimated parameter values.
• Maximum Entropy Estimation: Maximum entropy estimation is a method for estimating parameters from a given dataset by maximizing the entropy of the resulting probability distribution.

In summary, maximum likelihood is a statistical method used to estimate the parameters of a population distribution by finding values that maximize the likelihood of making the observations that were actually observed. Other approaches related to maximum likelihood include Bayesian estimation, MAP estimation, least squares estimation, and maximum entropy estimation. These methods use different techniques to estimate parameters from a given dataset.