As data science continues to grow in importance and relevance, so too does the need for tools and techniques that can help extract insights from large, complex datasets. One such tool that is becoming increasingly popular among data scientists is Maximum Likelihood Estimation (MLE). This is becoming more so important to learn fundamentals of MLE concepts as it is at the core of generative modeling (generative AI). MLE is a statistical method used to estimate the parameters of a probability distribution, based on a set of observed data points.
MLE is particularly important for data scientists because it underpins many of the probabilistic machine learning models that are used today. These models, which are often used to make predictions or classify data, require an understanding of probability distributions in order to be effective. By learning how to apply MLE, data scientists can better understand how these models work, and how they can be optimized for specific tasks.
In this blog, we will explore the concepts behind MLE and provide examples of how it can be used in practice. We will start with basic concepts of sample space, probability density, parametric modeling and then learn about likelihood and maximum likelihood estimation. We will also learn about how MLE is used in machine learning, before diving into the details of MLE and its applications. Whether you’re a seasoned data scientist or just starting out in the field, this blog will provide valuable insights into one of the key tools used in modern machine learning.
Key Concepts to Learn Prior Understanding Maximum Likelihood Estimation
Prior to learning the concepts of maximum likelihood estimation (MLE), it is important to understand some of the following concepts:
- Sample space: Given a random variable, sample space represents all values that the random variable can take. For example, let’s consider the example of a coin toss. When a coin is tossed, there are two possible outcomes: heads or tails. These outcomes represent the sample space for this experiment. The sample space, in this case, is simply the set of all possible outcomes. It can be represented as {Heads, Tails}. Let’s take another example. For instance, consider a survey that asks respondents to choose their favorite color from a list of 10 colors. The sample space in this case would consist of all 10 colors, i.e., {Red, Orange, Yellow, Green, Blue, Purple, Pink, Brown, Gray, Black}.
- Probability density function: Given a sample space, probability density function is a function that maps the value with a probability value between 0 and 1. Basically, probability density function, also termed as density function can be used to calculate probability of occurrence of that value. The example of probability distribution function includes normal, exponential, Poisson, uniform, binomial distribution, etc.
- Parametric modeling: Parametric models represents a set of density function with one or more parameters. For different values of parameters, there will be different density function. All of these density functions can be represented as the parametric models. We can term these parameters as \(\theta\). \(\theta\) can be used to represent different parameters such as \(\theta_1, \theta_2, \theta_3, \theta_4\), etc. For example, the probability density function of normal distribution has parameters such as mean and standard deviation. The example of parametric model can be family of density functions related to normal distribution, Poisson distribution, gamma distribution, binomial distribution, etc.
- Likelihood function: For a particular sample space, you can estimate different such parameters representing different density function. The Likelihood function represents the probability of observing or seeing the values in sample space if the true generating distribution was the model represented using the particular density function parametrized by \(\theta\). For a particular value x, the likelihood function is represented using the following:
\(L(\theta|x) = p_{\theta}(x)\) OR, \(L(\theta|x) = p(x|\theta)\)
When we take into consideration the entire data in the sample space X, the likelihood function becomes the following:
\(L(\theta|x) = \prod p_{\theta}(x)\)
Given the above would result in a very small number between 0 and 1 and gets computationally difficult to work with, due to probability value for each value in sample space lying between 0 and 1, the following is recommended and often used. It is called as log likelihood function. Basically, taking log results into summation of log of output of density function for each value in sample space.
What is Maximum Likelihood Estimation?
From previous section, we learned that the likelihood function is used to represent the probability of observing the data in sample space assuming the true data generating distribution was the model or density function parametrized by the \(\theta\).
Based on the above, we can thing that the goal becomes to find the optimal values of parameters of the model or density function, \(\theta\) that maximizes the likelihood of seeing or observing the data (X) in the sample space. This technique or method is called as Maximum Likelihood Estimation. The goal of the maximum likelihood estimation is to maximize the likelihood function. The formula below represents the maximum likelihood estimation function.
When working with neural networks, the loss function is typically minimized. Thus, we can go about finding the set of parameters that minimize the negative log-likelihood such as that given below:
Real-world Applications of Maximum Likelihood Estimation
Today, one of the most talked about areas in machine learning is generative modeling. Generative models, as implied by their name, are a class of models designed to generate new data points by accurately capturing the underlying probability distributions of the original data. MLE plays a key role in calibrating these models to the given data, thus enabling them to replicate and generate samples that mirror the structural and statistical properties of the dataset. The likelihood function is represented using log likelihood for the sake of computational ease and also ease of understanding. In MLE, what is maximized is this log likelihood function. By maximizing the log-likelihood, the MLE process effectively achieves the optimal parameter settings that enable the generative model to have the highest probability of producing the observed dataset. This, in turn, results in a model that can closely mimic the structure and patterns present within the data.
Conclusion
Maximum Likelihood Estimation (MLE) is a widely-used statistical method that helps us estimate the parameters of a probability density function which are used to assess the probability of observing the data in the sample space. At its core, maximum likelihood estimation is about finding the values for a set of parameters that provide the highest likelihood for the observed data. Through a variety of examples, we have explored how maximum likelihood estimation can be applied to real-world scenarios, such as predicting consumer behavior, understanding the effectiveness of medical treatments, and more. It is a powerful tool for statisticians, data scientists, and researchers that allows them to make informed decisions based on meaningful data. In short, maximum likelihood estimation is a fundamental concept in statistics that has immense practical applications in various fields, making it an essential technique to be learned and mastered by anyone interested in making data-driven decisions.
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