The point in which the parameter value that maximizes the likelihood function is called the maximum likelihood estimate. This principle was originally developed by Ronald Fisher, in the s. Which means, the parameter vector is considered which maximizes the likelihood function. The goal of maximum likelihood estimation is to make inference about the population, which is most likely to have generated the sample i.
Before proceeding further, let us understand the key difference between the two terms used in statistics — Likelihood and Probability which is very important for data scientists and data analysts in the world. Also Read: What is Machine Learning? How does it work? Problem: What is the Probability of Heads when a single coin is tossed 40 times. In some cases, after an initial increase, the likelihood percentage gradually decreases after some probability percentage which is the intermediate point or peak value.
The peak value is called maximum likelihood. The parameters of a logistic regression model can be estimated by the probabilistic framework called maximum likelihood estimation.
A probability distribution for the target variable labeled class must be assumed and followed by a likelihood function defined that calculates the probability of observing the outcome given the input data and the model.
The function can be optimized to find the set of parameters that results in the largest sum likelihood over the training dataset. The joint probability can also be defined as the multiplication of the conditional probability for each observation given the distribution parameters. As log is used mostly in the likelihood function, it is known as log-likelihood function.
It is common in optimization problems to prefer to minimize the cost function. Therefore, the negative of the log-likelihood function is used and known as Negative Log-Likelihood function. A random sample of 10 American female college students yielded the following weights in pounds :. Therefore, you might want to convince yourself that the likelihood function is:. It can be shown we'll do so in the next example! Note that the only difference between the formulas for the maximum likelihood estimator and the maximum likelihood estimate is that:.
Okay, so now we have the formal definitions out of the way. Now, let's take a look at an example that involves a joint probability density function that depends on two parameters. Now, that makes the likelihood function:. I'll again leave it to you to verify, in each case, that the second partial derivative of the log likelihood is negative, and therefore that we did indeed find maxima. They are, in fact, competing estimators. Well, one way is to choose the estimator that is "unbiased.
Breadcrumb Home 1 1. After today's blog, you should have a better understanding of the fundamentals of maximum likelihood estimation. In particular, we've covered:. She is an economist skilled in data analysis and software development. She has earned a B. You must be logged in to post a comment. Subscribe Now. In particular, we discuss: The basic theory of maximum likelihood. The advantages and disadvantages of maximum likelihood estimation.
The log-likelihood function. Modeling applications. What is Maximum Likelihood Estimation? This implies that in order to implement maximum likelihood estimation we must: Assume a model, also known as a data generating process, for our data.
Be able to derive the likelihood function for our data, given our assumed model we will discuss this more later. Advantages of Maximum Likelihood Estimation There are many advantages of maximum likelihood estimation: If the model is correctly assumed, the maximum likelihood estimator is the most efficient estimator. It provides a consistent but flexible approach which makes it suitable for a wide variety of applications, including cases where assumptions of other models are violated.
It results in unbiased estimates in larger samples. Efficiency is one measure of the quality of an estimator. An efficient estimator is one that has a small variance or mean squared error. Was this post helpful? Let us know if you liked the post.
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