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Differences Between OLS and MLE

OLS vs MLE

We often try to vanish when the topic is about statistics. For some, dealing with statistics is like a terrifying experience. We hate the numbers, the lines, and the graphs. Nevertheless, we need to face this great obstacle in order to finish schooling. If not, your future would be dark. No hope and no light. To be able to pass statistics, we often encounter OLS and MLE. “OLS” stands for “ordinary least squares” while “MLE” stands for “maximum likelihood estimation.” Usually, these two statistical terms are related to each other. Let’s learn about the differences between ordinary least squares and maximum likelihood estimations.

The ordinary least squares, or OLS, can also be called the linear least squares. This is a method for approximately determining the unknown parameters located in a linear regression model. According to books of statistics and other online sources, the ordinary least squares is obtained by minimizing the total of squared vertical distances between the observed responses within the dataset and the responses predicted by the linear approximation. Through a simple formula, you can express the resulting estimator, especially the single regressor, located on the right-hand side of the linear regression model.

For example, you have a set of equations which consists of several equations that have unknown parameters. You may use the ordinary least squares method because this is the most standard approach in finding the approximate solution to your overly determined systems. In other words, it is your overall solution in minimizing the sum of the squares of errors in your equation. Data fitting can be your most suited application. Online sources have stated that the data that best fits the ordinary least squares minimizes the sum of squared residuals. “Residual” is “the difference between an observed value and the fitted value provided by a model.”

Maximum likelihood estimation, or MLE, is a method used in estimating the parameters of a statistical model, and for fitting a statistical model to data. If you want to find the height measurement of every basketball player in a specific location, you can use the maximum likelihood estimation. Normally, you would encounter problems such as cost and time constraints. If you could not afford to measure all of the basketball players’ heights, the maximum likelihood estimation would be very handy. Using the maximum likelihood estimation, you can estimate the mean and variance of the height of your subjects. The MLE would set the mean and variance as parameters in determining the specific parametric values in a given model.

To sum it up, the maximum likelihood estimation covers a set of parameters which can be used for predicting the data needed in a normal distribution. A given, fixed set of data and its probability model would likely produce the predicted data. The MLE would give us a unified approach when it comes to the estimation. But in some cases, we cannot use the maximum likelihood estimation because of recognized errors or the problem actually doesn’t even exist in reality.

For more information regarding OLS and MLE, you can refer to statistical books for more examples. Online encyclopedia Websites are also good sources of additional information.

Summary:

  1. “OLS” stands for “ordinary least squares” while “MLE” stands for “maximum likelihood estimation.”

  2. The ordinary least squares, or OLS, can also be called the linear least squares. This is a method for approximately determining the unknown parameters located in a linear regression model.

  3. Maximum likelihood estimation, or MLE, is a method used in estimating the parameters of a statistical model and for fitting a statistical model to data.


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3 Comments

  1. Which one is better ols and mle. And why???

    • @Momna Riaz
      OLS and MLE are solving different extremum problems. OLS belongs to a class of methods called regression used to find the parameters of a model (I use “model” in the scientific sense. E.g. Newton’s laws of motion and Newton’s law of gravitation can be used to model the solar system, solve the resulting equations analytically or numerically and then use regression to extract orbital parameters from the observations. Einstein’s theory of Gravitation can also be used to model the solar system, but the set of parameters itself is different (e.g. Einstein’s theory includes an “extra” parameter, the speed of light), though there are approximations which lead to mappings between them.)

      Given observations and a model (linear, non-linear, differential, time series, whatever) regression answers “What values of the parameters of the model will minimize some (lower-bounded) error function of the difference between the observations and the model retrodictions?”. You get to choose what the error or cost function is: for engineering or science, it might well be squared error, for financial situations it might be the absolute error. Lots of other issues need to be addressed as well. OLS has a closed form solution.

      MLE applies to a much smaller subset of problems: one where you have observations corresponding to some distributed event (height of basketball players in some region or team, distance of a perfume molecule from the bottle, time at which an atom fissioned …) _and_ a stochastic model (which does only one thing: predict the probability of occurence of the event). Then, MLE answers a “Bayesian” like question “What are the values of the parameters of the stochastic model which will maximize the probability of getting the specific set of observations?” For simple stochastic models (normal, Laplace, exponential …), these have closed forms solutions.

      When you can use both (Regression of histogramed or binned data corresponding to some stochastic variable, and MLE for the same stochastic model) you will get different answers.

  2. The second paragraph perpetuates a misunderstanding common to non-scientists between the linearity of the regression and the linearity of the model. You can have non-linear models (y ~ t**2, p(r) ~ r**2 exp(-r**2/sigma**2)) for which the regression problem can be formulated as linear in the parameters and solved in closed form. Or you can have a linear model (y ~ x) but with L1 error or some other non L2 error like cosh and the regression equations are not linear in the parameters.

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