Mean Squared Error (MSE) is a commonly used metric in Machine Learning, specifically in regression problems, to measure the difference between predicted and actual values. It is defined as the average of the squared differences between predicted and actual values.

The main objective of using MSE is to quantify the performance of a regression model. In other words, it is used to evaluate how well a model performs in terms of predicting the outcomes. A smaller MSE value indicates that the model is more accurate and hence better at predicting the outcomes.

Mathematically, MSE is defined as:

MSE = 1/n * Σᵢ (yᵢ – ŷᵢ)²

where yᵢ is the actual value of the i-th observation, ŷᵢ is the predicted value of the i-th observation, and n is the total number of observations.

The formula for MSE can be broken down into two components: the squared difference between predicted and actual values, and the average of these squared differences. Since the squared values are always positive, MSE penalizes larger errors more than smaller errors.

For example, consider a housing price prediction problem, where the model is trained to predict the price of a house based on its features like the number of bedrooms, bathrooms, etc. Once the model is trained, it is tested on a set of unseen data. The predicted prices are then compared with the actual prices, and the difference between them is squared and averaged to give the MSE.

One of the limitations of using MSE is that it puts more emphasis on outliers. In other words, if the model has a few predictions that are significantly off, the MSE value can be skewed. To overcome this limitation, other metrics like Mean Absolute Error (MAE) or Root Mean Squared Error (RMSE) can be used instead.

In conclusion, MSE is a popular metric used in Machine Learning to evaluate regression models. It helps in quantifying the accuracy of the model by measuring the average of the squared differences between predicted and actual values. However, care should be taken while interpreting MSE results and other metrics should be considered depending on the problem at hand.