## Mean Squared Error (MSE) :

Mean squared error (MSE) is a widely used loss function in machine learning, which is often used to evaluate the performance of a model on a given dataset. MSE is defined as the average squared difference between the predicted values and the true values. In other words, it is the average squared error between the predicted values and the true values.

To better understand MSE, let’s look at an example. Imagine that you are building a model to predict the stock price of a company. Your model takes in the historical stock prices of the company as input and outputs a predicted stock price for the next day. If the true stock price on the next day is $100, and your model predicts a stock price of $105, then the squared error would be $(100 – 105)^2 = 25$. If your model predicts a stock price of $90, then the squared error would be $(100 – 90)^2 = 100$.

Now, let’s say you have a dataset with 10 historical stock prices. If your model predicts the following stock prices for the next 10 days: 105, 90, 110, 95, 100, 105, 100, 95, 105, and 110, then the MSE would be the average of the squared errors for each of these predictions. In this case, the MSE would be $(25 + 100 + 16 + 9 + 1 + 0 + 0 + 9 + 1 + 16)/10 = 21.4$.

MSE is commonly used in regression problems, where the goal is to predict a continuous value, such as a stock price or a temperature. It is a popular loss function because it is differentiable and easy to compute, which makes it well-suited for optimization algorithms like gradient descent.

Another advantage of MSE is that it is interpretable. Because MSE is the average squared error, the units of MSE are the same as the units of the predicted and true values. For example, if the predicted and true values are stock prices in dollars, then the MSE will be in dollars squared. This allows us to easily compare the MSE of different models and choose the one that has the lowest MSE.

However, MSE has some disadvantages as well. One limitation of MSE is that it is sensitive to outliers. For example, if there is a single outlier in our dataset, then the squared error for that outlier will be much larger than the squared errors for the other data points. This can cause the MSE to be disproportionately influenced by that outlier, which may not be representative of the overall performance of the model.

Another disadvantage of MSE is that it is not robust to skewed data. For example, if the distribution of the true values is heavily skewed, then the squared errors for the smaller true values will be much smaller than the squared errors for the larger true values. This can cause the MSE to be overly influenced by the smaller true values, which may not be representative of the overall performance of the model.

Despite its limitations, MSE is a widely used loss function in machine learning and can provide a useful measure of model performance for many regression tasks.