# Chapter 6: Loss Function: Measure of Prediction Accuracy

Welcome to Chapter 6! In this chapter, we will delve into the concept of a loss function, a critical component in training neural networks. We will start by understanding what a loss function is and why it is important. We will then explore the Mean Squared Error (MSE) loss function, which we will use in our model. Finally, we will walk through the Python code that defines our loss function, explaining each line in detail.

## 6.1: Understanding Loss Functions

In the world of machine learning, a loss function, also known as a cost function, is a method of evaluating how well your algorithm models your dataset. If your predictions are totally off, your loss function will output a higher number. If they're pretty good, it'll output a lower number. As you change pieces of your algorithm to try and improve your model, your loss function will tell you if you're getting anywhere.
Think of a loss function like a compass guiding a ship. The ship (our model) wants to reach the destination (the best prediction). The compass (loss function) tells the ship in which direction it should go. If the ship is off course, the compass will indicate the direction to get back on track.

## 6.2: Mean Squared Error: Definition and Application

There are many types of loss functions, each with its own use case. For our simple linear model, we will use the Mean Squared Error (MSE) loss function. The MSE function measures the average squared difference between the actual and predicted values. It's most commonly used in regression problems, which aim to predict a continuous value (like our model).
Let's consider an example. Suppose we have three samples with actual values of [2, 4, 6] and our model predicts the values [3, 5, 7]. The differences between the actual and predicted values are [1, 1, 1]. Squaring these differences gives us [1, 1, 1], and the average of these squared differences is 1. So, the MSE for these predictions is 1.
The MSE loss function is great for our model because it punishes larger errors more than smaller ones (due to the squaring), which is what we want when we're trying to minimize the error.

## 6.3: Code Explanation: Loss Function Definition

Now, let's go through the Python code that defines our loss function. We'll explain each line to ensure you understand how the code implements the concepts we've discussed.
def loss(Y_true, Y_pred):
return np.mean((Y_true - Y_pred) ** 2)
This code defines a function called `loss` that takes two arguments: `Y_true` and `Y_pred`. `Y_true` are the actual values we're trying to predict, and `Y_pred` are our model's predictions.
The function returns the mean squared error between the actual and predicted values. It calculates this by subtracting the predicted values from the actual values, squaring the result (using the `** 2` operation), and then taking the mean of these squared differences (using `np.mean`).
That's it for this chapter! You now understand what a loss function is, why it's important, and how to define one in Python. In the next chapter, we will build the training loop, where we will use our model and loss function to train our neural network. Stay tuned!