Why the hypothesis has to introduce two parameters, namely θ0 and θ1

Question

I was learning Machine Learning from this course on Coursera taught by Andrew Ng. The instructor defines the hypothesis as a linear function of the "input" (x, in my case) like the following:

h_θ(x) = θ₀ + θ₁(x)

In supervised learning, we have some training data and based on that we try to "deduce" a function which closely maps the inputs to the corresponding outputs. To deduce the function, we introduce the hypothesis as a linear function of input (x). My question is, why the function involving two θs is chosen? Why it can't be as simple as y⁽ⁱ⁾ = a * x⁽ⁱ⁾ where a is a co-efficient? Later we can go about finding a "good" value of a for a given example (i) using an algorithm? This question might look very stupid. I apologize but I'm not very good at machine learning I am just a beginner. Please help me understand this.

Thanks!

Answer 1

The a corresponds to θ₁. Your proposed linear model is leaving out the intercept, which is θ₀.

Consider an output function y equal to the constant 5, or perhaps equal to a constant plus some tiny fraction of x which never exceeds .01. Driving the error function to zero is going to be difficult if your model doesn't have a θ₀ that can soak up the D.C. component.