Mathematics – Joseph Woolf

Probability Distributions and Random Variables

Suppose I had two coins and I flipped both of them. The possible combinations can be two heads, two tails, or one of each. These combinations are all part of a sample space.

Now let’s take this a step further. Taking the above demonstration, we want to determine the probability that each combination would occur.

Assuming independence, we derive the following probabilities:

$P(\text{two heads}) = 0.25$
$P(\text{two tails}) = 0.25$
$P(\text{one of each}) = 0.5$

All of these probabilities belong in a probability distribution.

Deriving the Naive Bayes formula

In my previous post, I introduced a class of algorithms for solving classification problems. I also mentioned that Naive Bayes is based off of Bayes’ theorem. In this post, I will derive Naive Bayes using Bayes’ theorem.

Deriving the Cost Function for Logistic Regression

In my previous post, you saw the derivative of the cost function for logistic regression as:

$\frac{\partial}{\partial \theta_i} J(\theta_0,\theta_1,\ldots,\theta_n) = \frac{1}{m}\displaystyle\sum_{i=1}^{m}(g(x_i) - y_i)x_i, x_0=1$

I bet several of you were thinking, “How on Earth could you derive a cost function like this:

$J(\theta_1,\ldots,\theta_n) = -\frac{1}{m}\displaystyle\sum_{i=1}^{m}[(y_i)log(g(x_i)) + (1 - y_i)log(1-g(x_i))]$

Into a nice function like this:

$\frac{\partial}{\partial \theta_i} J(\theta_0,\theta_1,\ldots,\theta_n) = \frac{1}{m}\displaystyle\sum_{i=1}^{m}(g(x_i) - y_i)x_i$ ?”

Well, this post is going to go through the math. Even if you already know it, it’s a good algebra and calculus problem. Continue Reading