The purpose of this blog post is to introduce a
probabilistic classifier that is often implemented through computer software
called “Naive Bayes” which is essentially used for pattern recognition within
some data set. I will draw the majority of my understanding in order to write
this post from the this video.

He begins the video by explaining the structure of the
dataset necessary for the application of the Naïve Bayes classifier. From the video, the set up for the data
should follow this form.

D = ((x

^{(1)}, y_{1}), … , (x^{(n)},y_{(n)})) D is an algebraic expression for the data set._{ }In an attempt to make that more clear, the variable x^{(1) }represents a coordinate pair ( x_{1}^{(1)},x_{d}^{(1)}) where the superscript shows what point that coordinate belongs to and the subscript indexes that coordinate. X^{(i) }is a point in the space of R^{d}. Y_{i }belongs to some finite set. In the video, he states that y represents a finite set that will be the integers from 1 to n.
He mentions in
the video that there are several assumptions made when taking on a Naïve Bayes
approach to classification. Those assumptions are listed as the following.

1)
We assume we have a family of some set of
distributions parametrized by theta and these distributions will have the
following properties. Each of these is a joint distribution on x and y. So
here x is going to be in Rd and Y is a
class.

2)
P

_{Θ}(x,y) = P_{Θ}(x|y) P_{Θ}(y) = P_{Θ}(x_{1}|y) … P_{Θ}(x_{d}|y) P_{Θ}(y)
He mentions in the video that this second assumption
is very key to the Naïve Bayes classification approach and that what this
essential means is that the first expression of assumption 2 (P

_{Θ}(x,y) = P_{Θ}(x|y) P_{Θ}(y)) will factor to equal the second expression. What I essential draw from this is that we assume that P_{Θ}(x|y) factors out to be the probability of the first coordinate x given y up to the last coordinate of x given y.
3)
Assume that the points are independently,
identically distributed based on parameter Θ. He mentions that in this context, the coordinates x

_{1}… x_{n}are independent given y if (X,Y) ~ P_{Θ}_{.}
He finally makes
things a bit clearer at 7:35 in the video. He mentions that the main assumption
is a conditional independence assumption.

At this point,
he explains the “goal” of Naïve Bayes. Essentially he says that when some new x
enters the data set, we want to “predict” its y. He mentions that the algorithm
initiates by attempting to estimate the parameter Θ for which it is believed the distribution of the (x,y)s
follow. Theta is estimated from the data and then we compute the prediction of
y that maximizes over all possible classes the probability of that class given
the new x. Because we assume that P

_{Θ}(x|y) P_{Θ}(y) factors out to be = P_{Θ}(x_{1}|y) … P_{Θ}(x_{d}|y) P_{Θ}(y), we will attempt to maximize the prediction y across all x’s, and that value should give the new prediction for the value of y given to the new x. For a better understanding, please watch the video.