Algorithms

1 files changed, 1 insertions, 12 deletions

diff --git a/algorithm.tex b/algorithm.tex
index 9bc0fd6..1747227 100644
--- a/algorithm.tex
+++ b/algorithm.tex
@@ -20,15 +20,7 @@ distribution of the model is given by:
   P(\bx, y) =  \cN(\bx | \bar{\bx}_y, \Sigma) P(y),
   \label{eq:mixture of Gaussians}
 \end{align}
-where $P(y)$ is the probability of class $y$ and $\cN(\bx | \bar{\bx}_y,
-\Sigma)$ is a multivariate normal distribution, which models the density of
-$\bx$ given $y$. The mean of the distribution is $\bar{\bx}_y$ and the variance
-of $\bx$ is captured by the covariance matrix $\Sigma$. The decision boundary
-between any two classes is known to be is linear when all conditionals $\cN(\bx
-| \bar{\bx}_y, \Sigma)$ have the same covariance matrix \cite{bishop06pattern}.
-In this setting, the mixture of Gaussians model can be viewed as a
-probabilistic variant of the nearest-neighbor (NN) classifier in
-Section~\ref{sec:uniqueness}.
+where $\bx$ denotes an observation, $y$ is a class, $P(y)$ is the probability of the class, and $\cN(\bx | \bar{\bx}_y, \Sigma)$ is the conditional probability of $\bx$ given $y$. The conditional is a multivariate normal distribution. The mean of the distribution is $\bar{\bx}_y$ and the variance of $\bx$ is captured by the covariance matrix $\Sigma$. The decision boundary between any two classes is linear when all conditionals $\cN(\bx | \bar{\bx}_y, \Sigma)$ have the same covariance matrix \cite{bishop06pattern}. In this setting, the mixture of Gaussians model can be viewed as a probabilistic variant of the nearest-neighbor (NN) classifier in Section~\ref{sec:uniqueness}.
 
 The mixture of Gaussians model has many advantages. First, the model can be
 easily learned using maximum-likelihood (ML) estimation \cite{bishop06pattern}.
@@ -38,14 +30,12 @@ is computed as $\Sigma = \sum_y P(y) \Sigma_y$, where $\Sigma_y$ represents the
 covariance of $\bx$ given $y$. Second, the inference in the model can be
 performed in a closed form. In particular, the model predicts $\hat{y} =
 \arg\max_y P(y | \bx)$, where:
-
 \begin{align}
   P(y | \bx) =
   \frac{P(\bx | y) P(y)}{\sum_y P(\bx | y) P(y)} =
   \frac{\cN(\bx | \bar{\bx}_y, \Sigma) P(y)}{\sum_y \cN(\bx | \bar{\bx}_y, \Sigma) P(y)}.
   \label{eq:inference}
 \end{align}
-
 In practice, the prediction $\hat{y}$ is accepted when the classifier is
 confident. In other words, $P(\hat{y} | \bx) \! > \! \delta$, where $\delta \in
 (0, 1)$ is a threshold that controls the precision and recall of the
@@ -85,7 +75,6 @@ sequential hypothesis testing. Sequential hypothesis testing
 subject is sequentially tested for belonging to one of several classes. The
 probability that the sequence of data $\bx^{(1)}, \dots, \bx^{(t)}$ belongs to
 the class $y$ at time $t$ is given by:
-
 \begin{align}
   P(y | \bx^{(1)}, \dots, \bx^{(t)}) =
   \frac{\prod_{i = 1}^t \cN(\bx^{(i)} | \bar{\bx}_y, \Sigma) P(y)}