turns-00017.parquet:16238
5a96a841dfc56f67f400dcbbdegenerate_repetitionAbsentFinal dense release
USER
Explain more math detail using equations
ASSISTANT
Certainly! For a Gaussian classifier, the key concept is that the class-conditional probability density of the data can be modeled with a multivariate Gaussian distribution. Given that we are assuming a zero mean for all classes, we need only estimate the covariance matrices for each digit class. The probability density function (pdf) for a multivariate Gaussian where mean is zero vector can be expressed as follows:
\[ p(\mathbf{x} \mid \Sigma_i) = \frac{1}{(2\pi)^{\frac{n}{2}}|\Sigma_i|^{\frac{1}{2}}} e^{-\frac{1}{2} \mathbf{x}^\top \Sigma_i^{-1} \mathbf{x}} \]
Here, \( \mathbf{x} \) is a n-dimensional data vector (the image of a digit as a flattened vector), \( \Sigma_i \) is the covariance matrix for the digit class \( i \), \( |\Sigma_i| \) denotes the determinant of \( \Sigma_i \), and \( \Sigma_i^{-1} \) is the inverse of the covariance matrix.
For classification, you want to find the class \( i \) that maximizes this pdf for a given observation \( \mathbf{x} \). That is, you wish to assign \( \mathbf{x} \) to the class \( \hat{i} \) that maximizes the likelihood or, equivalently, the log-likelihood:
\[ \hat{i} = \arg \max_{i} \log \left(p(\mathbf{x} \mid \Sigma_i)\right) \]
Since the logarithm is a monotonically increasing function, maximizing the log-likelihood is equivalent to maximizing the likelihood. Converting our pdf to log-likelihood, we get:
\[ \log \left(p(\mathbf{x} \mid \Sigma_i)\right) = -\frac{n}{2} \log(2\pi) - \frac{1}{2}\log|\Sigma_i| - \frac{1}{2} \mathbf{x}^\top \Sigma_i^{-1} \mathbf{x} \]
When comparing which class \( \hat{i} \) to assign a data point to, we can ignore the \( (2\pi)^{\frac{n}{2}} \) term because it is constant for every class. Thus, the discriminant function we actually use for classification simplifies to:
\[ g_i(\mathbf{x}) = - \frac{1}{2}\log|\Sigma_i| - \frac{1}{2} \mathbf{x}^\top \Sigma_i^{-1} \mathbf{x} \]
The classifier assigns \( \mathbf{x} \) to the class for which value of \( g_i(\mathbf{x}) \) is the largest. This function takes into account the spread and shape of each digit's distribution by looking at the covariance matrix \( \Sigma_i \). Even though the means are zero, the shape and spread of the data vary for each digit, which is encoded in the covariance matrix.
A digit '0' may have a different pattern of pixel variations compared to a digit '1', for example, with pixels varying together in a circular pattern for '0's and along a line for '1's. It is these patterns of variation and covariation that the covariance matrices capture, and these differences are what allow you to discriminate between the digits even when their means are the same.