The Gaussian Distribution

Figure 2.1: Gaussian distribution

The Gaussian distribution is one of the most widely used probability distributions in practical problems, as it models a significant portion of the probability distribution in real-world events. In this document, it is specifically utilized in filters (section 2.12) and Bayesian classifiers (section 4.2), as well as in LDA (section 4.3).

Definition 2   The standard Gaussian distribution, denoted by the symbol $\mathcal{N}(0;1)$, is characterized by the density

$$p(x) = \frac{1}{\sqrt{2\pi}} e ^ { \left( - \dfrac{1}{2}x^{2} \right) }$$ (2.13)

Definizione 5   The general Gaussian distribution $\mathcal{N}(\mu;\sigma^{2})$, with $\mu,\sigma \in \mathbb{R}, \sigma^{2} \geqslant 0$, is the one obtained from the standard distribution through the transformation $x \mapsto \sigma x + \mu$.

In the univariate case (univariate Gaussian), the Gaussian has the following distribution function:

$$p(x) = \frac{1}{\sigma \sqrt{2 \pi} } e^{ -\dfrac{1}{2} \left( \dfrac{x - \mu}{ \sigma } \right)^2}$$ (2.14)

where $\mu$ is the mean and $\sigma^2$ is the variance. Within $\pm \sigma$ from $\mu$, 68 The multivariate Gaussian distribution (multidimensional Gaussian) is characterized by a vector $\boldsymbol\mu$ of dimension $n$, representing the mean value of the various components, and a covariance matrix $\boldsymbol\Sigma$ of dimensions $n \times n$:

$$p(\mathbf{x}) = \frac{1}{ (2 \pi)^{\frac{n}{2}} \sqrt{\vert\... ...)^{\top} \boldsymbol\Sigma^{-1} (\mathbf{x}-\boldsymbol\mu) }$$ (2.15)

normal distribution with mean $\boldsymbol\mu = \left[ \mu_1, \mu_2, \dots \mu_n \right]^{T}$ and covariance $\boldsymbol\Sigma = \begin{bmatrix} \sigma_{11} & \cdots & \sigma_{1n} \\ \vdots & \ddots & \vdots \\ \sigma_{n1} & \cdots & \sigma_{nn} \\ \end{bmatrix}$.

It can be anticipated that the exponent quantity in equation (2.15) is the Mahalanobis distance (section 2.4) between $\mathbf {x}$ and $\boldsymbol\mu$.

When the random variables are independent and have equal variance, the matrix $\boldsymbol\Sigma$ is a diagonal matrix with all values equal to $\sigma^{2}$, and the multivariate normal distribution simplifies to

$$p(\mathbf{x}) = \frac{1}{ (2 \pi \sigma^{2} )^{n/2} } e^{ - ... ...ac{ \vert \mathbf{x} - \boldsymbol\mu\vert^2}{2 \sigma^{2}} }$$ (2.16)

.