wiki/scratch.page

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title: Scratch Page
toc: no
format: markdown
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# Math Stuff

On observation of $\mathcal{D}$, the *likelihood* of hypothesis
$\mathcal{R}_{\alpha}$ is
$\mathit{P}(\mathcal{D}|\mathcal{R}_{\alpha})$.

## Fingerprint Variance

Additionally, we associate a collective **RSS Variance**
$\sigma_{F_s}$ with each fingerprint, which is a weighed average of
the RSS values of each of the vector elements using $C_i$ as the
weight. It is calculated in this manner:

$$
\sigma_{F_s} = \frac{\sum_{i \in F_s} \sigma_i\, C_i}
                    {\sum_{i \in F_s} C_i}
$$

# Bayesian Regression

First, specify a set of probabilistic models of the data.

Let a member of this set be denoted by $\mathcal{R}_\alpha$

$\mathcal{R}_\alpha$ has a *prior* probability $P(\mathcal{H}_\alpha)$

On observation of $\mathcal{D}$, the *likelihood* of hypothesis
$\mathcal{R}_{\alpha}$ is
$\mathit{P}(\mathcal{D}|\mathcal{R}_{\alpha})$.

The *posterior* probability of $\mathcal{R}_{\alpha}$ is then given by
$\mathit{P}(\mathcal{H}_{\alpha})\mathit{P}(\mathcal{D}|\mathcal{H}_{\alpha})$

This follows from **Bayes' Theorem** which says

$$
P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}
$$

# Matrix Stuff
This is a column vector:
$$\vec v = \left(\begin{matrix}
1\\
3\\
7
\end{matrix}\right)$$

The *vector sum* of $\vec u$ and $\vec v$ is:

$$\vec u + \vec v =
\left(\begin{matrix}
u_1\\
\vdots\\
u_n
\end{matrix}\right) + \left(\begin{matrix}
v_1\\
\vdots\\
v_n
\end{matrix}\right) = \left(\begin{matrix}
u_1 + v_1\\
\vdots\\
u_n + v_n
\end{matrix}\right)
$$

The *scalar multiplication* of the real number $r$ and the vector $\vec v$ is:

$$
r \cdot \vec v = r \cdot \left(\begin{matrix}
v_1\\ \vdots \\ v_n
\end{matrix}\right) = \left(\begin{matrix}
rv_1 \\ \vdots \\ rv_n
\end{matrix}\right)
$$

This system:
$$\begin{alignedat}{4}
 x & {}-{} &  y & {}+{} &  z & = 1\\
3x & {}+{} &    &       &  z & = 3\\
5x & {}-{} & 2y & {}+{} & 3z & = 5
\end{alignedat}$$

reduces

$$
\left(\begin{array}{rrr|r}
1&-1&1&1\\
3&0&1&3\\
5&-2&3&5\\
\end{array}\right) \longrightarrow_{-5\rho_1+\rho_3}^{-3\rho_1+\rho_2}
\left(\begin{array}{rrr|r}
1&-1&1&1\\
0&3&-2&0\\
0&3&-2&0\\
\end{array}\right) \longrightarrow^{-\rho_2+\rho_3}
\left(\begin{array}{rrr|r}
1&-1&1&1\\
0&3&-2&0\\
0&0&0&0\\
\end{array}\right)
$$

to a one parameter solution set:

$$
\{ \begin{pmatrix}1\\ 0\\ 0\end{pmatrix} +
   \begin{pmatrix}-1/3\\ 2/3\\ 1\end{pmatrix} z \mid z \in \Bbb{R} \}
$$