# General Bayesian Model for Signals

We accept that every sampled piece of data contains both the "true" signal and noise accumulated in the channel.

Let:

$d_i = f(t_i) + e_i$

Where,

 $d_i \equiv$ sampled data $f(t_i) \equiv$ expected signal $e_i \equiv$ noise at the given sample

As with traditional methods, we let:

$f(t_i) = \sum\limits^{m}_{j=1} B_j G_j (t,\{w\})$

Where,

 $B_j \equiv$ a type of amplitude $G_j \equiv$ a function of some kind $t \equiv$ continuous time $\{ \omega \} \equiv$ a set of parameters; could be frequencies, chirp ratios, decay rates, or some other quantity

Recall the product rule:

$P(H|DI) = \frac{P(H|I)P(D|HI)}{P(D|I)}$

$H \equiv$ hypothesis

$D \equiv$ data

$I \equiv$ prior information

Let the hypothesis be the signal: $H = f(t)$

Accept all possible amplitudes by treating them as nuanced parameters:

$P(\omega |DI) = \int P(\omega B|DI) dB$

where $\omega \equiv \{ \omega_1 , \omega_2 , ... \omega_n \}$

Product rule now shows after direct substitution:

$P(H|DI) = \frac{P( \omega |I)P( D | \omega I)}{P(D|I)}$

By using Maximum Entropy method, we find a solution for the probability of the signal:

$P(\{\omega\} | D \sigma I) \propto exp(\frac{-N \bar{d^2} + m \bar{h^2} }{2 \sigma^2 })$

Where

 $h_j \equiv \sum\limits^{N}_{i=1} d_i H_j (t_i)$ and ($1 \leq j \leq m$) $\bar{A^2} \equiv$ the arithmetic mean of $A^2$ for each value of $A \equiv \{A_1 , A_2 , A_3 , ... A_m\}$ More of a discussion what $H_j$ represents in a later revision $\sigma \equiv$ the second moment; which is usually known prior to a calculation

If $\sigma$ is not known, then

$P({\omega} | DI) \propto (1 - \frac{m \bar{h^2} }{N \bar{d^2}})^{\frac{m-N}{2}}$

$N \equiv$ total number of samples

If a sinusoidal model is expected (for example, a real and complex wave) then the model would be as follows:

$P(\omega | DI) \propto 1 - \left( \frac{ \frac{R(\omega)^2}{c} + \frac{I(\omega)^2}{s} }{N \bar{d^2}} \right) ^{\frac{2-N}{2}}$

Where

 $R(\omega) \equiv \sum\limits^{N}_{i=1} d_i \cos{\omega t_i}$ $I(\omega) \equiv \sum\limits^{N}_{i=1} d_i \sin{\omega t_i}$ $c \equiv cos(\omega) \frac{N}{2} + \frac{sin(N \omega)}{2sin( \omega )}$ $s \equiv sin(\omega) \frac{N}{2} - \frac{sin(N \omega)}{2sin( \omega )}$