%% goes over the formula for bayes theorem, todo have it’s applications, questions and links to other ideas. colorise. # bayes_theorem $$\definecolor{c1}{RGB}{255, 0, 0} \definecolor{c2}{RGB}{0, 0, 255} \definecolor{c3}{RGB}{255, 165, 0} \definecolor{c4}{RGB}{75, 0, 130} \definecolor{c5}{RGB}{220, 220, 0} \definecolor{c6}{RGB}{238, 130, 238} \definecolor{c7}{RGB}{0, 128, 0} \definecolor{c8}{RGB}{100,100,100} \definecolor{c9}{RGB}{45,177,93} \definecolor{default}{RGB}{10,20,20} \color{default}$$
$$ \color{c1} \pi (\theta | x) \color{default}= \frac{ \color{c2} f(x | \theta) \color{c3}\pi_{\Theta}(\theta)} {\color{c4}\int_{\Theta}f(x|\theta)\pi_{\Theta}(\theta) d\theta} \color{default}= \frac{f(x|\theta) \pi_{\Theta}(\theta)}{\pi_M(x)} \propto f(x | \theta) \pi_{\Theta}(\theta). $$
$$ \color{c1} \text{posterior distribution} $$
$$ \color{c2} \text{Likelihood function} $$
$$ \color{c3} \text{Prior distribution} $$
$$ \color{c4} \text{Marginalisation constant or marginal likelihood} $$
$$ \color{c1} \text{ an updated guess } \color{default} \text{ is equal to } \color{c2} \text{ the probability of the data given the parameter } \color{default} \text{ times the } \color{c3} \text{ best guess before the experiment } \color{c4} \text{ normalised } $$
To remember, integrate the numerator with respect to θ