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Bayes' theorem¶
$ \newcommand{\thetavec}{\boldsymbol{\theta}} \newcommand{\pr}{\textrm{p}}$
Bayes' theorem with $\thetavec$ the vector of parameters we seek.
$$ \overbrace{\pr(\thetavec \mid \textrm{data})}^{\textrm{posterior}} = \frac{\color{red}{\overbrace{\pr(\textrm{data} \mid \thetavec)}^{\textrm{likelihood}}} \times \color{blue}{\overbrace{\pr(\thetavec)}^{\textrm{prior}}}} {\color{darkgreen}{\underbrace{\pr(\textrm{data})}_{\textrm{evidence}}}} $$If we view the prior as the initial information we have about $\thetavec$, summarized as a probability density function, then Bayes' theorem tells us how to update that information after observing some data: this is the posterior pdf.
We will go through this post, "Bayes' Theorem with Lego".
Discussion: the Bayesian Brain hypothesis.¶
Here is an interesting article, which might serve as a starting point, about this idea.
Further material¶
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