70 lines
2.5 KiB
Plaintext
70 lines
2.5 KiB
Plaintext
{
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"## Bayes' theorem\n",
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"\n",
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"\n",
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"$ \\newcommand{\\thetavec}{\\boldsymbol{\\theta}} \\newcommand{\\pr}{\\textrm{p}}$\n",
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"\n",
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"Bayes' theorem with $\\thetavec$ the vector of parameters we seek.\n",
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"\n",
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"$$\n",
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" \\overbrace{\\pr(\\thetavec \\mid \\textrm{data})}^{\\textrm{posterior}} =\n",
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" \\frac{\\color{red}{\\overbrace{\\pr(\\textrm{data} \\mid \\thetavec)}^{\\textrm{likelihood}}} \\times\n",
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" \\color{blue}{\\overbrace{\\pr(\\thetavec)}^{\\textrm{prior}}}}\n",
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" {\\color{darkgreen}{\\underbrace{\\pr(\\textrm{data})}_{\\textrm{evidence}}}}\n",
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"$$\n",
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"\n",
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"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 <em>update</em> that information after observing some data: this is the posterior pdf.\n",
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"\n",
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"\n",
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"We will go through [this post, \"Bayes' Theorem with Lego\"](https://www.countbayesie.com/blog/2015/2/18/bayes-theorem-with-lego).\n",
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"\n",
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"## Discussion: the Bayesian Brain hypothesis.\n",
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"\n",
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"Here is an [interesting article](https://towardsdatascience.com/the-bayesian-brain-hypothesis-35b98847d331), which might serve as a starting point, about this idea.\n",
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"\n",
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"## Further material\n",
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"\n",
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"* [Bayesian Statistics the Fun Way](https://katalog.ub.uni-freiburg.de/opac/RDSIndex/Search?lookfor=Bayesian%20Statistics%20the%20Fun%20Way%20&source=homepage)\n",
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"\n",
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"* [Doing Bayesian Data Analysis : A Tutorial with R, JAGS, and Stan\n",
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"John Kruschke](https://katalog.ub.uni-freiburg.de/opac/RDSIndex/Search?lookfor=Bayesian+Data+analysis+Kruschke&type=AllFields&limit=10&sort=py+desc)\n",
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"\n",
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"* https://www.3blue1brown.com/, especially https://www.youtube.com/watch?v=HZGCoVF3YvM\n"
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