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Bayesian Statistics

Foreword

Started 06/06/2024

So, as you may know if you’re taking a gander at my website is I do molecular bio and AI.

Rationality is about knowing which facts are relevant, not knowing the facts.

Statistical inference is the logical framework whichwe can use to trial our beliefs about the noisy world against data. We formalise our beliefs in models of probability. The models are probabilistic bc we are ignorant of many of the interacting parts of a system, meaning we cannot say with certainty whether something will, or not, occur.

  • ¬\lnot is used as not in probability, e.g. ¬10\lnot 10
  • | means given in probability, e.g. Pr(1,1∣ rigged casino )Pr(1,1| \text{ rigged casino })

Bayes’ theorem is the rule or theorem that allows us to find the causation behind the effect

Pr(effect∣cause)→bayes’ theoremPr(cause∣effet)Pr(\text{effect}|\text{cause}) \xrightarrow{\text{bayes' theorem}} Pr(\text{cause}|\text{effet})

Which can be written as

Pr(cause∣effet)=Pr(cause∣effect)⋅Pr(cause)Pr(effect)Pr(\text{cause}|\text{effet}) = \dfrac{Pr(\text{cause}|\text{effect}) \cdot Pr(\text{cause})}{Pr(\text{effect})}