# Bayes' rule — Log-odds form We can take the log of both sides of [[Bayes' rule — Odds form]] to get the log-odds form: $ \log\left(\frac{P(H | e)}{P(\neg H | e)}\right) = \log\left(\frac{P(e | H)}{P(e | \neg H)}\right) + \log\left(\frac{P(H)}{P(\neg H)}\right) $ This form makes it easier to visualize belief updates, since each time you come across a $(1:10)$ observation in favour of H, your belief shifts by one unit to the right of the log-odds line (if you use a base 10 log).  The statement "[[Extraordinary claims require extraordinary evidence]]" is then even easier to understand: if you have a prior of $(1:100,000)$ for H, your belief starts at 6 bits on the right. If you then see a $(100:1)$ piece of evidence, then your belief is still 4 bits on the right, so it does not change your position that much. Finally, observe that probability 0 or 1 corresponds to $-\infty$ or $+\infty$ on the log-odds line. This means that if you assign probability 0 or 1 to a hypothesis, no amount of evidence will make you change your mind about it. --- ## 📚 References - [Bayes' rule: Log-odds form](https://arbital.com/p/bayes_log_odds/?pathId=77121)