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BTW can anybody share a link to a really simple explanation of the Bayes' Theorem? I've once seen one, it was a size of a twit and would let you understand it in a matter of seconds, all the "super-duper intuitive explanations" around are too huge and complex actually.
BTW can anybody share a link to a really simple explanation of the Bayes' Theorem? I've once seen one, it was a size of a twit and would let you understand it in a matter of seconds, all the "super-duper intuitive explanations" around are too huge and complex actually.
P(A|B) = P(B|A)P(A)/P(B)
P(prior|data) = P(data|prior)P(prior)/P(data)
P(prior|data) proportional to P(data|prior)P(prior)
This is probably not what you were looking for but this is it.
BTW can anybody share a link to a really simple explanation of the Bayes' Theorem? I've once seen one, it was a size of a twit and would let you understand it in a matter of seconds, all the "super-duper intuitive explanations" around are too huge and complex actually.
BTW can anybody share a link to a really simple explanation of the Bayes' Theorem? I've once seen one, it was a size of a twit and would let you understand it in a matter of seconds, all the "super-duper intuitive explanations" around are too huge and complex actually.
P(Hypothesis|Data) = P(Hypothesis) * evidence_factor
P(Hypothesis) is the prior probability of the Hypothesis being true, in other words the probability we gave to the Hypothesis before seeing any of the data we are using in the theorem. When new data is observed, we use Bayes' theorem to update our believe in the hypothesis, which in practice means multiplying our prior probability by a number that depends on how well the new data fits our hypothesis. More precisely:
evidence_factor = P(Data|Hypothesis)/P(Data)
So it is the ratio of how likely our data is if our hypothesis is true, compared to (divided by) how likely it is in general. If it is more likely to occur in our Hypothesis, our probability of it being true increases, if it is more likely in general (and thus also more likely in case our hypothesis is not true, you can prove mathematically that those two statements are the same), then our believe in the hypothesis decreases.
TLDR: Prob(Hypothesis after I have seen new data) = Prob(Hypothesis before I saw the new data) * (how likely I am to see the data if my hypothesis is true, compared to in general)
BTW can anybody share a link to a really simple explanation of the Bayes' Theorem? I've once seen one, it was a size of a twit and would let you understand it in a matter of seconds, all the "super-duper intuitive explanations" around are too huge and complex actually.
So a prior's probability increases to the degree that it predicts an observation better than alternative priors.
Earlier quoted context omitted.
NUTS and Stan are quite old at this point! Here's a more recent advance https://arxiv.org/pdf/1711.09268.pdf
Stan supports Hamiltonian Monte Carlo. https://arxiv.org/pdf/1701.02434.pdf
'vanilla' HMC uses detailed balance to guarantee that the stationary distribution of the chain is the one you want, causing the process to behave like a random walk. So although the Hamiltonian bit of HMC lets you take these great big steps through state space, you end up retracing your steps quite a lot.
Hence NUTS (No-U-turn sampler) et. al
BTW can anybody share a link to a really simple explanation of the Bayes' Theorem? I've once seen one, it was a size of a twit and would let you understand it in a matter of seconds, all the "super-duper intuitive explanations" around are too huge and complex actually.
But it’s worth noting that the most important branch of the physical science today, quantum theory, is manifestly non-Bayesian.