What motivates Bayes' Theorem?

[Dale]

That the frequentist interpretation is a mistake is pushing your luck

That is not what I said was a mistake. What is a mistake is the belief that the frequentist interpretation defines probability. Long run frequencies and uncertainties are both valid examples of probabilities. But what defines probability is the axioms.

 

[Agent Smith]

Sometimes there is no sample. So a Bayesian can still choose a prior that represents their uncertainty.

Choose as opposed to compute? This would matter I feel.

 

[Agent Smith]

Which is precisely why I mention the axioms. A lot of frequentists make the same mistake, but it is a mistake.

So frequentism (I hope that's the correct term) is just one version of probability.

 

[Dale]

Choose as opposed to compute? This would matter I feel.

Yes. Choose. With no data how would you compute it? This is one of the best things about Bayesian statistics, and it is also one of the things that most bothers people learning about it. It allows you to include nebulous things like expert opinion that may not be something that you can compute.

I dive into this topic a bit in one of my insights articles, but I don’t remember which one

So frequentism (I hope that's the correct term) is just one version of probability.

Yes. Just like an arrow is one version of a vector.

 

[Agent Smith]

@Dale I'm a bit lost here, Bayes' theorem is $\text{P(Hypothesis|Evidence)}$. Isn't the evidence a measurement with/without computation, a/the data? However I can see how the prior probability can be chosen (it's just a guess???). The posterior probability would be the result of a computation, using Bayes' Theorem.

 

[Agent Smith]

Also @Dale , can we reduce Bayesian uncertainty to a frequentism, give it a frequentist interpretation? So if I'm 90% certain the earth is flat, it would "mean" 9 out 10 scenarios I'm right and 1 out 10 I'm wrong.

 

[Dale]

@Dale I'm a bit lost here, Bayes' theorem is $\text{P(Hypothesis|Evidence)}$. Isn't the evidence a measurement with/without computation, a/the data? However I can see how the prior probability can be chosen (it's just a guess???). The posterior probability would be the result of a computation, using Bayes' Theorem.

So in this context Bayes theorem is $$P(Hypothesis|Evidence)=\frac{P(Evidence|Hypothesis) \ P(Hypothesis)}{P(Evidence)}$$ What is chosen is $P(Hypothesis)$. This is called the prior. It represents the uncertainty before looking at the evidence. It can be based on any prior studies and any prior data, but if there really is not any evidence then it can be based on things like expert opinion or rough estimation. Whatever your knowledge is, from whatever source you have, before looking at your new data, the prior is chosen to reflect that knowledge.

Also @Dale , can we reduce Bayesian uncertainty to a frequentism, give it a frequentist interpretation? So if I'm 90% certain the earth is flat, it would "mean" 9 out 10 scenarios I'm right and 1 out 10 I'm wrong.

In situations where frequentist probabilities exist, they will match Bayesian probabilities. There reverse is not necessarily true. There are situations that there is no reasonable definition of frequentist probability that the Bayesian probability models just fine.

Also, Bayesian statistical tests are usually equivalent to frequentist statistical tests when the Bayesian test is performed using an uninformative prior.

 

[Agent Smith]

@Dale muchas gracias.

 

[Agent Smith]

The universe is either flat or it isn't. Probability doesn't apply.

@Dale

Not intending to start a fight, but in what way does Bayesian thinking (statistics/probability) inform Quantum Physics? Last I checked, from the very layman's discussions I've had, for a particle it isn't the case that it is "either flat or it isn't". Some say it's both here and there (position-wise). The (Bayesian) uncertainty is a an actual feature of the world of particles I believe.

 

[Dale]

in what way does Bayesian thinking (statistics/probability) inform Quantum Physics?

I don’t know. I am not a big quantum mechanics guy. My physics knowledge runs to classical physics, relativity, and biomedical engineering.

I do know that there is a quantum Bayesian interpretation called qbism, but I don’t know the details. I tend to be interpretation-agnostic in many things, so I probably would not be a particularly strong adherent of it.

 

[Agent Smith]

@Dale , but per the axioms of probability (gracias for that), the sum of the probabilities of a particle's position must add up to $1$. Doesn't that mean they're mutually exclusive? Perhaps you can answer from a mathematician's point of view.

 

[Dale]

@Dale , but per the axioms of probability (gracias for that), the sum of the probabilities of a particle's position must add up to $1$.

Yes. This means that the probability that the particle is somewhere is 1.

Doesn't that mean they're mutually exclusive? Perhaps you can answer from a mathematician's point of view.

Again, I am not a big QM guy, but what are you referring to by mutually exclusive?

 

[Agent Smith]

Yes. This means that the probability that the particle is somewhere is 1.

Again, I am not a big QM guy, but what are you referring to by mutually exclusive?

It's ok, I'll try and work it out on my own. Gracias,