Bayesian statistics

Continuation of https://matchmaticians.com/questions/qse0ka .
Still struggling with understanding Bayesian statistics, so I need comparsion of MLE and Bayesian method(or whatever it is called) please.
The thing which confuses me the most is parameter.
In MLE I have pdata - some unknown distribution which generated samples.
pmodel with parameters - model which I believe describes samples in the best way with it's parameters.
So my goal in MLE is to find parameters for my model(distribution) which maximizes the probability of observing the given samples under the assumed distribution with parameter values θ.

Let's consider an example which @Mathe showed for Bayesian:

Suppose we are interested in knowing the efficacy of a treatment to heal a person, that is, we are interested in the probability pp of a person being cured after following the treatment. We know that the treatment was administered to 100 patients, and 60 were cured.
Now, we want to follow a Bayesian approach, so we treat the unknown probability p as a random quantity. Based on previous studies, we believe the efficacy of the treatment could be modeled by a beta distribution with parameters α=3,β=5. We want to see how the recent sample of treated patients can update our beliefs about the treatment.

So we start from describing our prior observations and believe that they can be described by a beta distribution with parameters α=3,β=5, right?

The prior then takes the form of $p_{prior}(p) \propto p^{3-1} (1-p)^{5-1}$ and the likelihood function is $p_{data}(x|p) \propto p^{60}(1-p)^{40}$.

Here what confuses me. Why probability is a parameter? I understand a parameter as a part of some parametric probability distribution.
Also @Mathe mentioned at the start of answer that parameter is a part of pmodel
, what is pmodel here?

Other calculations are clear for me, I mean that I can calculate Byesian, but still don't understand how it works.

  • Kav10 Kav10
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    Low bounty!

    • @Kav10 I will add 3.44 for you ahah

    • Kav10 Kav10
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      Still low for all your questions!

    • I mean 3.34

    • @Kav10 how much do you want

    • you accepted to answer while saying low bounty, why?

  • Kav10 Kav10
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    This will take at least 30-45 minutes to explain and write. So, $25-35. You can tip me after, please.

    • If this explanation will be sufficient I will do it with great pleasure

    • Kav10 Kav10
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      Sounds good. Can you extend the deadline or you want it in ~12 hours?

    • I want in 12 hours

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Kav10 Kav10
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  • "So, the important thing here is that probabilities themselves are treated as distributions. In the example you provided, as you said, the parameter of interest is the probability p that a person is cured after the treatment, right? So, in the Bayesian approach, this parameter is treated as a random variable with its own probability distribution, which is called the prior distribution." So there is a probability of probability? How should I understand it?

  • "So, in Bayesian statistics, probability is treated as a parameter because it represents our belief about the true value of the parameter of interest." Still not clear. What you are describing must be called random variable, or variable of interest, but why is it called a parameter?

  • Please add some examples for explanation. Why text above took you more than 10 minutes to write? You didn't use any examples, formulas etc. Just generic description of Bayesian approach.

    • Kav10 Kav10
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      Seriously?

    • Kav10 Kav10
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      You are not paying for type writing!! You are paying for knowledge and skill. Its not a ready text to write! Needs thinking, …

    • Kav10 Kav10
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      You provided an example yourself and asked questions about that example!!

    • When you are explaining something to other people you should use at lest some little examples, formulas etc imho. I payed 10$ for your answer and not more since it wasn't sufficient for me.

    • Kav10 Kav10
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      Nope! You already had an example. You clearly need to learn a lot of things about this :-)

    • Okay thanks.

    • Kav10 Kav10
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      Learn that you are not paying for type writing, you are paying for skills and knowledge that you don't have :-)

    • @Kav10 so that you don’t have to say “it will take me 30 minutes to answer” and accept my question before any agreement to pay you extra

    • Kav10 Kav10
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      That's called trust. I have seen users like you here who say they pay and they ask million questions after and don't pay. That's why I did not answer your questions.

    • Don’t lie. I payed for all your previous answers an extra amount of money. But this time you decided yourself that I’m gonna pay you and accepted to answer without even previously asking me :)

    • Kav10 Kav10
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      I have already wasted my time to reply to your msgs here. Just FYI, this is the reason I did not respond to your questions: "Why text above took you more than 10 minutes to write?" This shows you don't even have the basic udnerstanding that "You don't pay for typewriting, you pay for KNOWLEDGE that you lack" :-)

  • Ok nvm. I think I managed to understand it after covering examples from "Bayesian Data Analysis"

    • Kav10 Kav10
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      So, not paying bc explanation was insufficient?!

    • Yes. Answer was insufficient

    • Kav10 Kav10
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      Well well, you seriously think with $10 you can ask a lengthy technical question and then ask million questions and expect an answer? lol

    • For 10$ not, I wanted to pay you 15$ extra for answering my questions in comments section, but you didn’t so I found an answer on my own

    • Kav10 Kav10
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      That's fine. Just remember that you don't pay for typewriting when you say "Why text above took you more than 10 minutes to write?" You pay for the knowledge that you don't have :-)

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