$Bayesian statistics$

Question

$Bayesian statistics$

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.

Probability Statistics

Babaduras

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Kav10

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Low bounty!
- Babaduras
  
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  @Kav10 I will add 3.44 for you ahah
- Kav10
  
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  Still low for all your questions!
- Babaduras
  
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  I mean 3.34
- Babaduras
  
  0
  
  @Kav10 how much do you want
- Babaduras
  
  0
  
  you accepted to answer while saying low bounty, why?
Kav10

0

This will take at least 30-45 minutes to explain and write. So, $25-35. You can tip me after, please.
- Babaduras
  
  0
  
  If this explanation will be sufficient I will do it with great pleasure
- Kav10
  
  0
  
  Sounds good. Can you extend the deadline or you want it in ~12 hours?
- Babaduras
  
  +1
  
  I want in 12 hours

Answer

The answer is accepted.

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Kav10

0

Low bounty!

Babaduras

0

@Kav10 I will add 3.44 for you ahah

Kav10

0

Still low for all your questions!

Babaduras

0

I mean 3.34

Babaduras

0

@Kav10 how much do you want

Babaduras

0

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

0

you accepted to answer while saying low bounty, why?
Kav10

0

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

Babaduras

0

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

Kav10

0

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

Babaduras

+1

I want in 12 hours
Babaduras

0

If this explanation will be sufficient I will do it with great pleasure
Kav10

0

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

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

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. This prior distribution represents your initial beliefs about the parameter “before observing any data”.

In the case of the efficacy of the treatment, you've assumed a beta distribution with parameters you mentioned as the prior.

This distribution captures your prior uncertainty about the true value of p, representing the belief that p is more likely to be around some number based on previous studies.

The likelihood function represents how probable the observed data (60 out of 100 patients cured) is, given a specific value of p. This function is derived from the binomial distribution.

The Bayesian approach then combines the prior distribution and the likelihood function to get the posterior distribution:

Prior Distribution & Likelihood Function —> Posterior Distribution

This represents your updated beliefs about the parameter p after observing the data. This posterior distribution is proportional to the product of the prior and the likelihood.

So, in Bayesian statistics, probability is treated as a parameter because it represents our belief about the true value of the parameter of interest. The prior distribution captures our initial beliefs, the likelihood incorporates the data, and the posterior distribution represents the updated beliefs after observing the data.

As far as I know, the term "pmodel" isn't standard terminology in Bayesian statistics. The typical terms used are "prior distribution," "likelihood function," and "posterior distribution” as I described above. Key components of Bayesian analysis: incorporating prior beliefs, updating those beliefs with observed data, and obtaining a refined distribution for the parameter of interest. In the context of the example you provided, the term "pmodel" doesn't directly apply.

So, in general, there are key differences between the two approaches. MLE seeks to find the values of parameters that make the observed data most likely under a given model. Bayesian methods focus on updating prior beliefs about parameter values based on observed data. MLE treats parameters as fixed, unknown constants. Bayesian methods treat parameters as random variables with associated probability distributions.

Best,
Kav10

Kav10

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Babaduras

0

"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?
Babaduras

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"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?
Babaduras

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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
  
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  Seriously?
- 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
  
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  You provided an example yourself and asked questions about that example!!
- Babaduras
  
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  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
  
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  Nope! You already had an example. You clearly need to learn a lot of things about this :-)
- Babaduras
  
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  Okay thanks.
- 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 :-)
- Babaduras
  
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  @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
  
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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.
- Babaduras
  
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  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
  
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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" :-)
Babaduras

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Ok nvm. I think I managed to understand it after covering examples from "Bayesian Data Analysis"
- Kav10
  
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  So, not paying bc explanation was insufficient?!
- Babaduras
  
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  Yes. Answer was insufficient
- 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
- Babaduras
  
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  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
  
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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 :-)

$Bayesian statistics$

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