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I am studying this answer, by user kjetil b halvorsen, on likelihood functions. He says the following:

We model $n$ independent Bernoulli experiments, leading to data $X_1, \dots, X_n$, each with a Bernoulli distribution with (probability) parameter $p$. This leads to the likelihood function $$ \prod_{i=1}^n p^{x_i} (1-p)^{1-x_i} $$

How does one come to the conclusion that the likelihood function is

$$ \prod_{i=1}^n p^{x_i} (1-p)^{1-x_i}? $$

I would greatly appreciate it if people would please take the time to explain this.

The Pointer
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    The definition of a likelihood function (of a parameter) is the probability function of the sample where the random variable $X$ is set to the observed value $x$. For *independent* random variables, their *joint* probability is the product of their marginal pmfs. For a Bernoulli RV, the pmf is $p^X(1-p)^{1-X}$. Does that answer your question? – AdamO Feb 19 '20 at 16:59
  • @AdamO Thank you. Almost. I'm still unclear on the connection between the likelihood function and the joint probability; I think that is where my confusion lies. – The Pointer Feb 19 '20 at 17:03
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    Likelihood Function in this case: $$P(X_1,\dots,X_n|p)=\prod_{i=1}^n P(X_i|p)$$ – gunes Feb 19 '20 at 17:04
  • @gunes Ahh, that makes sense. How does that definition relate to the definitions provided here https://en.wikipedia.org/wiki/Likelihood_function#Definition ? I checked the Wikipedia page before posting this question, but I wasn't able to see the connection (which was the impetus for posting this question). – The Pointer Feb 19 '20 at 17:06
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    $P_\theta(X=x)$ means the same thing as $P(X=x|\theta)$. Sometimes, $\theta$ is just an unknown parameter, sometimes it's modeled as a RV in which the given notation makes more sense. – gunes Feb 19 '20 at 17:08
  • @gunes Ok, I understand now. Thank you both for taking the time to clarify this. – The Pointer Feb 19 '20 at 17:09

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