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I know that the very similar question has been asked many times, but I hope that somebody can explain the mechanics in simple terms.

Let's say there is a population of N=1000 people. A certain share p of them are honest, and 1-p are liars. This share is unknown to the individuals, but each individual has a prior belief about p. Let's say a specific individual i has p = 0.56. At time t=1 he met a person, and with the probability equals p = 0.56 he decided to trust him. It turned to be that the person whom he met, was a liar. What will be his p_(t-1) if he updates his beliefs based on this one encounter? Again sorry for a naive question.

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  • $\begingroup$ You could clarify a few points. For instance, when you say that a specific individual has $p=0.56$, if this means that for this individual $\Pr\{p=0.56\}=1$, then according to Bayes Theorem he will never change his mind, no matter what new information he gets. $\endgroup$ – Zen Oct 15 '17 at 22:29
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What you are describing is the beta-binomial model. You have sample of $n$ individuals, from whom $k$ are liars, you want to learn about the proportion $p$ of the lairs in the population. When applying Bayes theorem to probabilities of events, you update the probabilities directly. When dealing with continuous (or with discrete probability distributions), you deal with functional forms of probability distributions. First, you would need to define the likelihood function of your data, the distribution describing number of "successes" in $n$ trials is a binomial distribution. So far so good, you can describe your data in terms of binomial distribution and you want to learn about the probability of "success" $p$. Next, you need to assume some probability distribution for $p$, your prior. Then you would need apply Bayes theorem

$$ \pi(p | y) = \frac{f(y|p) \; \pi(p)}{ \int f(y|p) \; \pi(p) \; dp } $$

where $f(y|p)$ is your likelihood function and $\pi(p)$ is your prior. In this case, $f(y|p)$ is a binomial distribution parametrized by known parameter $n$ and unknown $p$. There is many possible choices for prior distribution $\pi(p)$ (it can be any continuous distribution bounded in $[0,1]$), but beta distribution is very convenient choice, since it is a conjugate prior, so it has a closed-form solution. If your model is

$$ \begin{align} p &\sim \mathcal{Beta}(\alpha, \beta) \\ y &\sim \mathcal{Bin}(n, p) \end{align} $$

then posterior distribution for $p$ is simply

$$ p \sim \mathcal{Beta}(\alpha + k, \beta + n-k) $$

If you want to use "uninformative" prior, you can use e.g. $\alpha=\beta=1$ what leads to uniform prior distribution for $p$.

If this is still unclear, check the links I provided in the text and one of the multiple handbooks on Bayesian inference.

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