# Likelihood function and Posterior Probability

Cited from wikipedia

The likelihood function $L(θ|x)=f(x|θ)$ is not the same as the probability that those parameters are the right ones, given the observed sample. Attempting to interpret the likelihood of a hypothesis given observed evidence as the probability of the hypothesis is a common error, with potentially disastrous real-world consequences in medicine, engineering or jurisprudence. See prosecutor's fallacy for an example of this.

So, is the posterior probability the right ones? Can I think this like posterior probability is just likelihood taking account of prior probability as stated in the Bayes rules?

What confuses more is why maximum-likelihood estimation can be a proper estimate of the model parameters.

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Yes, the posterior probability is more or less like the likelihood but taking the prior probability into account. From a Bayesian perspective, maximum likelihood is like using a prior proportional to 1 on the entire real line. At best this is an improper prior (e.g. for the mean of a Normal distribution). At worst it doesn't make sense at all--for example, if you are estimating a proportion $p \in [0,1]$, your "prior" is that $p$ is as likely to be in the interval $[0.4,0.5]$ as it is to be in the interval $[6.4,6.5]$, which is clearly nonsense.