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I'm trying to interprete the Example 1 from the wikipedia page: the likelihood function of a coin flip with a single parameter p expressing how likely a head will come up.

The likelihood is defined as "the plausibility of a parameter value of the statistical model assumed to describe the observed data, given specific observed data".

It's clear to me that the plausibility of a coin being fair (p = 0.5) flipping the coin twice and observing two heads (likelihood = 0.25) is lower than the plausibility of any other model that assumes a higher probability for the head to come up.

But how does one interprete a likelihood = 1 with p = 1, i.e. a model that assumes a head will ALWAYS come up? I'm confident it doesn't mean we are 100% sure that that's the correct model but I cannot really state why not.

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Likelihoods express the probability, that the observed event (two heads up) occurs given some parameters ($p = 1$). If you have a coin that always lands on head, two heads in a row have a probability of 1. However, the likelihood does not equal the probability, that these parameters / models are the correct ones. It is crucial to distinguish between plausibility and probability here.

As you yourself stated in the second to last paragraph, likelihoods can be used to compare different parameter choices / models. The one with a higher likelihood is to be prefered.

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  • $\begingroup$ Thanks, now I understand. The 100% certainty is that with that specific parameter we would see that outcome. $\endgroup$
    – Dario R
    Jan 4, 2019 at 9:32
  • $\begingroup$ Yes, that's the right way to think of it $\endgroup$
    – nope
    Jan 4, 2019 at 10:03
  • $\begingroup$ If likelihood is not equal to probability then why do we write this $L(\theta|O) = P(O|\theta)$? Doesnt $P(O|\theta)$ integrate to 1? $\endgroup$
    – GENIVI-LEARNER
    Jan 23, 2020 at 13:30

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