In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values. Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics.
In non-technical parlance, "likelihood" is usually a synonym for "probability." But in statistical usage, a clear technical distinction is made depending on the roles of the outcome or parameter.
- Use probability when describing a function of the outcome given a fixed parameter value.
- “Given that I have flipped a coin 100 times and it is a fair coin, what is the probability of it landing heads-up every time?"
- Use likelihood when describing a function of a parameter given a fixed outcome.
- "Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?"
Read more about Likelihood Function: Definition, Log-likelihood, Likelihood Function of A Parameterized Model, Example 1, Example 2, Likelihoods That Eliminate Nuisance Parameters, Historical Remarks
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