Likelihood function
In the English language, the word likelihood means the same as, or almost the same as, the word 'probability'.
In probability theory the technical term likelihood is a concept related to, but distinct from, that of probability. (Confusingly, likelihood is also used in its informal English sense in some discussions of the interpretation of mathematical probability.)
Unlike probability, which estimates the frequency of or degree of belief in unknown consequences of known causes, likelihood works backwards, from observed results to hypothetical models and parameters.
The likelihood function is the function which specifies the probability of the sample observed on the basis of a known model, in terms of the model's parameters.
Note: This is not the same as the probability of those parameters being 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.
Example
For example, if I toss a coin, with a probability pH of landing heads up ('H'), the probability of getting two heads in two trials ('HH') is pH2. If pH = 0.5, then the probability of seeing two heads is 0.25.
In symbols, we can say the above as
- p('HH' | pH = 0.5) = 0.25
Another way of saying this is to reverse it and say that "the likelihood of pH = 0.5 given the observation 'HH' is 0.25". But this is not the same as saying that the probability of pH = 0.5 given the observation is 0.25.
To take an extreme case, on this basis we can say "the likelihood of pH = 1 given the observation 'HH' is 1". But this is clearly not the case: the event 'HH' can occur for any pH > 0 (and often does, in reality, for p roughly 0.5).
In fact, since the parameter pH is continuous, we should think of the likelihood function as a likelihood density function.
The likelihood function does not in general follow all the axioms of probability: for example, the integral over the likelihood density function does not in general sum to 1.
In this example, the integral of the likelihood density over the interval [0, 1] in pH is 1/3, clearly demonstrating that the likelihood density function cannot be interpreted as a probability density function for pH.
See also: