Likelihood

Dina statistik, fungsi likelihood mangrupa fungsi conditional probabilitas dumasar kana "pertimbangan" fungsi alesan kadua nu mana fungsi mimiti dianggap angger, ahirna:

${\displaystyle B\mapsto P(A|B),}$

salian ti éta fungsi séjén proporsional saperti halna fungsi likelihood. ku sabab kitu, fungsi likelihood keur B mangrupa kelas ékivalénsi tina fungsi

${\displaystyle L(b)=\alpha \;P(A|B=b)}$

keur unggal babandingan konstanta α > 0. Mangka nilai numerik L(b) teu bisa dijéntrékeun; sakabéh nu aya bakal mibanda babandingan L(b₂)/L(b₁), ku sabab ieu konstanta bakal gumantung kana babandingan konstanta tadi.

Likelihood nyaéta istilah nu dipaké keur fungsi likelihood. Dina basa nu ilahat dipaké sapopoe, "likelihood" nyaéta salah sahiji harti nu sarua keur istilah "probability", tapi dina artikel ieu bakal ngagunakeun dina harti sacara téhnik.

Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantuanna didagoan pikeun narjamahkeun.

In a sense, likelihood works backwards from probability: given B, we use the conditional probability P(A | B) to réason about A, and, given A, we use the likelihood function P(A | B) to réason about B. This mode of réasoning is formalized in Bayes' theorem; note the appéarance of a likelihood function for B given A in:

${\displaystyle P(B|A)={\frac {P(A|B)\;P(B)}{P(A)}}}$

since, as functions of B, both P(A|B) and P(A|B)/P(A) are likelihood functions for B given A.

For more about making inferences via likelihood functions, see also the method of maximum likelihood, and likelihood-ratio testing.

Likelihood function of a parametrized model

Among many applications, we consider here one of broad théoretical and practical importance. Given a parametrized family of probability density functions

${\displaystyle x\mapsto f(x\mid \theta ),}$

where θ is the paraméter (in the case of discrete distributions, the probability density functions are probability "mass" functions) the likelihood function is

${\displaystyle L(\theta )=f(x\mid \theta ),}$

where x is the observed outcome of an experiment. In other words, when f(x | θ) is viewed as a function of x with θ fixed, it is a probability density function, and when viewed as a function of θ with x fixed, it is a likelihood function.

Note: This is not the same as the probability that those paraméters 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 réal-world consequences in medicine, engineering or jurisprudence. See prosecutor's fallacy for an example of this.

Conto

For example, if I toss a coin, with a probability p_H of landing héads up ('H'), the probability of getting two héads in two trials ('HH') is p_H². If p_H = 0.5, then the probability of seeing two héads is 0.25.

In symbols, we can say the above as

${\displaystyle P({\mbox{HH}}\mid p_{H}=0.5)=0.25}$

Another way of saying this is to reverse it and say that "the likelihood of p_H = 0.5 given the observation 'HH' is 0.25", i.e.,

${\displaystyle L(p_{H}=0.5\mid {\mbox{HH}})=P({\mbox{HH}}\mid p_{H}=0.5)=0.25}$.

But this is not the same as saying that the probability of p_H = 0.5 given the observation is 0.25.

To take an extreme case, on this basis we can say "the likelihood of p_H = 1 given the observation 'HH' is 1". But it is cléarly not the case that the probability of p_H = 1 given the observation is 1: the event 'HH' can occur for any p_H > 0 (and often does, in réality, for p_H roughly 0.5).

The likelihood function does not in general follow all the axioms of probability: for example, the integral of a likelihood function is not in general 1. This is because integration of the likelihood density function L is performed over all possible values of the modél paraméters (in this case, p_H), while integration of a probability density function f is performed over the random variables (which in this case take on the four pairs of values 'TT', 'TH', 'HT' and 'HH'). In this example, the integral of the likelihood density over the interval [0, 1] in p_H is 1/3, demonstrating again that the likelihood density function cannot be interpreted as a probability density function for p_H. On the other hand, given any particular value of p_H, e.g. p_H=0.5, the integral of the probability density function over the domain of the variabel acaks is 1.

See also:

• Bayesian inference
• Likelihood principle
• Likelihood-ratio test
• Maximum likelihood method
• The principle of maximum entropy is also related to the maximum likelihood method
• Minimum entropy method
• score (statistics)