In statistics, the use of<unk> s is a<unk> ian alternative to classical hypothesis testing.
Given a model selection problem in which we have to choose between two models M1 and M2, on the basis of a data vector x. The<unk>  K is given by
where 
p
(
x
|
M
i
)
{\displaystyle p(x|M_{i})}
 is called the marginal likelihood for model i. This is similar to a likelihood-ratio test, but instead of maximising the likelihood,<unk> ians average it over the parameters. Generally, the models M1 and M2 will be parametrised by vectors of parameters θ1 and θ2; thus K is given by
The logarithm of K is sometimes called the weight of evidence given by x for M1 over  M2, measured in bits, nats, or bans, according to whether the logarithm is taken to base 2, base e, or base 10.
A value of K > 1 means that the data indicate that M1 is more strongly supported by the data under consideration than M2. Note that classical hypothesis testing gives one hypothesis (or model) preferred status (the 'null hypothesis'), and only considers evidence against it. Harold Jeffreys gave a scale for interpretation of K:
The second column gives the corresponding weights of evidence in decibans (tenths of a power of 10).  According to I. J. Good a change in a weight of evidence of 1 deciban (ie a change in an odds ratio from evens to about 5:4) is about as finely as humans can reasonably perceive their degree of belief in a hypothesis in everyday use.
The use of<unk> s or classical hypothesis testing takes place in the context of inference rather than decision-making under uncertainty. That is, we merely wish to find out which hypothesis is true, rather than actually making a decision on the basis of this information. Frequentist statistics draws a strong distinction between these two because classical hypothesis tests are not coherent in the<unk> ian sense.<unk> ian procedures, including<unk> s, are coherent, so there is no need to draw such a distinction. Inference is then simply regarded as a special case of decision-making under uncertainty in which the resulting action is to report a value. In a decision-making context<unk> ian statisticians might use a<unk>  as part of making a choice, but would also combine it with a prior distribution and a loss function associated with making the wrong choice. In an inference context the loss function would take the form of a scoring rule. Use of a logarithmic score function for example, leads to the expected utility taking the form of the Kullback-Leibler divergence. If the logarithms are to the base 2 this is equivalent to Shannon information.
Bayes factor
Absolute deviation
Potential energy
Free energy
Quantum yield
Ancillary statistic
P-value
Cmax