Where does likelihood collide?

In statistics the likelihood is the probability of the observed data, read as a function of a model's unknown parameters rather than of the data: for fixed data you vary the parameters and ask which values would have made what you saw most probable. Fisher's 1922 On the Mathematical Foundations of Theoretical Statistics used the word to mark this quantity off from probability, and built maximum likelihood estimation and the likelihood ratio on it. Famously the likelihood is not a probability distribution over the parameters: it need not integrate to one and carries no prior, so it measures the relative support the data give to each parameter value, not the chance that a value is true. This is a precise mathematical object, unlike the trademark law test of consumer confusion or the everyday feeling that something is probable.[1]

Unit

One likelihood value for a parameter setting given fixed data

Recovery means

Not applicable

Measured by

Computed from the model's density evaluated at the observed data, then maximised or compared as a ratio to estimate parameters

In trademark and unfair-competition law the likelihood of confusion is the central test for infringement: whether an ordinary, reasonably prudent consumer would probably be confused about the source or sponsorship of goods that carry a mark. Courts do not compute a number but weigh a checklist of factors, such as the similarity of the marks, the closeness of the goods, and any evidence of actual confusion. Beebe's 2006 An Empirical Study of the Multifactor Tests for Trademark Infringement examined how United States courts actually apply these multi-factor tests in practice. Here likelihood names a legal standard about a consumer's probable state of mind, not the statistician's function of parameters and not a bare degree of plausibility.[2]

Unit

One judicial finding that confusion is, or is not, likely for a pair of marks

Recovery means

Not applicable

Measured by

Assessed by courts weighing multi-factor tests, sometimes aided by consumer surveys of actual confusion

In its ordinary and philosophical meaning, likelihood is simply how probable or plausible a claim is: the degree to which the evidence leads us to expect that something is true. Keynes's 1921 A Treatise on Probability developed this as a logical relation, the rational degree of belief that a body of evidence warrants in a conclusion, which may be strong, weak, or in some cases not comparable at all. On this view a likely event is one we are more entitled to expect, and everyday standards, such as deciding a matter on the balance of probabilities, appeal to this graded expectation. This plain degree of plausibility is not the statistician's likelihood function, which is deliberately not a probability, and it is not the trademark law's specific test of consumer confusion.[3]

Unit

A graded degree of expectation, running from unlikely to near-certain

Recovery means

Not applicable

Measured by

Judged informally, or represented as a probability or a degree of rational belief warranted by evidence