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## 2.2 Modeling Uncertainty: The Likelihood Principle

Assume that the value of a random variable z must be inferred from a certain observation or subjective information S. If z has been drawn from p(z) and the measurement or judgment process is characterized by a conditional p(S|z), our knowledge on z is updated according to p(z|S)=p(z) p(S|z)
/ p(S), where p(S) = ∫Z p(S|z) p(z) dz (see Fig. 4).

The likelihood function fS(z) ≡ p(S|z) is the probability density ascribed to S by each possible z. It is an arbitrary nonnegative function over z that can be interpreted in two alternative ways. It can be the "objective" conditional distribution p(S|z) of a physical measurement process (e.g. a model of sensor noise, specifying bias and variance of the observable S for every possible true value z), also known as *error model*. It can also be a "subjective" judgment about the chance of the different z values (e.g. intervals, more likely regions, etc.), based on vague or difficult to formalize information. The dispersion of fS(z) is directly related to the uncertainty associated to the measurement process. Following the *likelihood principle* (Berger 1985), we explicitly assume that all the experimental information required to perform probabilistic inference is contained in the likelihood function fS(z).

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