Assigning a Likelihood Function

As scientists, we want to parameterise our models and compare them against alternative models. For these purposes, measurement data are needed, but how exactly should we use the data? The answer is always the same: in the likelihood function. The likelihood is a conditional probability density function, denoted as \(p[y|\theta ]\) (or more succinctly as \(L[\theta ]\) ), where as usual y can be multi-dimensional. It is the answer to the question: ’what is the probability of measuring y if the true value is \(f(x,\theta )+\epsilon _{\text{model}}\) ?’. This can be written formally as follows: \(\displaystyle \begin{aligned} \begin{aligned} &\underline{\text{LIKELIHOOD FUNCTION:}} \\ &L[\theta] = p[y|\theta] = p[f(x,\theta) - y = \epsilon_y + \epsilon_{\text{model}}]. \end{aligned} \end{aligned}\)