0.0.1 Likelihood Inference
Likelihood inference is based upon the likelihood principle which is the proposition that, given a specific statistical model, all of the evidence in a sample relevant to model parameters is contained in the likelihood function.
0.0.2 Likelihood Function
A likelihood function arises from a [[probability-density-function|probability density function]] considered as a function of its distributional parameterization argument. For example consider a model which gives the probability density function \(f_X(x;\theta)\) of observable random variable \(X\). For a specific value \(x\) of \(X\) the function \[ \mathcal{L}(\theta;x)=f_X(x;\theta) \] is a likelihood function of \(\theta\) as it gives a measure of how likely any particular value of \(\theta\) is given that \(X\) took a value \(x\). Two likelihood functions are equivalent if one is a scalar multiple of the other.
0.0.3
The likelihood principle is this:
All information from the data that is relevant to inferences about the value of the model parameters is in the equivalence class to which the likelihood function belongs.
The strong likelihood principle applies this same criterion to cases such as sequential experiments where the sample of data that is available results from applying a to the observations earlier in the experiment.