A note on Likelihood and Likelihood Based Inference

Statistical inference aims to draw conclusions about unknown parameters using observed data. Among the various inferential frameworks, likelihood-based inference occupies a central position because it provides a direct measure of the support that observed data give to competing parameter values. This paper presents a concise review of the likelihood function and its role in statistical inference. The concept of likelihood is introduced for both discrete and continuous probability models, emphasizing its interpretation as a measure of relative evidence rather than a probability distribution for the parameter. Several illustrative examples, including binomial, normal, and exponential models, demonstrate the construction and interpretation of likelihood functions. The paper discusses important properties of likelihood, including invariance under multiplication by positive constants and the combination of independent likelihoods through multiplication. Connections between likelihood and Bayesian inference are examined, showing how posterior information can be viewed as the product of prior and current likelihood information. Fundamental principles of likelihood-based inference, including the Likelihood Principle and the Stopping Rule Principle, are reviewed with classical examples. The role of maximum likelihood estimation as a practical method for parameter estimation is also highlighted. The discussion illustrates how the likelihood function serves as a compact and informative summary of statistical evidence, providing a unified framework for estimation and inference.