Editing Statistical inference (section)

===Likelihood-based inference===
{{Main|Likelihoodism}}Likelihood-based inference is a paradigm used to estimate the parameters of a statistical model based on observed data. [[Likelihoodism]] approaches statistics by using the [[likelihood function]], denoted as <math>L(x | \theta)</math>, quantifies the probability of observing the given data <math>x</math>, assuming a specific set of parameter values <math>\theta</math>. In likelihood-based inference, the goal is to find the set of parameter values that maximizes the likelihood function, or equivalently, maximizes the probability of observing the given data.

The process of likelihood-based inference usually involves the following steps:

# Formulating the statistical model: A statistical model is defined based on the problem at hand, specifying the distributional assumptions and the relationship between the observed data and the unknown parameters. The model can be simple, such as a normal distribution with known variance, or complex, such as a hierarchical model with multiple levels of random effects.
# Constructing the likelihood function: Given the statistical model, the likelihood function is constructed by evaluating the joint probability density or mass function of the observed data as a function of the unknown parameters. This function represents the probability of observing the data for different values of the parameters.
# Maximizing the likelihood function: The next step is to find the set of parameter values that maximizes the likelihood function. This can be achieved using optimization techniques such as numerical optimization algorithms. The estimated parameter values, often denoted as <math>\bar{y}</math>, are the [[Maximum likelihood estimation|maximum likelihood estimates]] (MLEs).
# Assessing uncertainty: Once the MLEs are obtained, it is crucial to quantify the uncertainty associated with the parameter estimates. This can be done by calculating [[standard error]]s, confidence intervals, or conducting [[hypothesis test]]s based on asymptotic theory or simulation techniques such as [[Bootstrapping (statistics)|bootstrapping]].
# Model checking: After obtaining the parameter estimates and assessing their uncertainty, it is important to assess the adequacy of the statistical model. This involves checking the assumptions made in the model and evaluating the fit of the model to the data using goodness-of-fit tests, residual analysis, or graphical diagnostics.
# Inference and interpretation: Finally, based on the estimated parameters and model assessment, statistical inference can be performed. This involves drawing conclusions about the population parameters, making predictions, or testing hypotheses based on the estimated model.