---
title: "Statistical inference"
chunk: 4/4
source: "https://en.wikipedia.org/wiki/Statistical_inference"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:46:05.679193+00:00"
instance: "kb-cron"
---

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 
  
    
      
        
          
            
              y
              ¯
            
          
        
      
    
    {\displaystyle {\bar {y}}}
  
, are the 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 errors, confidence intervals, or conducting hypothesis tests based on asymptotic theory or simulation techniques such as 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.

=== AIC-based inference ===

The Akaike information criterion (AIC) is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.
AIC is founded on information theory: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.)

=== Other paradigms for inference ===

==== Minimum description length ====

The minimum description length (MDL) principle has been developed from ideas in information theory and the theory of Kolmogorov complexity. The (MDL) principle selects statistical models that maximally compress the data; inference proceeds without assuming counterfactual or non-falsifiable "data-generating mechanisms" or probability models for the data, as might be done in frequentist or Bayesian approaches.
However, if a "data generating mechanism" does exist in reality, then according to Shannon's source coding theorem it provides the MDL description of the data, on average and asymptotically. In minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximum-entropy Bayesian priors). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling.
The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining.
The evaluation of MDL-based inferential procedures often uses techniques or criteria from computational complexity theory.

==== Fiducial inference ====

Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious. However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments. An attempt was made to reinterpret the early work of Fisher's fiducial argument as a special case of an inference theory using upper and lower probabilities.

==== Structural inference ====
Developing ideas of Fisher and of Pitman from 1938 to 1939, George A. Barnard developed "structural inference" or "pivotal inference", an approach using invariant probabilities on group families. Barnard reformulated the arguments behind fiducial inference on a restricted class of models on which "fiducial" procedures would be well-defined and useful. Donald A. S. Fraser developed a general theory for structural inference based on group theory and applied this to linear models. The theory formulated by Fraser has close links to decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.

== Inference topics ==
The topics below are usually included in the area of statistical inference.

Statistical assumptions
Statistical decision theory
Estimation theory
Statistical hypothesis testing
Revising opinions in statistics
Design of experiments, the analysis of variance, and regression
Survey sampling
Summarizing statistical data

== Predictive inference ==
Predictive inference is an approach to statistical inference that emphasizes the prediction of future observations based on past observations.
Initially, predictive inference was based on observable parameters and it was the main purpose of studying probability, but it fell out of favor in the 20th century due to a new parametric approach pioneered by Bruno de Finetti. The approach modeled phenomena as a physical system observed with error (e.g., celestial mechanics). De Finetti's idea of exchangeability—that future observations should behave like past observations—came to the attention of the English-speaking world with the 1974 translation from French of his 1937 paper, and has since been propounded by such statisticians as Seymour Geisser.

== See also ==
Algorithmic inference
Induction (philosophy)
Informal inferential reasoning
Information field theory
Population proportion
Philosophy of statistics
Prediction interval
Predictive analytics
Predictive modelling
Stylometry

== Notes ==

== References ==

=== Citations ===

=== Sources ===

== Further reading ==
Casella, G., Berger, R. L. (2002). Statistical Inference. Duxbury Press. ISBN 0-534-24312-6
Freedman, D.A. (1991). "Statistical models and shoe leather". Sociological Methodology. 21: 291–313. doi:10.2307/270939. JSTOR 270939.
Held L., Bové D.S. (2014). Applied Statistical Inference—Likelihood and Bayes (Springer).
Lenhard, Johannes (2006). "Models and Statistical Inference: the controversy between Fisher and Neyman–Pearson" (PDF). British Journal for the Philosophy of Science. 57: 69–91. doi:10.1093/bjps/axi152. S2CID 14136146.
Lindley, D (1958). "Fiducial distribution and Bayes' theorem". Journal of the Royal Statistical Society, Series B. 20: 102–7. doi:10.1111/j.2517-6161.1958.tb00278.x.
Rahlf, Thomas (2014). "Statistical Inference", in Claude Diebolt, and Michael Haupert (eds.), "Handbook of Cliometrics ( Springer Reference Series)", Berlin/Heidelberg: Springer.
Reid, N.; Cox, D. R. (2014). "On Some Principles of Statistical Inference". International Statistical Review. 83 (2): 293–308. doi:10.1111/insr.12067. hdl:10.1111/insr.12067. S2CID 17410547.
Sagitov, Serik (2022). "Statistical Inference". Wikibooks. http://upload.wikimedia.org/wikipedia/commons/f/f9/Statistical_Inference.pdf
Young, G.A., Smith, R.L. (2005). Essentials of Statistical Inference, CUP. ISBN 0-521-83971-8

== External links ==

Statistical Inference – lecture on the MIT OpenCourseWare platform
Statistical Inference – lecture by the National Programme on Technology Enhanced Learning
An online, Bayesian (MCMC) demo/calculator is available at causaScientia
Statistical Inference – interactive Coggle diagram