Jump to navigation Jump to search Not to be confused with Statistical interference. Statistical inference is the statistical data for analysis of using data analysis to deduce properties of an underlying probability distribution.
Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling. Kitagawa state, «The majority of the problems in statistical inference can be considered to be problems related to statistical modeling». Relatedly, Sir David Cox has said, «How translation from subject-matter problem to statistical model is done is often the most critical part of an analysis». The conclusion of a statistical inference is a statistical proposition.
Any statistical inference requires some assumptions. A statistical model is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference. Fully parametric: The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters. Non-parametric: The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal. Semi-parametric: This term typically implies assumptions ‘in between’ fully and non-parametric approaches.
For example, one may assume that a population distribution has a finite mean. Incorrect assumptions of ‘simple’ random sampling can invalidate statistical inference. More complex semi- and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions. Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these. With indefinitely large samples, limiting results like the central limit theorem describe the sample statistic’s limiting distribution, if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples.
In frequentist inference, randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments. Objective randomization allows properly inductive procedures. Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures. The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model. However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.
It is standard practice to refer to a statistical model, often a linear model, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme. Different schools of statistical inference have become established. These schools—or «paradigms»—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms. By considering the dataset’s characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging. The frequentist procedures of significance testing and confidence intervals can be constructed without regard to utility functions.
Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions. There are several different justifications for using the Bayesian approach. Many informal Bayesian inferences are based on «intuitively reasonable» summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. Bayes rule’ is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides optimal decisions in a decision theoretic sense. You can help by adding to it.
Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. 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. 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. 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 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. Developing ideas of Fisher and of Pitman from 1938 to 1939, George A. The topics below are usually included in the area of statistical inference. According to Peirce, acceptance means that inquiry on this question ceases for the time being. In science, all scientific theories are revisable. Probability and Statistics: The Science of Uncertainty.