Inferential Statistics

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Statistical inference or inferential statistics is the process of using data analysis to infer properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example, by testing hypotheses and deriving estimates. The observed data set is assumed to be drawn from a larger population.

Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics deals only with the properties of the observed data and is not based on the assumption that the data comes from a larger population. In machine learning, the term inference is sometimes used instead to mean "making a prediction, evaluating an already trained model"; in this context, the inference of model properties is called training or learning (rather than inference ), and the use of a model for prediction is called inference (rather than prediction ); see also predictive inference.

Introduction

Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling. Given a hypothesis about a population for which we wish to make inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deriving propositions from the model.

Konishi and Kitagawa state that "most statistical inference problems can be considered problems related to statistical modeling". In this connection, Sir David Cox has said: "The way in which [the] translation of the matter problem into the statistical model is done is often the most critical part of an analysis."

The conclusion of a statistical inference is a statistical statement. Some common forms of statistical proposition are as follows:

a point estimate, that is, a particular value that best approximates some parameter of interest;
an interval estimate, for example, a confidence interval (or a pool estimate), that is, an interval constructed using a data set drawn from a population such that, on repeated sampling of said data sets, said intervals would contain the true value of the parameter with the probability at the established confidence level;
a credible interval, that is, a set of values containing, for example, 95% of the posterior belief;
rejection of a hypothesis;
grouping or classifying data points into groups.

Models and assumptions

Any statistical inference requires some assumptions. A statistical model is a set of assumptions related to the generation of the observed and similar data. Descriptions of statistical models often emphasize the role of the population quantities of interest, about which we wish to make inferences. Descriptive statistics are typically used as a preliminary step before drawing more formal inferences.

Degree of models / assumptions

Statisticians distinguish between three levels of modeling assumptions;

Fully parametric – The probability distributions that describe the data generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters. For example, it can be assumed that the distribution of values in the population is truly Normal, with unknown mean and variance, and that the data sets are generated by "simple" random sampling. The family of generalized linear models is a flexible and widely used class of parametric models.
Nonparametric – The assumptions about the process that generates the data are much smaller than in parametric statistics and can be minimal. For example, every continuous probability distribution has a median, which can be estimated using the sample median or the Hodges-Lehmann-Sen estimator, which has good properties when the data arise from simple random sampling.
Semi-parametric : This term normally implies "intermediate" assumptions full and non-parametric approaches. For example, a population distribution can be assumed to have a finite mean. Furthermore, one can assume that the mean response level in the population depends in a truly linear way on some covariate (a parametric assumption) but make no parametric assumption describing the variance around that mean (i.e., on the presence or possible form of any heteroscedasticity). More generally, semiparametric models can often be separated into "structural" and "random variation" components. One component is treated parametrically and the other nonparametrically.

Importance of valid models/assumptions

Whatever level of assumption is made, properly calibrated inference generally requires that these assumptions be correct; that is, that the data generation mechanisms have actually been correctly specified.

Incorrect assumptions from "simple" random sampling can invalidate statistical inference. More complex semi- and fully parametric assumptions are also of concern. For example, incorrectly assuming the Cox model can, in some cases, lead to erroneous conclusions. Incorrect assumptions of population normality also invalidate some forms of regression-based inference. The use of any The parametric model is viewed skeptically by most experts on human population sampling: "most sampling statistics, when dealing with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples." large, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal." In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption if we were dealing with any kind of economic population." Here, the central limit theorem states that the distribution of the sample mean "for very large samples" has an approximately normal distribution, if the distribution is not heavy-tailed.

Approximate distributions

Given the difficulty of specifying exact distributions of the sample statistics, many methods have been developed to approximate them.

With finite samples, approximation results measure how closely a limiting distribution approaches the sampling distribution of the statistic: for example, with 10,000 independent samples, the normal distribution approximates (with two digits of precision) the distribution of the mean sample for many population distributions, according to the Berry method –Esseen Theorem. However, for many practical purposes, the normal approximation provides a good approximation to the distribution of the sample mean when there are 10 (or more) independent samples, depending on simulation studies and the experience of statisticians. Following the work of Kolmogorov in the 1950s, advanced statistics uses approximation theory and functional analysis to quantify approximation error. In this approach, the metric geometry of probability distributions is studied; this approach quantifies the approximation error with, for example, the Kullback-Leibler divergence, the Bregman divergence, and the Hellinger distance.

With indefinitely large samples, limiting results such as the central limit theorem describe the limiting distribution of the sample statistic, if it exists. Limiting results are not statements about finite samples and, in fact, are irrelevant for finite samples. However, the asymptotic theory of limiting distributions is often invoked to work with finite samples. For example, limiting results are often invoked to justify the generalized method of moments and the use of generalized estimating equations, which are popular in econometrics and biostatistics. The magnitude of the difference between the limiting distribution and the true distribution (formally, the "error" of the approximation) can be evaluated by simulation.

Randomization-based models

For a given data set that was produced by a randomization design, the randomization distribution of a statistic (under the null hypothesis) is defined by evaluating the test statistic for all plans that could have been generated by the randomization design. In frequentist inference, randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is especially important in survey sampling and design of experiments. Statistical inference from randomized studies is also simpler than many other situations. In Bayesian inference, randomization is also important: in survey sampling, the use of sampling without replacement ensures interchangeability of the sample with the population;

Objective randomization allows adequately inductive procedures. Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures. (However, it is true that in fields of science with developed theoretical insights and experimental control, randomized experiments can increase the costs of experimentation without improving the quality of inferences.) Similarly, leading statistical authorities recommend the results of randomized experiments as allowing inferences with greater reliability than observational studies of the same phenomena. However, a good observational study may be better than a bad randomized experiment.

Statistical analysis of a randomized experiment can be based on the randomization scheme established in the experimental protocol and does not require a subjective model.1

However, at any point in time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, these randomized studies are wasteful or unethical.

Model-Based Analysis of Randomized Experiments

It is standard practice to refer to a statistical model, for example a linear or logistic 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. Seriously misleading results can be obtained by analyzing data from randomized experiments ignoring the experimental protocol; Common errors include forgetting the lock used in an experiment and mistaking repeated measurements on the same experimental unit for independent replicates of the treatment applied to different experimental units.

Model-Free Randomization Inference

Model-free techniques provide a complement to model-based methods, which employ reductionist strategies of simplifying reality. The former combine, evolve, assemble and train algorithms by dynamically adapting to the contextual affinities of a process and learning the intrinsic characteristics of the observations.

For example, simple linear regression without a model is based on

a random design , where the pairs of observations $(X_{1},Y_{1}),(X_{2},Y_{2}),\cdots,(X_{n},Y_{n})$ are independent and identically distributed (iid), or
a deterministic design , where the variables $X_{1},X_{2},\cdots ,X_{n}$ are deterministic, but the corresponding response variables $Y_{1},Y_{2},\cdots,Y_{n}$ are random and independent with a common conditional distribution, that is, $P\left(Y_{j}\leq y |X_{j}=x\right)=D_{x}(y)$ , which is independent of index $j$ .

In any case, model-free randomization inference for the features of the common conditional distribution $D_{x}(.)$ is based on some regularity conditions, eg, functional smoothness. For example, the model-free randomization inference for the conditional mean characteristic population , $\mu(x)=E(Y|X=x)$ , can be consistently estimated by a local average or a local polynomial fit, under the assumption that it $\mu(x)$ is smooth. Furthermore, based on asymptotic normality or resampling, we can construct confidence intervals for the population characteristic, in this case the conditional mean , $\mu(x)$ .

Inference paradigms

Different schools of statistical inference have been established. These schools, or "paradigms," are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms.

Bandyopadhyay and Forster describe four paradigms: "(i) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) Akaikean information criteria-based statistics". The classical (or frequentist) paradigm, the Bayesian paradigm, the plausible paradigm, and the AIC-based paradigm are summarized below.

frequentist inference

This paradigm calibrates the plausibility of propositions by considering (theoretical) repeated sampling of a population distribution to produce data sets similar to the one at hand. By considering the characteristics of the data set in repeated sampling, the frequentist properties of a statistical proposal can be quantified, although in practice this quantification can be challenging.

Examples of frequentist inference

p -value
Confidence interval
Null hypothesis significance test

Frequentist inference, objectivity and decision theory

One interpretation of frequentist inference (or classical inference) is that it is applicable only in terms of frequency probability; that is, in terms of repeated sampling of a population. However, Neyman's approach develops these procedures in terms of prior probabilities to the experiment. That is, before undertaking an experiment, one decides on a rule for reaching a conclusion such that the probability of its being correct is adequately controlled: such a probability need not have a repeated-sampling or frequentist interpretation. By contrast, Bayesian inference works in terms of conditional probabilities (that is, probabilities conditional on the observed data),

Frequentist significance testing and confidence interval procedures can be constructed without taking utility functions into account. However, some elements of frequentist statistics, such as statistical decision theory, incorporate utility functions. In particular, frequentist developments of optimal inference (such as least variance unbiased estimators or uniformly most powerful tests) make use of loss functions, which play the role of (negative) utility functions. Loss functions do not need to be stated explicitly for statistical theorists to show that a statistical procedure has an optimality property. However, loss functions are often useful for setting optimization properties: for example,

While statisticians using frequentist inference must themselves choose the parameters of interest and the estimators/test statistics to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to be widely considered 'objective'. '.

Bayesian inference

Bayesian calculus describes degrees of belief using the "language" of probability; the beliefs are positive, they are integrated into one and obey the axioms of probability. Bayesian inference uses available posterior beliefs as a basis for making statistical propositions. There are several different justifications for using the Bayesian approach.

Bayesian inference examples

Credible interval for interval estimation
Bayes factors for model comparison

Bayesian inference, subjectivity and decision theory

Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, posterior mean, median and mode, highest posterior density intervals, and Bayes factors can be motivated in this way. While it is not necessary to state a user's utility function for this type of inference, all of these summaries depend (to some extent) on prior stated beliefs and are generally considered subjective conclusions. (Pre-construction methods that do not require external input have been proposed, but have not yet been fully developed.)

Formally, Bayesian inference is calibrated with reference to an explicitly stated utility or loss function; the 'Bayes rule' is the one that maximizes the expected utility, averaged over the posterior uncertainty. Thus, formal Bayesian inference automatically yields optimal decisions in a decision-theoretic sense. Given the assumptions, the data, and the utility, Bayesian inference can be performed for virtually any problem, although not all statistical inferences need a Bayesian interpretation. Analyzes that are not formally Bayesian may be (logically) inconsistent; a feature of Bayesian procedures that use proper priors (that is, those integrable to one) is that they are guaranteed to be consistent. must take place in this decision-theoretic framework, and that Bayesian inference must not end with the evaluation and summary of subsequent beliefs.

Likelihood-based inference

Likelihood approaches statistics by using the likelihood function. Some verisimilists reject the inference, considering statistics as only computer support of the evidence. Others, however, propose inferences based on the likelihood function, of which the best known is the maximum likelihood estimate.

AIC-based inference

The Akaike Information Criterion (AIC) is an estimator of the relative quality of statistical models for a given data set. 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 based on information theory: it provides 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 addresses the trade-off between model goodness-of-fit and model simplicity.)

Other paradigms for inference

Minimum description length

The principle of minimum description length (MDL) has been developed from ideas in information theory and Kolmogorov complexity theory. The (MDL) principle selects statistical models that compress the data as much as possible; The inference proceeds without assuming "data-generating mechanisms" or non-falsifiable or counterfactual probability models for the data, as might be done in the frequentist or Bayesian approaches.

However, if a "data generation mechanism" actually exists, then, according to Shannon's source coding theorem, it provides the MDL description of the data, averaged and asymptotically. By minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum posterior estimation (using maximum-entropy Bayesian priors). However, MDL avoids assuming that the underlying probability model is known; The CDM principle can also be applied without assuming that, for example, the data arise from independent sampling.

The MDL principle has been applied in communication coding theory, information theory, linear regression, and data mining.

The evaluation of inferential procedures based on MDL 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 "fiducial distribution". In subsequent works, this approach has been described as ill-defined, extremely limited in applicability, and even fallacious. However, this argument is the same as showing that the 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 the conclusions drawn from the fiducial arguments. An attempt was made to reinterpret Fisher's early work on the fiducial argument as a special case of a theory of inference using superior and inferior probabilities.

Structural inference

Building on the ideas of Fisher and Pitman from 1938 to 1939, George A. Barnard developed "structural inference" or "fundamental inference", an approach that uses invariant probabilities on families of groups. Barnard recast the arguments behind fiducial inference about a restricted class of models in which "fiducial" procedures would be well defined and useful. Donald AS Fraser developed a general theory for structural inference based on group theory and applied it to linear models. The theory formulated by Fraser has close links with decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.

Inference Topics

The topics below generally fall into the area of statistical inference .

Statistical assumptions
Statistical decision theory
Estimation theory
Statistical hypothesis test
Review of opinions in statistics
Design of experiments, analysis of variance and regression.
Survey Sampling
Summarizing Statistical Data

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