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📚 Statistical Inference for Everyone
Chapters ▾
1 Introduction to Probability2 Applications of Probability3 Random Sequences and Visualization4 Introduction to Model Comparison5 Applications of Model Comparison6 Introduction to Parameter Estimation7 Priors, Likelihoods, and Posteriors8 Common Statistical Significance Tests9 Applications of Parameter Estimation and Inference10 Multi-parameter Models11 Introduction to MCMC12 Concluding Thoughts13 Appendix A: Computational Analysis14 Appendix B: Notation and Standards15 Appendix C: Common Distributions and Their Properties16 Appendix D: Tables

1.2 Models and Data

There are two main aspects of statistical inference: description of data and model analysis. In the description of data, one attempts to summarize a set of data with a smaller set of numbers. Grades in the classroom are summarized by the average, votes in a state are summarized by a percentage, etc... This smaller description of the data is useful for both practical and theoretical reasons. It is more expedient to communicate a small set of numbers than the entire data set, and it is almost always the case that the detailed properties of a set of data are not relevant to the questions that you are asking.

A model refers to a mathematical structure which is used to approximate the underlying causes of the data, and unify seemingly unrelated problems. One may have a (mathematical) model for a coin flip which ignores all of the details of the flip, the bounce, and the catch and summarizes the possible results by a single number: the chance that the coin will come up heads. You may then use that same model to describe the voting behavior of citizens during a presidential election, or to describe the radioactive decay of particles in a physics experiment. The mathematics is identical, but the interpretation of the components of the model will be different depending on the problem. Models simplify, by summarizing data with a small set of causes, and they are used for inference, allowing one to predict the outcome of subsequent events.

The goal of statistical inference is then to take data, and update our knowledge about various possible models that can describe the data. This often means deciding which of several models is the most likely. It can also entail the refinement of a single model, given the new data. All of these activities are closely related to (and perhaps identical to) the methods in science. What we are trying to do is make the best inferences from the data, improve our inferences as new data come in, and plan what data would be the most useful to improve our inferences. In a nutshell, the approach is:

Initial Inference + New Data → Improved Inference

In order to deal with a wide variety of problems, we require a minimal amount of mathematical structure and notation, which we introduce in this chapter.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.