This is the second part of a series of writings talking about the difficulties of probabilistic reasoning; here, we zoom in the problem of applying probability to actual data, examining the paradigms and techniques of the field of statistics.

Our focus is, similarly to the first article, on the hidden assumptions people make, the effects these assumptions have on the validity of their inferences, and the absence of perfect solutions — in short, why statistics is difficult. Along the way, we’ll be introducing lots of statistical frameworks, techniques, and models, generally at a relatively rigorous level. You don’t necessarily have to follow the more rigorous aspects of each argument and derivation to get the gist of it, but a familiarity with calculus helps immensely. Useful properties of some common distributions are given in part A of the Appendices, while the most technical and/or tedious derivations are stowed away in part B.


Table of Contents

The Uncertainty Series

Part 1: Probability is Difficult

Part 2: Statistics is Difficult

Part 3: Causality is Difficult

Planned

Part 4: Modeling is Difficult

Planned

Part 5: Prediction is Difficult

Planned

Part 6: Experimentation is Difficult

Planned

Part 7: Science is Difficult

Planned

Epilogue: The Past, Present, and Future of Uncertainty

Planned

Appendix: Uncertain Appendices

Expanding


The Questions of Statistical Inference

In the previous article, we covered the basic interpretations of probability:

Now, we’ll put probability to the test by figuring out how to use it to understand the world. That it is useful is clear, for it is used all the time — to determine the reliability of industrial processes, to predict the fluctuations of the stock market, to verify the accuracy of measurements, and, in general, to help us make informed decisions. Hence, we must understand why, how, and when it is useful, and how to use it correctly.

To paraphrase Bandyopadhyay’s Philosophy of Statistics, there are at least four different motivations for the use of statistical techniques:

  1. To determine what belief one should hold, and to what degree to hold it to;
  2. To understand whether some data constitutes evidence for or against some hypothesis;
  3. To figure out what action to take in order to achieve some end;
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