- Discussion of issues
in JH's
Notes and assignment on C&H Ch01 [prob. Models] and Ch02 [conditional Prob. Model]

Answers to be handed in for: Supplementary exercises 1.2, 2.1, 2.2, 2.3, 2.5, 2.10

**Remarks**:

**Chapter 1 of C&H**introduces some ways of looking at statistical entities and concepts that you may not have met, as well as some terminology that is used in a more specific way in epidemiology. You might want to look at section 1 of JH's notes, from earlier years, on Concepts involved in Occurrence Measures in Epidemiology. JH has also included the first page of this section (mostly definitions) in the notes that annotate the C and H chapters: he has placed it under the heading**'Important: Concepts and Terms in Epidemiology'**after his notes on 1.2 Binary data, and before 1.3 The binary probability model.

JH's notes on Section 1.4 of C&H (and Supp. exercise 1.2) are intended to 'shake you up a bit' and force you to think outside the box as for how you used to estimate the parameters of a simple linear regression. This model is usually shown as a 2-parameter (slope, intercept) model, but JH has deliberately reduced the model to a 1-parameter version, with the "line" going through the origin [other examples might be trying to estimate (from error-containing measurements of the volumes of 2 spheres of different radii: radii measured withut error!) the constant in the relation: Volume of a sphere = "some constant" times the cube of its diameter.] The fewer the elements involved, the more chance there is to really master the fundamentals and 'join the dots.'

**Chapter 2 of C&H**is -- to JH at least -- a very elegant and simple and graphic way to introduce probabilities, and particularly those that are linked to each other in time, or by additional pieces of knowledge. And notice how many probabilities of interest go from right to left, i.e., from after to before. It is worthwhile to work through C&H's own exercises and then check your answers agains the solutions they provide at the end of Ch 2.

Fig 2 in JH's Notes on Ch 2 has several simple but educational examples showing the different 'directionalities'. It also emphasizes that products of probabilities are like 'fractions of fractions' but that sometimes, the probabilities depend on what has gone before, and sometimes do not.

The 2 stories accompanying the Notes on section 2.2 should serve as a stark and frightening reminder that P(theta|data) is a very different 'animal' than P(data|theta) and that the consequences of mixing them can be enormous.

If you want a topical example, think of the difference between P(A|B) and P(B|A), where A = the hypothesis that Higgs Boson particles exist, and B = the bump in the curve. Btw, JH likes to label the elements in what appears to be the best 'logical' or 'chronological' or 'causal' order, i.e., A -> B, but notices that many textbooks teach the concepts using arbitrary letters.

JH's notes on Section 2.3 have a genetics (haemophilia) example that is still very relevant. But, since he first encountered it 40 years ago, medical science has advanced , and so one doesn't not now need to wait until the woman has one or more offspring before learning about her carrier status. JH would be grateful for a different example where one would still need to wait.

At a debate a few years ago, JH came up with the challenge of estimating/judging a person's age from various pieces of information. You might like to take a quick look at the example & pieces of information provided

**Supplementary Exercise 2.2**('The Monty Hall Problem') can be very frustrating and is easily misunderstood. JH has had to break up fights between people who are over-confident but under-listening. Key is the fact that Monty Hall KNOWS which door contains which: sometimes (how often?) he has a choice of 2 doors that he could open to reveal nothing, and sometimes (how often?) he only has 1 choice.

In Exercise**2.3**, it is equally important to be very precise as to the information provided.

In Exercise**2.4**, we have another good example of the difference between P(H|data) and P(data|H). Notice here that we are not examining a range of possible H's, just 2 specific H's. Notice further that in the Bayesian approach we do not consider data values that have not been observed; in contrast, the p-value does consider data values that have not been observed (we should not call such unobserved values 'data', but rather, potential data values.

JH finds that diagrams, especially 'tree' diagrams, can be very helpful in these types of problems, and again when we revisit the Binomial.

Q**2.5**was new in 2015, so the wording hasn't had the same beta-testing as 2.1-2.4.

Its a pity that in the otherwise clever 'left brain' article, the BMJ messed up on the 'teaser' introduction. JH finds The Economist graphics clearer and simpler. What about you?