William M. Briggs

Statistician to the Stars!

Page 151 of 414

Teaching Journal: Day 3

In the real, physical class we learned to count yesterday. Elementary combinatorics, I mean. Figured out what “n!” and “n choose k” and the like meant and how that married with probability and produced a binomial distribution. Pure mechanics. (We begin Chapter 4 today.)

I trust, dear reader, if you don’t already know these things you can read over the class notes to learn, or can find one of hundreds of internet sites which have this sort of information. It is of some interest and we will later use the binomial for this and that. If my Latex renderer is working, you should be able to see this formula for the binomial (if not, even Wikipedia gets this one right):

     Binomial = {n\choose k} p^k (1-p)^{n-k}

Now the thing of interest for us is that we must have some evidence, or list of premises, or propositions taken “for the sake of argument”, that we assume are true and which state, E = “There is some X which can and must take one of two states, a success or a failure, and the chance X is a success is always p; plus, X can be a success anywhere from k = 0, 1, 2, …, n times.” Then, given this E, the formula above gives the probability that in n attempts we see k successes.

An example of an X can be X = “A side shows 6.” Now given the evidence (we have seen many times before) Ed = “There is a six-sided object to be tossed, just one side is labeled ’6′, and just one side can show” we know that

     Pr(X | Ed) = 1/6 = p.

Notice that if n = k = 1, then the binomial formula just reproduces p.

Here is where it gets tricky, and where mistakes are made. Notice that Ed is evidence we assume is true. Whether it really is true (with respect to any other external evidence) is immaterial, irrelevant. Also notice—and here is the juice; pay attention—Ed says nothing about a real, physical die. We are still in the realm of pure logic. And logic is just a study of the relation between propositions: it is silent on the nature of the propositions themselves.

So for instance, let Em = “Just one-sixth of all Martians wear a hat” and let Y = “The next Martian to pass by wears a hat.” Thus

     Pr(Y | Em) = 1/6 = p.

For the next n Martians that pass by, we could calculate the probability that k = 0, or k = 1, or … k = n of them wear a hat. Even though, of course, given our observational evidence that there are no Martians, no Martian will ever pass by.

(If you are sweating over this, remove “Just one-sixth of” from Em; Pr(Y|Em) = 1 and then we can still us the binomial to calculate the probability that the next k of n Martians wears a hat.)

Probability, like all logical statements, are measures of information, and information between propositions. The propositions do not have to represent real, physical objects. Pick up any book of introductory logic to convince yourself of this.

Where people go wrong in statistics in not starting with the reminder that probability is logical, a branch of logic. Thus they confuse Ed with saying something about real dice. They ask questions like, “How do we know the die isn’t weighted? How do we know how it’s tossed? How much spin is imparted? What kind of surface is the die tossed onto? What about the gravitational field into which the die is tossed? Is there a strong breeze?”

All of those (and many, many more) are excellent questions to ask about real dice, but all of them are absolutely irrelevant to Ed and to our absolute, deduced knowledge that Pr(X | Ed) = 1/6.

It is only later, after we learn the formal rules of probability, some basic mechanics, but more importantly after we have fully assimilated the interpretation of probability, do we invert things and ask questions like, “Given that we have seen so many real-life tosses of this real-life die, in this certain real-life situation, what is the probability we will see a 6 on the very next roll?”

Homework

Make sure you see the difference between that last question and the one above using the binomial using Ed or Em.

Read over Chapter 2 and be sure you understand the four most basic rules of probability (the mechanics stuff).

Correct any typos in this post.

Teaching Journal: Day 2

As might have been obvious from yesterday, the truth, falsity, or the somewhere-in-betweenness of any conclusion-hypothesis-proposition can only be assessed with reference to a list of assumptions, premises, data. That is, you cannot know the status of any proposition without some list of premises. Different premises lead to different statuses.

In particular, this means that you cannot ask, “What is the probability of X?”, where X is some proposition. For example, you cannot ask, X = “What is the probability that I roll a six on a die?” This probability does not exist. Similarly, you cannot ask, Y = “What is the probability that Socrates is mortal?” This probability also does not exist. There are no unconditional probabilities, no unconditional arguments of any kind.

If I assume that E = “All men are mortal and Socrates is a man” then I can claim that “It is certain, given E, that Y”, or that “If I assume it is true, regardless whether or not it is or that I can know it is, that all men are mortal and Socrates is a man, then the probability that Socrates is mortal is 1.” Or I can write:

    Pr( Y | E ) = 1.

But I cannot write:

    Pr( Y ) = something,

for that is forever unknown. There just is no such thing as an unconditional probability, just as there are no such things as unconditional logical arguments, just as there are no such things as unconditional mathematical theorems, and so on. If you find yourself disagreeing, have a go at creating the probability of some hypothesis that does not make reference to any assumptions/premises/data.

(It’s only true that in most textbooks probability is written as if it were unconditional. While this makes life for the author and for typesetters, it ends up producing confusion about the nature of probability.)

It’s important to understand that E is only that which we assume is true. It matters not one whit whether E—with respect to some other set of premises—really is true, or false, or somewhere in between. Logic concerns itself only with the connections between premises and conclusions. The premises and conclusions are something exterior, something given to us.

Another hoary example. Let Ed = “A six-sided object, just one side of which is labeled 6, will be tossed and only one side can show.” Then if X = “A 6 shows”,

    Pr( X | Ed ) = 1/6.

We have deduced—just as we do with all probabilities—the probability that X will be true. Notice that this says nothing about real dice in any real situation. This is just a logical argument, no different in nature from the premise “All Martians wear hats and George is a Martian” which lets us deduce that “George wears a hat.” This conclusion with respect to this evidence is true, it’s probability is 1; and this is so even though we know, with respect to observational evidence, that there are no Martians.

Now if we write X = “A Buick shows”, we can write

    0 < Pr( X | Ed ) < 1

We are stuck because our evidence says nothing about a Buick. There may be a Buick on one of the other five sides, there may not. The evidence is mostly mute on this subject. Except if we suppose there is an implicit call to the contingent nature of this object being tossed. If we assume that, then we can at least say the probability is not 0 and not 1, but it may be anywhere in between. But we can also make the argument that Ed should be interpreted more strictly. If it is the case, then the best we can do is this:

    Pr( X | Ed ) = unknown.

Probability cannot be relative frequency. For example, given “Half of all Martians wear hats and George is a Martian” which lets us deduce that the probability “George wears a hat” is 0.5. But there is no relative frequency of this “experiment.” This one counter-example is enough to show that the relative frequency interpretation of probability is false (it doesn’t show it has things backwards; for that, read the book, paying attention to the references).

Probability cannot be subjective in the following sense. If we accept that “All men are mortal and Socrates is a man” then the probability that “Socrates is mortal” is 1. Even if we don’t want it to be. And above the probability that “George wears a hat” cannot be anything but 0.5. Probability only appears to be subjective in some instances because we often are bad at listing the premises we hold when assessing probabilities. However, if we agree on the exact list of premises (and on the rules of logic) then we must agree on the probabilities deduced.

Probability cannot always be quantified. If we accept that “Some men are mortal and Socrates is a man” then the probability that “Socrates is mortal” is something between 0 and 1, but the exact number cannot be deduced.

Homework

Why doesn’t adding “the six-sided object is weighted” or “the six-sided object is fair” to Ed change the probability that Pr( X | Ed ) = 1/6?

List the exact premises you hold in your Pr (“Barack Obama will be re-elected” | your premises).

Change the evidence “All men are mortal and Socrates is a man” so that the probability of “Socrates is mortal” is bounded between two fixed numbers.

In a tearing hurry today. Did not have time to check for typos!

Teaching Journal: Day 1

Truth exists. Therefore, so does falsity. That truth exists is one of the many things we know to be true based on no external evidence. Naturally there are a very large number of things we know to be true given external evidence: most of what we believe to be true is this way.

Incidentally, though I can’t lay my hand on the book, and therefore must paraphrase, a statement by Roger Scruton in his Modern Philosophy: An Introduction and Survey is useful to us. Scruton cogently argues that relativists, in saying that it is certainly true that there are no truths, invite us to disbelieve them. Consequently, I shall not here argue against this (ever) seductive but obviously false philosophy.

If evidence for truth isn’t external, it must be internal. For example, we know via introspection, in our heart of hearts, or, even better, by accepting it on faith, that the following proposition is true: A = “For all numbers x,y = 1, 2, 3, …, if x = y then y = x.” Another way to put it: Given our faith, A is true. Or given our belief, A is true.

We know A is true relative to our faith, our belief. In this way the truth of proposition A is relative, but it’s a special kind of relativity, as we shall see. Contrast A with the proposition G = “George wears a hat” (yes, an old example, well known to regular readers, but still a good one). We have no sense that G is true or false, while A appears true after even momentary reflection. Why? Well, it turns out we are equipped—somehow, never mind here how (but it cannot be an empirical process which gives us this knowledge)—with the knowledge that A is so. But none of us comes pre-made with knowledge of G.

Suppose I give you this evidence, E1 = “All Martians wear hats and George is a Martian.” Now given E1 G is true. Another way to say it: Conditional on E1, G is true. Another: The probability G is true given E1 is 1, or 100%. Another: Accepting for the sake of argument E1, then G.

The truth of G is, just as A was, relative. But we had to supply external evidence which made G true, while we all of us come equipped-from-the-factory with the evidence that makes A true. This is made even clearer by supposing we supply evidence E2 = “Most Martians wear hats and George is a Martian.” Given E2, G is not true, but neither is it false. It is somewhere in between. That somewhere in between falsity and truth is where probability lies.

G then can be true, false, or merely probable depending on the evidence with which it is assessed. It is important to understand that for most everyday propositions it is we who supply the evidence. For example, your enemy may insist that G is false because he holds that E3 = “No Martian wears a hat and George is a Martian.” The argument then (ideally, anyway) moves away from G and to which of E1, E2, or E3 is true.

But how do we know that any of these are true? Well, they aren’t any of them obviously true, as A was, so we need to supply further evidence with which to gauge each E. What if we take F = “There are no Martians”? What happens to G for each E? What if F = “Observations suggest there are no Martians”? How about F = “I have no idea if there are any Martians”? And F = “There are either Martians or there aren’t”?

That’s it for the post; much more in the book, which you must read to keep up (Chapters 1 and 2 today). But there’s also material in this post which isn’t in the book. Which means I have to finish re-writing the book. Time…

Homework

Read then do the questions in Chapter 1, paying especial attention to the challenge to name other propositions which you “just know” are true. Are they really true, or are there hidden or tacit assumptions which you are using? You have to be careful because it is easy to fool yourself. However, this is not necessarily a bad thing.

For example, a person may hold that B is true “just because”, or by faith, or relative to introspection, or whichever synonymous phraseology you prefer. But it might turn out we can prove that B is true only because W, X, Y, … and so on are true; and they true only because some fundamental axiom C is true. But none of this makes B false. Indeed, suppose B = “Four divided by two equals two.” Well, it’s just true! It isn’t, but if you were to believe it, you would not be making an error, at least about the truth of B.

Here is an exercise to prove that probability is not subjective. The United Nations Conference on Sustainable Development, or Rio+20 Earth Summit begins this week (guess where: no, really, guess: why is your guess right or wrong?). Canada’s Environment Minister Peter Kent said “We just aren’t seeing people arriving in the frame of mind to make significant progress towards significant commitments. And we clearly need that.”

What is the probability that H = “The Rio+20 Leaders make significant progress towards significant commitments”? Supply the exact evidence and chain of argument you use to specify this probability. Try not to be facetious and you will learn something. This is a real assignment, incidentally.

The moral of today’s lesson is: know what you’re arguing about.

Teaching Journal: Day 0

This begins my two week tarriance at Cornell, teaching ILR-ST somenumberorother, a course in the Masters of Professional Studies, which is not unlike, though not entirely the same as, an MBA. Students are mostly working professionals who have not seen the inside of an algebraic equation in many a year. Though to learn these things they’ve come to the right guy, for

I’m very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I’m teeming with a lot o’ news —
With many cheerful facts about the square of the hypotenuse.

And from these theorems it’s a straight line to the, well, to the straight line, the second of only two equations with which I re-familiarize students. The other is Bayes’s theorem, so named after the Most Reverend Thomas Bayes. Nothing but simple multiplication and division, really. Even you, dear reader, can do it.

I invite you to play along with us this year. You’ll need a protractor, a compass, a new box of crayons, or whatever writing implement you prefer, and my class notes. The latest version of the book/class notes, free for the asking, can be found at my book page. The paper copy is passable, but (somehow) my enemies have surreptitiously ferreted into it an array of embarrassing typos.

This blog will thus function sort of, kind of, a little like, but not really, an on-line class. But with no registrations or promises. And with most of the onus on you to keep up.

Like I mentioned a few weeks back, I run the class with a lot of back and forth between me and the students. Mostly questions and answers and more questions and more answers. Trouble is, I don’t have a list of preprepared questions, just ideas. Everything changes depending on what students already have in their heads.

Speaking of that: the easiest students to teach, incidentally, are those who haven’t had a statistics course before. Everything is strange to them, but new and congenial. The hardest to influence are those who anchor their minds to a fixed point: for them, everything is interpreted in the context of the old; the new is never new. Actually, of course, I don’t run into the latter type in this classroom.

Class officially starts tomorrow, 9 am sharp. When I lectured undergraduates, I’ve been known to begin speaking at the strike of the bell even into an empty room. This is an old, and now unbreakable habit inculcated into me by my Uncle Sam, who taught me that “on time” meant 15 minutes early, that “late” meant on time, and that there is no allowable third category. But you, dear reader, may show up whenever it is convenient, or even not at all.

Homework today is to mediate on the reasons, motivations, and other character flaws that would lead somebody to want to take a class in statistics in the middle of summer vacation, in the (forecasted) ninety-degree heat.

Today is my day to struggle to recall what it is I want to teach. I usually review my notes, knowing that for the first day I never go much beyond Chapter 1.

Oh yes. The first reminder. I require all students to collect their own data. It should look something like the appendicitis.csv file on the books page. You can read all about the structure of this file in the book (can’t recall the chapter).

This should be data of interest to you, to answer a question you want answered. My students generally pick something work-related, but many choose hobbies or other interests. I’ve had weather in golf tournaments, Top Chef (who’s favored, New York- or LA-based chefs), boxing (who’s favored, righties or south-paws), wine tasting, and on and on.

It’s up to you to figure this out. But if you want to play, start thinking about this now. Two weeks isn’t enough time to procrastinate.

I’ll try and answer whatever questions you have, but be warned that I probably won’t be able to answer everything. I’ll just refer you to old blog posts or relevant sections in the book.

The only bad part is that since you’re doing this remotely, you won’t be able to join us for the wine tour and tasting next Saturday. I’ll post pictures.

Get a good night’s sleep. See you in the morning.

What Do You Think Of The New Look? (Back To Old: Temporarily)

Everybody gets to vote. Though, like in any socialist paradise, there is only one candidate. This one. This new design, I mean.

The old one was growing stale, it was hard to read in some contexts, and much of the site wasn’t used. I’ve eliminated all but three pages: the main-post one, which you are seeing now and which of course must stay; the old Start Here, which is a list of fundamental posts; and a modified About, which combines contact information with my CV and so on.

This style focuses on the posts, or rather, wordy posts, which is all we really do here. Larger font, wider separations between paragraphs, lighter colors, easier on the eye. No having to click on a post to read it, either, as before.

The links to other blogs and sites and other functionality are now at the bottom of the main-post page. These items were scarcely used by anybody, but they are still helpful to have, so they stay but are tucked away.

Comments should be easier, at least in the sense that there is a guide showing which HTML commands are allowed. I still do not allow nested comments. I find them ugly; and they are too apt to become squished.

I’m particularly interested in hearing from anybody who reads the site on a mobile device or tablet computer. It looks great on a/my laptop, anyway. There are some tweaks of the code I know of, but I’m sure there are others you will suggest. What do you think of the within-posts linking? Little dots as underline. No color.

The theme is a (minorly) modified version Esquire theme by Matthew Buchanan. This explains the colors, which for now I’m leaving default. Still some problems with the blockquotes. My mug is covering the search drop down. I’m working on it.

Update Thanks for the comments everybody. I’ve temporarily switched back to the original format to give me time to work on the new. There are lots more problems with this old version, but we’re all used to them and three’s something to be said about familiarity.

The “extra” whitespace on the test theme wasn’t extra. You can’t have text running from left end of the screen to the right. All the text has to exist in a box to be readable. You don’t notice it in the old theme because it’s not white but blue gray space.

Also, this old theme is even worse for mobile viewers…

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