William M. Briggs

Statistician to the Stars!

Page 152 of 415

Teaching Journal: Day 4

Today is the quietest day, a time when all is still, a moment when nary a voice is raised and, quite suddenly, appointments are remembered, people have to be seen, the room empties. Because this is the day I introduce the classical confidence interval, a creation so curious that I have yet to have a frequentist stick around to defend it.

Up until now we have specified the evidence, or premises, we used (“All Martians wear hats…”) and this evidence has let us deduce the probabilities of the conclusions (which we have also specified, and always will, and always must; e.g. “George wears a hat”).

But sometimes we are not able to use the premises (data, evidence) in a direct way. We still follows the rules and dictates of logic, of course, but sometimes the evidence is not as clear as it was when we learned that “Most Martians wear hats.”

The game of petanque is played by drawing a small circle into which one steps, keeping both feet firmly planted. A small wooden ball called a cochonette is tossed 6 to 10 meters downstream. Then opposing teams take turns throwing manly steel balls, or boules, towards the cochonette trying to get as close as possible to it. It is not unlike the Italian game of bocce, which uses meek wooden balls.

Now I am interested in the distance the boule will be from the cochonette. I do not know, before I throw, what this distance will be. I therefore want to use probability to quantify my uncertainty in this distance. I needn’t do this in any formal way. I can, as all people do, use my experience in playing and make rough guesses. “It’s pretty likely, given all the games I have seen, the boule will be within 1 meter of the cochonette.” Notice the clause “given all the games I have seen”, a clause which must always appear in any judgment of certainty or uncertainty, as we have already seen.

But I can do this more formally and use a store-bought probability distribution to quantify my uncertainty. How about the normal? Well, why not. Everybody else uses it, despite its many manifest flaws. So we’ll use it too. That I’m using it and accepting it as a true representation of my uncertainty is just another premise which I list. Since we always must list such premises, there is nothing wrong so far.

The normal distribution requires two parameters, two numbers which must be plugged in else we cannot do any calculations. These are the “m = central parameter” and the “s = spread parameter.” Sometimes these are mistakenly called the “mean” and “standard deviation.” These latter two objects are not parameters, but are functions of other numbers. For example, everybody knows how to calculate a numerical mean; that is just a function of numbers.

Now I can add to my list of premises values for m and s. Why not? I already, quite arbitrarily, added the normal distribution to the list. Might as well just plug in values for m and s, too. That is certainly legitimate. Or you can act like a classical statistician and go out and “collect data.”

This would be in the form of actual measurements of actual distances. Suppose I collect three such measurements: 4cm, -15cm, 1cm. This list of measurement is just another premise, added to the list. A frequentist statistician would say to himself, “Well, why don’t I use the mean of these numbers as my guess for m?” And of course he may do this. This becomes another premise. He will then say, “As long as I’m at it, why don’t I use the standard deviation of these numbers as my guess for s?” Yet another premise. And why, I insist, not.

We at least see how the mistake arises from calling the parameters by the names of their guesses. Understandable. Anyway, once we have these guesses (and any will do) we can plug them into our normal distribution and calculate probabilities. Well, only some probabilities. The normal always—as in always—gives 0 probabilities for actual observable (singular) events. But skip that. We have our guesses and we can calculate.

The frequntist statistician then begins to have pangs of (let us say) conscience. He doubts whether m really does equal -3.3cm (as it does here) and whether s really does equal 10.2cm (as it does here). After all, three data points isn’t very many. Collecting more data would probably (given his experience) change these guesses. But he hasn’t collected more data: he just has these three. So he derives a statement of the “uncertainty” he has in the guesses as estimates of the real m and s. He calls this statement a “95% confidence interval.” That 95% has been dictated by God Himself. It cannot be questioned.

Now the confidence interval is just another function of the data, the form of which is utterly uninteresting. In this example, it gives us (-10cm to 3.3cm). What you must never say, what is forbidden by frequentist theory, is to say anything like this, “There is a 95% chance (or so) that the true value of m lies in this confidence interval.” No, no, no. This is disallowed. It is anathema. The reason for this proscription has to do with the frequentist definition of probability, which always involves limits.

The real definition of the CI is this: if I were to repeat this experiment (where I measured three numbers) an infinite number of times, and for each repetition I calculated a guess for m and a confidence interval for this guess, and then I kept track of all these confidence intervals (all of them), then 95% of them (after I got to infinity) would “cover”, or contain, the real value of m. Stop short of infinity, then I can say nothing.

The only thing I am allowed to say about the confidence interval I actually do have (that -10cm to 3.3cm) is this: “Either the real value of m is in this interval or it isn’t.” That, dear reader, is known as a tautology. It is always true. It is true even (in this case) for the interval (100 million cm, 10 billion cm). It is true for any interval.

The interval we have then, at least according to strict frequentist theory, has no meaning. It cannot be used to say anything about the uncertainty for the real m we have in front of us. Any move in this direction is verboten. Including finite experiments to measure the “width” of these intervals (let he who readth understand).

Still, people do make these moves. They cannot help but say something like, “There is (about) a 95% chance that m lies in the interval.” My dear ones, these are all Bayesian interpretation. This is why I often say that everybody is a Bayesian, even frequentists.

And of course they must be.

Homework

Typo patrol away!

Find, in real-life, instances where the normal has been used with confidence intervals. Just you see if whoever used the interval interpreed it wrong.

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.

« Older posts Newer posts »

© 2014 William M. Briggs

Theme by Anders NorenUp ↑