## William M. Briggs

### Statistician to the Stars!

#### Page 152 of 692

Way my dad and I used to play is that when somebody won they got to grab the hand of the loser and then, with the fore- and middle-fingers only, slap the loser on the wrist. My dad would lick his fingers (“to cut down on air resistance”) before whacking me.

Say, these days that would be child abuse. Well, the government knows best, right?

Briefly, you and simultaneously your opponent select rock, paper, or scissors. Rock breaks scissors, paper covers rock, and scissors cut paper. There are 9 outcomes: 3 of which are rock for you, and rock, paper, or scissors for your enemy; et cetera. Three of the 9 outcomes are ties, in which nobody gets slapped; there are 3 ways for you to lose, and 3 ways for you to win.

Now, given just those premises, and none other, what is the probability you don’t get slapped? Well, “don’t get slapped” means tying or winning; and since there are 6 ways for these things to happen, the chance is 2/3. Similarly, there is a 1/3 chance for you to lose, and 1/3 probability you get to slap.

These deduced probabilities are correct assuming only the premises describing the rules of the game. Which implies, and it is true, that the probabilities are not likely to be correct assuming other premises. What other premises? There are an infinite number of premises we might choose, but what we’re after are premises that help us win the game.

The thing to emphasize is that we know with certainty the non-human premises, and so know with certainty the non-human probabilities. Rock paper scissors is thus similar to poker where we have a good handle on the probabilities; but where in poker they are harder to memorize, yet in poker we know there are consistent and good players.

There are also consistently winning Rock Paper Scissors players. Like 2008 champion Sean “Wicked Fingers” Sears. That means, like poker, human premises must exist that change the odds.

In any number of places you’ll read that the way not to get beat is to make your pick “randomly.” This is impossible. No matter what, you must cause your pick, and no cause in the universe is ontologically “random.” Suppose you decide to divide up a minute into thirds and pick based on the secondhand of your watch. If your opponent does not know this, he has no way of guessing what you’ll do (except that you must choose, of course), and so to him your guess is “random”—which means only unknown. But to you, the choice is determined, caused by your decision and the state of the time.

If your opponent catches you sneaking peeks before each round, then he’ll too know what you’re going to do—creating a new and probative and deductive premise for him—and you’ll consistently lose.

That’s the way to win, too. By searching for patterns, i.e. premises, which your opponent is using, knowingly or not. Bin Xu and pals think they’ve discovered patterns many people use. In their arxiv paper “Cycle frequency in standard Rock-Paper-Scissors games: Evidence from experimental economics” they posit that winners often repeat their winning pick, and losers select the next object in the cycle (lose with scissors and move to rock). Armed (handed?) with these premises allows you to change the odds.

Until your opponent figures out his mistake; and when he does, he can use that knowledge against you. Figuring you are figuring on a rock (since your opponent just lost on a scissors), and thus you’d pick paper, your opponent upends the algorithm and sticks with scissors.

Noted RPS expert Sharisa Bufford, a member of the prestigious USA Rock Paper Scissors League, is sure that “Girls always throw scissors first. Guys always throw rock.” If that’s so, these are winning premises. Unless your opponent knows these rules, too. And you know they know. And you know they know you know…you know?

Now it’s true you cannot “pick randomly” but must always cause a choice. But that does not mean it’s impossible to discover a strategy which your opponent cannot guess—where your opponent may be some “sophisticated computer” (computers are only dumb unthinking distillations of fractions of human brains). All it takes to create impossible-to-discover picks are to create picks which your opponent cannot guess beyond the standard premises (a tautology, really).

Maybe that’s memorizing a stream of numbers generated by some process which remains unknown to your opponent. Or maybe that means changing your picking algorithm based on circumstance. All that matters is that your opponent cannot guess.

Incidentally and curiously, computer programs which test “randomness” (i.e. unpredictability) always turn a blind eye to the algorithm which generated the sequence they’re testing.

Lastly, to prove (as I’ve done before) that I can pick a number you cannot guess with certainty, no matter what your resources, I’m thinking of a number between 0 and 4. What is it? (It’s hidden in the code to this page.)

This may be proved in three ways. The first…

See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles. All posts are under the category SAMT.

Previous post.

Today some (I think the term is) theological brush clearing. Three chapters which prove, in effect, that God is superior to man. Only post-moderns and fallen angels doubt this, so we needn’t spend too much time here. Take this time to review, then, because next week new metaphysical terms are introduced.

Chapter 29: Of The Likeness Of Creatures

4 Dionysius is in agreement with this argument, for he says (Div. Nom. ix.): The same things are like and unlike to God; like, according as they imitate Him, as far as they can, Who is not perfectly imitable; unlike, according as effects fall short of their causes.

5 However,[3] according to this likeness, it is more fitting to say that the creature is like God than vice versa. For one thing is like another when it possesses a quality or form thereof. Since then what is in God perfectly is found in other things by way of an imperfect participation, that in which likeness is observed is God’s simply but not the creature’s. And thus the creature has what is God’s, and therefore is rightly said to be like God. But it cannot be said in this way that God has what belongs to His creature: wherefore neither is it fitting to say that God is like His creature; as neither do we say that a man is like his portrait, although we declare that his portrait is like him…i

Chapter 30: What Terms Can Be Predicated Of God

1 AGAIN in sequel to the above we may consider what can and what cannot be said of God; also what is said of Him alone, and what is said of Him together with other beings.

2 For since every perfection of creatures is to be found in God, albeit in another and more eminent way, whatever terms denote perfection absolutely and without any defect whatever, are predicated of God and of other things; for instance, goodness, wisdom, and so forth. But any term that denotes suchlike perfections together with a mode proper to creatures, cannot be said of God except by similitude and metaphor, whereby that which belongs to one thing is applied to another, as when a man is said to be a stone on account of the denseness of his intelligence.ii

Such are all those terms employed to denote the species of a created thing, as man and stone: for its proper mode of perfection and being is due to each species: likewise whatever terms signify those properties of things that are caused by the proper principles of the species, therefore they cannot be said of God otherwise than metaphorically. But those which express these perfections together with the mode of supereminence in which they belong to God, are said of God alone, for instance the sovereign good, the first being, and the like…

Chapter 31: That The Divine Perfection And The Plurality Of Divine Names Are Not Inconsistent With The Divine Simplicity

1 FROM what has been said we are also able to see that the divine perfection and the various names applied to God are not inconsistent with His simplicity.iii

1 For we asserted that all the perfections to be found in other things are to be ascribed to God in the same way as effects are found in their equivocal causes:[1] which causes are in their effects virtually, as heat is in the sun. Now this virtue unless it were in some way of the genus of heat, the sun acting thereby would not generate its like. Wherefore by reason of this virtue the sun is said to be hot, not only because it causes heat, but because the virtue whereby it does this, is something in conformity with heat. Now by this same virtue by which the sun causes heat, it causes also many other effects in lower bodies, such as dryness. And so heat and dryness, which are distinct qualities in fire, are ascribed to the sun in respect of the one virtue.iv

And so too, the perfections of all things, which are becoming to other things in respect of various forms, must needs be ascribed to God in respect of His one virtue. And this virtue is not distinct from His essence, since nothing can be accidental to Him, as we have proved.[2] Accordingly God is said to be wise not only because He causes wisdom, but because in so far as we are wise, we imitate somewhat the virtue whereby He makes us wise. He is not however called a stone, although He made the stones, because by the term stone we understand a definite mode of being, in respect of which a stone differs from God.[3] But a stone imitates God as its cause, in respect of being, goodness and so forth, even as other creatures do.v

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[3] Sum. Th., l.c., ad 4. [Referencing chapter 29] [1] Coel. Hier. ii. 3. [Referencing chapter 30] [1] Ch. xxix. [Referencing chapter 31] [2] Ch. xxiii.

iThis is plain enough. It is only some academics and politicians who think they are gods themselves. I skipped lightly over the obvious that causes are greater than their effects, etc.; that a cause does not have to give the whole of itself, as it were, etc. Review!

iiSpeaking of the same individuals… Anyway, I believe the modern terms are “brick”, “thick”, or just “dense.”

iiiPlease don’t forget simplicity is here a technical term, meaning not made of parts, with no potentiality, and those kinds of things. It does not mean easy nor lacking nor any other colloquial shade of the word.

ivIn other word, the sun can do more than one thing even though it itself is only one thing. What thing it does “at the moment” depends on focus. We would never make the mistake of saying, “The sun is warming me therefore it can’t also be lighting my path.” The sun, of course, also provides a point for us to rotate about; it also “hardens” the atmosphere and allows radio waves to propagate over long distances; and many other things. You get the idea.

vHe is also not called a (human) body, even though He made bodies, because by the term body we understand a definite mode of being. It is also significant that wisdom, or rather being wise or unwise, is something we do with our intellects, and our intellects are not material.

It’s a lazy Saturday, so some musings today on entropy and information and probability. It’s about time we started tying these things together.

Things like the following are heard: “Given a list of premises, I judge the probability of rain tomorrow at 0%”. Swap in for “rain tomorrow” your favorite prediction: “the team wins”, “the customer buys”, “the man defaults”, and so on.

Now zero-percent, a probability of 0, is mighty strong. The strongest. It means the proposition is impossible: not unlikely, impossible. Impossible things cannot happen. Even God cannot do the impossible.

Yet impossible events occur all the time. It rains when it’s not supposed to, the team loses, the customer leaves, the man honors. And so on. Something has gone wrong.

The failure, as always, is not to keep assiduous track of our language. Equivocation creeps in unaware. Impossible doesn’t mean impossible but conditionally impossible. Why?

All (as in all) probability is conditional. That means some list of premises existed which allowed the judgement of a probability of 0. That is, the forecaster had some model—a model is a list of premises—which spit out “probability 0” for his proposition.

Since it did later rain, we have falsified this model, i.e. this list of premises. But we haven’t shown which of the main premises are false: we have only proved that at least one premise of the model isn’t so. The “model”, i.e. collection of premises as a whole is false, it is wrong, but there may be pieces of it which are true. So much we already know.

Introduce the idea of “probabilistic surprise.” A surprising event is a rare one. If our model—you simply must get into the habit of thinking of any model as a list of premises; however “list of premises” is not as compact as “model”—says the probability of some proposition is low, then, conditional on that model, is the proposition is observed to be true, we are surprised. Rarer events are more surprising.

Think how you’d feel winning the lottery. The model easily lets us deduce the probability for the proposition “I win.” This is small; your shock at winning is correspondingly large.

Propositions which are certain conditional on the model are not surprisingly; indeed, since they must happen, since they have a probability of one, they are inevitable. Who could be surprised but what must occur? (Insert your own joke here.)

Lotteries and other dichotomous situations are great examples since we can easily track possibilities. Tracking possibilities isn’t always easy, nor even always possible with every model (list of premises!). Tracking means deducing every (as in every) proposition which has a probability relative to the model. In formal situations, we’re fine; informally, not so fine. So let’s stick with dichotomy, which is free beer.

Turns out, with this definition of “surprise”, or rather believing the notion that we can quantify surprise, a move which is open to debate, we can deduce a formula for the amount of surprise we can expect given we have tracked the model and identified all the propositions and deduced their conditional probabilities (pi). This formula is:

$H(\mbox{model}) = -\sum p_i \log{p_i}$.

Yes, entropy. Now, isn’t that curious?

Update See below for comments about calculating entropies. I originally had calculations here (something I don’t normally do) and because of my sloppiness (and laziness) I distracted us from the main points.

What about impossible propositions? Once again, our habitual slackness with language leads to mistakes. If a conditionally impossible event happens, according to our derivation, our surprise should be infinite.

Infinite surprise!

In one way, this is right. If something truly impossible happened—these events are defined just as we define necessary truths, i.e. from indubitable first principles and irresistible deduction—then our surprise would surely be infinite. And deadly. Who could handle the shock? And since entropy is often given physical interpretation, impossible events imply the Trump of Doom.

But it’s obviously wrong. Impossible events which really occur always mean a broken model. They always imply that a false premise has snuck in and been believed.

The lesson (the only one we can do today) models which assign zero probabilities to contingent events are inherently flawed. (Contingent events are those which are not logically necessary or—you guessed it—logically impossible. It all ties together!)

Well, that’s it today. We haven’t done information nor scores nor a world of other things.

Yessir, this sweet baby of an algorithm can predict anything from NFL games to the Spanish GDP.

I’ve never seen the show, but I’ve heard that the protagonist on the X Files used to have a desk sign which read, “I want to believe.” That sentiment characterizes most buyers and users of predictive or explanatory algorithms, i.e. statistics.

Really, there is no scientific promise too large that it will not be at least hoped for. It doesn’t matter how many failures or unrealized dreams are met, the newest thing always excites. We saw something of this Tuesday with a new algorithm that claimed to be able to forecast the stock market two years in advance.

Experience suggests that this purported marvel, or any new gee-whiz algorithm, will liquefy the rationality centers of those people in charge of securing predictions (financial services, governments, marketers, academics, etc.). The authors of that paper now have a window of opportunity to cash in on their method.

Meanwhile, grumpy naysayers warning against enthusiasm will meet with a lesser fate.

A tale. I was dealing with a potential client who believed they discovered a means to increase their predictive accuracy to astonishing heights. R-squareds (a poor measure; don’t use them) northward of 95% were seen in tests! I was to automate the process.

Turns out they were smoothing two sets of time series which originally had no relationship to one another, and then correlating the smoothed series—which suddenly showed a remarkable correlation! Regular readers will know this is a huge no-no. Smoothing artificially boosts correlation and predictive accuracy. I tried showing the client how this works, proving it with several examples using made-up data that looked like theirs.

Funny thing is that I convinced a business type I was right, but he was overruled by their algorithm person. This algorithm person was basing his excitement on a sterling certified peer-reviewed paper by an academic from an institution of world renown. Regular readers know all about peer-reviewed papers from credentialed academics.

In what is now the theme of my career, I didn’t get the job.

Another anecdote, necessarily vague because I cannot betray any confidences. This didn’t happen to me, but to somebody I know. Major company wanted to understand how “social media”, specifically Twitter data, predicted their income. My colleague correctly noted how noisy this data is.

My colleague thus warned that, while something might be learned, whatever it would be wouldn’t be earthshaking. Certainly not much money should be spent on the idea. This advice was rejected and the major company sought bids from algorithm firms which could take on the job. One was found. I cannot tell you how much money was asked for or given, but if I did you would faint dead away. I can only say that whoever runs this algorithm company could easily find a position in government.

I offered to do the job for an order of magnitude less. My bid was rejected, but then I, like my colleague, cautioned that not much would come of the analysis.

The sequel? You already know what happened so there’s no point going into it.

If you want to set up business as a data scientist (the newfangled term by which statisticians are beginning to call themselves), the lesson is this: promise the moon and charge like you’re actually going there. Failure is rarely punished and never remembered.

Uncertainty is the same tough sell in science. The way to do statistics (or machine learning, or AI, or whatever) properly, like I’m always saying, is to use whatever model you have to predict new, never-before-seen data. If your model works, you’ll make good predictions. If not, not. Problem is, this method is necessarily less certain than the old ways of doing things.

Doing it the right way makes it look like you know a hell of a lot less. Fireworks are rare. Everybody hates this. Science is supposed to make us more, not less, certain!

It also does no good proving that if you get uncertainty right that, even though you will be less sure of yourself, you will and must make better decisions, and that better decisions mean greater rewards. The allure of certainty is too strong. People want easy answers and can’t abide the fogginess which attends uncertainty.

I have only given you two anecdotes, but they can be multiplied indefinitely (especially in academia). It’s thus rational to believe that nothing will ever change and that people will continue to be over-certain.

Update Breaking news! I have just developed a zero-point energy super computerized big data artificial intelligent learning prediculator. Investors should use my contact page and send me money.