On Probability Theory and Theories of Probability

(This expands on some ideas I touched on in the post about the single-event probability fallacy. If you have a sense of deja vu, that’s why. It’s a different angle in the same ideas.) 

Probability theory is abstract mathematics. It has the same axioms as measure theory (plus one that says the measure of the whole space is 1, but that’s really just a convention), and it focuses on different things. As a theory, it has applications.

One is to the frequencies of outcomes of repeated events, such as rolling a dice, making a component by machine tool, or the path of a small particle surrounded by fast-moving smaller particles. With a suitably set-theoretic understanding of what ‘events’ and ‘outcomes’ are, probability theory can be shown to apply to such frequencies.

Another application is to betting odds, though here probability theory does not apply as a description but rather as a prescription. If the betting odds are to be ‘fair’, that is, if the odds don’t favour the bookmaker or the customer, those odds must follow the laws of probability.

The same applies to the idea of ‘degree of belief’, whatever that means and however we measure it. If those degrees of belief are to be consistent, they must follow the laws of probability. Betting-odds and degrees of belief are sometimes called subjectivist probability.

In earlier and less enlightened times, there were heated arguments over which was the ‘real theory of probability’, and both sides missed the point that they were discussing different applications of the same abstract theory, and as a result were having an argument about whether over-easy or well-done was the correct way of cooking eggs.

In addition, there was something called ‘The Principal Principle’ stating that the rational degree of belief in the outcome of a repeated event is its frequency. The result is that, if we are talking about repeated outcomes, probability means frequencies.

This leaves the question about what we might mean by the probability of single events and how it might be measured. The ingenuity of some answers rival the madder interpretations of Quantum Mechanics. Some of them turn out to be frequencies in disguise, as is the Possible Worlds interpretation. (I’m not going to describe that: it’s like the Multiverse and just as non-empirical.) It’s not that those interpretations don’t work: it’s that only about forty people at any given time can understand them, and none of them work as statisticians. So whatever the working statisticians might mean, it’s not what the ingenious people suggest.

Personally, I think that phrases like ‘I don’t think that’s very likely’ or ‘I wouldn’t be surprised’ or ‘That’s probably what happened’ are figures of speech, referring, if to anything, to something that does not have to obey the probability calculus. There is no obligation on the figurative speech of ordinary people to obey rules made up by mathematicians. People do believe things, and that belief may be a bodily sensation, as the disappointment of a belief often is. Maybe those figures of speech are about the strength of those belief-sensations. We can, of course, say that if those belief-sensations are to be rational, they need to obey the probability calculus, but what we can’t say is that if they don’t, then ordinary people should not use probability-words to express their beliefs. Ordinary language got there first.

Similar issues affect the idea of the expected value. The expected value, or expectation, or prevision in French, or average in GCSE arithmetic, is a mathematical construction. It’s the sum of the probability-weighted outcome values. The formal expected value of a single roll of a fair dice is 1/6+2/6+3/6+4/6+5/6+6/6, which is 3.5 and that’s never going to appear on any roll of a six-sided dice. (A fair dice has no modal value - or perhaps it has six - and its median is any value between 3.00000000...1 and 3.999999999999... ) as well: half the throws will be below a number between 3 and 4 and the other half will be above it.

In a game with payoffs of £0 and £100, with equal odds, the expected value is £50, but that will never be the result of an individual trial: the payoffs are £0 or £100. It is what we would expect to be the long run average value of the payoff per trial. However, an actual sequence of trials that ever reached and stabilised at £50 after a ‘reasonable number’ of trials would be quite rare: what we should really expect is that the actual average payoff per trial should appear to converge to £50 as the number of trials increased. Measuring an expected value in practice is much more complicated than calculating it.

We can always make a formal calculation and, rightly, call that the expected value. But we must ask how that value is to be measured, and if it can’t be, or only has a meaning in some series of counter-factual logical universes, then it remains a formal calculation with no practical application. We can calculate the expected value of a one-off event, but we can’t measure it. Measuring expected values is a process that refers implicitly to a run of outcomes. The formal calculation for a single event is correct, but formal correctness is no guarantee of empirical application.

Since the formal expected value of our game has no empirical meaning for one event, it can’t be a guide to any decision we make. This has, as I’ve discussed before, some consequences for so-called rational economics.

The Tit-For-Tat Conjecture

Suppose you and I are going to play a co-operation game. There are many strategies for these, but the most beneficial is tit-for-tat: start by co-operating, then repeat the previous move of the other player. It’s simple, but it doesn’t dominate all the others. But it does give the maximum reward. If I know you’re going to use it, I may as well climb on board for those maximum rewards as well. Tit-For-Tat is a fairly rare strategy: it works even when the other person knows you’re using it. In fact, it works especially when the other person knows you’re using it.

One that fails if it’s public knowledge is the Secretary Strategy. In this, an employer is hiring, and has a limited time to pick a new person. It turns out that the most effective strategy is for them to look at the first third of the candidates, and then hire the first candidate better than all the ones they have already seen. This will get them the best candidate in 37% of hires. Some, of course, will never hire anyone, because the best was in the first third. It’s not a reliable strategy.

In this economy, recruitment is done through agencies, and they get to know the habits of the recruiter. If the agency know the employer uses the Secretary Strategy, they will arrange for the employer to see lesser-quality applicants at first, so they can place a reasonable one rapidly. The Secretary Strategy fails because the employment agent invalidates one of the assumptions, which is that candidates arrive at random. But then that’s the point of strategies and gaming. The only way out for an employer is to recruit directly, like they used to. Even then, in a small world, which some industries are, an interviewee finding she is the first might politely decline, on the grounds that ‘everyone knows you never hire the first person you see’. This denies the employer the opportunity to calibrate that the Strategy provides. The only way out of that is to lie to the candidates about their place in the queue: that’s not such a smart idea.

Most strategies are like this: they work as long as the other side don’t know. What makes Tit-For-Tat different? The Secretary Strategy predicts the future behaviour of its user, which allows others to game it. Tit-For-Tat can also be predicted, but the prediction is based on the other person’s behaviour, not its user’s intentions.

Strategies are, amongst other things, formalised intentions. If we know the strategy, we have a good idea about the objectives it is intended to achieve, and if we know that, we can make more informed guesses about the other ploys the other side might use.

Here’s a Conjecture: any strategy that works even when the other side knows you’re using it is equivalent to Tit-for-Tat.

If this is true, the immediate consequence is: unless you’re using Tit-for-Tat, your strategy can be gamed to your disadvantage.

Best Drop Ever!

I came across a video of what was supposed to be the "Best Drop Ever" from Carl Cox, and then Skrillex. Horrible stuff with all the subtlety of a slap on the back. "Oi mate, here's the beat."

The 'Drop' in case you don't go near certain types of dance music, is the moment the beat kicks back in after a floaty, beatless period of chords or other music designed a) to give the dancers a rest, and b) to make them wait for the beat so they appreciate it more. 'Bangers' hardly ever have drops, or if they do, the floaty bit is very short.

The longest build-up and deepest drop I know is not by a DJ at all. Not even The Great Digweed has matched the drop on this track. I play this over headphones at work, and it's just plain awesome.

The Single-Event Probability Fallacy

Here’s a choice. I can give you £50. Or you can play a game which gives you a 50% chance of winning £100. Which do you want to do?

Mostly people take the money. All behavioural economists, including a couple of guys with the ‘Nobel’ in Economics, think that makes us irrational. Well, maybe not.

Here’s the really simple rebuttal.

When the outcomes have equal expected values (probability-weighted values), as this game does, then, by definition, expected values can’t help us decide. In that case, we bring on some other idea or rule. Many of these will work in some circumstances, but will fail for some cunningly-constructed example. That failure doesn’t mean the rule is useless in every case, just those cases constructed by a behavioural economist to make a point. Here’s my rule: “I know what you’re thinking: which way will the coin fall? Well, seeing as how this is a behavioural economics experiment with a very limited budget, you gotta ask yourself: do I feel lucky? Well, do you? Do you feel lucky?” Yep. Thought so. Take the £50 instead.

When this example is presented, at least by a cute French PhD pitching their start-up, there’s a suggestion that given two options with equal expected values, we ‘should’ choose the one that offers the largest pay-off. What that proves is that I’m conservative, and the cute French PhD is a risk-taker, which is why she’s in a start-up and I’m a wage-slave.

If the algorithm her company is developing treats my decision as a guide to my temperament, I almost don’t have a problem.

Except. There’s always an unstated assumption that you have one shot at playing the game. Investment decisions are one-shot, unless we keep changing our minds, when the fees will eat up any gains we make. Now here’s the catch: the kind of probability that applies to single events is not the kind of probability we can use to calculate expected values.

We talk in an ordinary way about single events being likely or unlikely, and while it’s never exactly clear what we mean by it, one way to think of it is that we should devote our limited resources to the the option we consider ‘most likely’ and that we can also do something about. Whatever we do mean by talking about the outcomes of a single event being likely or unlikely, it isn’t the frequency of those outcomes over a run of that event. Because we’ve only got one.

We can have ‘degrees of belief’ about the outcome of single events - because we can have ‘degrees of belief’ about anything. But it make no sense to multiply degrees of belief by pay-offs and treat the result as something real.

Suppose you’re playing a game where outcome A pays off £50 and outcome B pays nothing. The odds of A to B are 50:50. The expected pay-off is £25 per play. In the long run. This is frequencies. It makes sense to multiply frequencies by pay-offs, because that’s a shorthand way of playing the game in the long run. The calculation corresponds to something you can observe.

Now suppose you multiply the pay-offs by your degrees of belief. The result will be £25, with a degree of belief of 100% (that it’s the correct expected value, not that it’s what you will win per play). But that’s not a pay-off of the game. It corresponds to nothing real. You cannot believe that on one shot of the game you will win £25, since the only outcomes are £0 or £50. So multiplying degrees of belief by pay-offs is not always meaningful: in fact, it is meaningful only when those beliefs are about frequencies, and it’s only accurate when the degrees of belief correspond with the long-run frequencies.

That’s the mistake the behavioural economists make in the first example. They think that we should calculate an expected value for the one-shot we have at the £100-or-Nothing offer, see that it is £50 and say that it's equivalent to the other offer. Except it isn't. The expected £50 is a fiction. The other £50 is a certainty. Somewhere in the back of our minds, most of us can see the difference. The behavioural economist cannot.

So does the example we started with really prove that I am more risk-averse than a cute French PhD? No. Because one of the choices does not involve any risk at all. The example is testing my preference for a sure thing over a game of chance. Only a degenerate gambler takes a risk over a sure thing. What would measure risk-aversion?

Here’s a different game. You can either spin Wheel One, which will pay off £50 with 50% probability, or Wheel Two, which will pay off £100 with 25% probability. Same expected value of £25. That is a choice between risks.

Does this matter? Financial regulators have lapped up behavioural economics. They think it’s telling them about we make irrational decisions, or maybe just bad ones, and how they can make regulations to stop the Nasty Banks from cheating us. Start-ups are designing programs which incorporate, in a Regulator-approved manner, these ‘insights’ for banks and advisors to use when selling you investments, and they have big-name clients. So yes, this stuff matters.

Is Dr Greg House a Dreysfus Expert?

My 2018 Box Set is all eight seasons of House. I watched my way through S1 during the Great Flu, and have despatched the first disc of S2 already. Dr Greg House is supposed to be the best diagnostician in the USA, and the question that will occur to you and me is, well, is he a Dreyfussian ‘expert’.

Hubert Dreyfus is a phenomenologist whose most well-known work came from trying to understand the failure of so-called ‘expert systems’ that, in the 1960’s and 1970’s, were going to replace everyone by now. And yet haven’t. The reason that rule-based, as opposed to computationally-heavy algorithms, failed, Dreyfus suggested, was that experts don’t use rules. More than that, the process of becoming an expert involved abandoning detached rule-based decisions and replacing it with a huge stock of special cases and a sophisticated pattern-matching process that the human brain seems to be good at creating but can’t be reduced to rules.

In Dreyfus’ model there are five stages to acquiring and practicing a skill. The novice is given simple techniques and processes to use and explicit rules about when these can be used; the advanced beginner has a larger stock of techniques and processes, and has added an amount of judgement in how these are used to deal with the tasks they are given; competency begins when the practitioner has too many rules and conditions and must make a decision about how to solve the problem in front of them (and it becomes problem-solving because it’s not immediately obvious which rules apply and which techniques will work). This stage must involve emotional engagement to be effective: the practitioner must experience hope and upset during its progress, and satisfaction or disappointment at the result. The emotional involvement is what seems to set the more complex learning processes going.

At the proficient stage, the act of recognising what kind of problem it is becomes more and more automatic, but the practitioner still needs to think about what to do, and may apply rules to determine the action. (“Ah yes, it’s Watkins’ Irritation: do we use hapagobulin or leave it to pass on its own?”) The Expert passes straight from seeing the situation to prescribing the action: to the Novice, the Expert “just seems to know what to do”.

So back to Dr Greg House. Each episode of House has a period of Shakespearean comic relief, called the Clinic. House has to see regular patients and treat them. These he dispatches within the minute: a glance at the person, listen to the symptoms, and he has the answer. “Your husband in cheating on you, you’re losing sleep and taking too many pills” or something similar. We’ve seen this before: it’s Sherlock Holmes. Holmes combined acute observation with a huge amount of background knowledge and experience to tell people things about themselves they thought were hidden. Conan Doyle called it ‘deduction’, but it wasn’t. It was Dreyfusian expertise, reconstructed as deduction for the benefit of the audience. Faced with regular life, House is smart as a whip.

Each episode of House also has the Puzzling Patient. This is the patient whose symptoms don’t make sense. House and his team spend the episode trying and rejecting one hypothesis after another, often with dramatic effects on the patient. The Puzzling Patient defeats the experts because their symptoms don’t fit one paradigm or another: there’s always a symptom missing or a symptom left over. Sometimes the patient has been treated before for something that matched some of their symptoms. In other words, the Puzzling Patient disables the Expert’s expertise.

So now there’s nothing for it but experiment. When the experiment goes wrong, the team react immediately and correctly. Faced with an episode that their Expert can recognise, they respond as Experts. And if the experiments don’t work, it’s time for the oddest feature of the show, the break-in to the patient’s living quarters. House’s catchphrase is ‘Everybody Lies’. It’s a way for him to remember that the patient may not be telling him everything they need to. In one episode, a Mexican handyman never mentions that he works at an illegal cock-fighting event Saturday nights. If he had, House would have immediately suspected a reasonably well-known problem. So the break-ins and other devices are there to find something the patient isn’t saying, or may not even know is relevant.

And when the final piece of information is found, the diagnosis is immediate and dealt with Expertly. Each episode of House is about the search for the missing fact that will make the Puzzling Patient no longer Puzzling, but an instance of one of the thousands of special cases in House’s Expert list.

And it all, so far, ends happily ever after.

DISCLAIMER: Yes I know the hospital in House bears the same resemblance to reality as the CSI lab in CSI: Vegas does to real CSI Labs; and that any doctor who behaved like House would be out on his ear in a month. This is TV, not a reality show.

December 2017 Review

The month started on a high note. I saw the Basquiat exhibition Boom For Real at the Barbican, followed by lunch at the Cote Brasserie across the road. The exhibition deserves its status as “the one to see” and that many people new to art will remember for a long while. I find the Barbican itself now doesn’t deserve the brickbats that used to get thrown at it. And I had no idea there was an ornamental lake in the middle.

I read Joshua Ferris’ To Rise Again At A Decent Hour; Michael Lewis’ The Undoing Project, about Daniel Kahneman and Amos Tversky; Svetlana Katok’s p-adic Analysis Compared With Real, which is very clear, should you want to learn about p-adic numbers; started Charles Hugh Smith’s Money and Work Unchained; and finished Richard Yates’ Eleven Kinds of Loneliness. And I continued my bedtime reading of the first volume of G W F Hegel’s Aesthetics. I read and looked at the pictures in Julian Schnabel’s C.V.J. Nicknames of Maitre D’s and Other Excerpts From Life, and read the book of the Basquiat exhibition, also titled Boom For Real.

I saw S1 of House; 1992, a series about that pivotal year in Italian history; and Suicide Squad and Fandango on DVD.

Sis and I dined at the Marleybone Picture, which was an experience we will be repeating. Sis cooked Christmas lunch and we all fell asleep watching TV.

Then the sickness and fever hit, and I was not really recovered until about the 31st. I made myself go to the gym every morning from the 28th to the 31st, and was pretty darn weak the first couple of days.

Now here’s why I do this. Because in my memory, December was a horrible write-off. But in fact, while I was well, I did a reasonable amount of reading and was not totally inactive. It was a horrible flu, and no consolation that others had it as well.

Thoughts at the Start of 2018

Entrepeneurs in Cars had a list of six lessons learned in 2017. Three caught my attention:

1. Only a very small proportion of men are willing to do the work to make themselves a better version of themselves
2. Manage your energy - give a frak only when it’s going to benefit you
3. If it’s not Hell Yeah, it’s Frak No

And I’m going to take the first and third as somewhere to start.

Am I one of that small proportion of men prepared to do the work to be a better version of myself? Ask what I would look like if I wasn’t sober, exercising, single and employed. I’d be a mess. I’d be on filthy drugs with names ending in ‘statin’ or ‘formin’. I would have no muscle tone, I would have a gut, with even worse body-fat than I have now. I would be paying alimony, living in a squalid flat because that’s all I could afford after the divorce. I would be alone in an empty marriage. If I wasn’t sober, I’d be dead. And don’t doubt that. Ever. I would be mostly unemployed, with occasional breaks of clerical work through agencies: if I was lucky, I’d have a postal round.

You know what? I pretty much am a better version of myself. Actually, as long as I’m sober, I a better version of myself. Two things.

First, maintenance is a bitch. When you’re ‘better’, very few of your options take you to ‘even better’, and those are hard work. Enough is enough. Better is the enemy of the good (old Russian proverb). And so on.

Second, I’m not having as much fun as I could be, in fact, I’m rather good at denying myself fun. Some of that is due to the habits of denial acquired by wearing dental braces for eighteen months. Some of it is about logistics, and I’ll get to those.

Next. If it’s not Hell Yeah, it’s Frak No. Now I like this. The idea is that if I’m not immediately up for whatever it is, I don’t talk myself into it because what else am I doing? What this made clear to me is that there are some things I want to do, but talk myself out of, usually on the basis of cost and logistics. I may need to reverse the maxim: if it’s not Frak No, it’s Hell Yeah.

My idea of fun is cultural consumption. That includes good food in nice restaurants. A good few years ago, the culture industry jacked the prices up: £18 for an art movie? £72+ for Shen Yun at the Dominion Theatre? £33 for a concert at the Wigmore Hall? I don’t demur at the prices for modern dance at Sadlers Wells, but that’s because it’s My Thing and there’s no alternative. For £33 I can get a top-end Wagner opera on CD. Wait a few months and I can get the movie on DVD for £10 or less in Fopp. (You can tell I’m a Pricing Guy.) And usually there’s more than the ticket price. If it’s mid-week, I park in Richmond so I can avoid the ultimate buzz-kill of travelling from Waterloo after 20:00. That’s nearly £14. And there’s a supper as well, say £10 or so even for a pizza. Going out in London is expensive.

To which I tell myself, either spend the money or find some other kind of fun.

Logistics. I wake up at 05:30, do a day’s work, and you expect me to be looking forward to a two-hour show at 19:30 and a return commute at 22:00? Especially with a session at the gym and a light meal to pass the time between leaving work and the start of the show. That’s a long, long day, and there ain’t no sleeping-in the next morning. Last year I trapped myself into trying to wake up refreshed and relaxed. As if I could do that with an extra sleep cycle. But dragging one’s ass out of bed into the kitchen every morning can lead to the belief that There Must Be Something I Can Do. There isn’t, but I fell for it. Trying to get enough sleep and wake up at 05:30 means no activity in the evening at all. Late afternoon at the latest. Movies that start at 18:30 in town are verboten. Even 18:30 locally could be a bit late.

I think the objection-by-logistics is an excuse for not doing something. I live where I live, not in central London. This is one of those suck-it-up things. And right now, it’s damn cold, and that makes any kind of going out feel like a really bad idea.

No pretty, full-of-motivation conclusions, but for the moment that’s where I am.