William M. Briggs

Statistician to the Stars!

Category: Philosophy (page 1 of 110)

The philosophy of science, empiricism, a priori reasoning, epistemology, and so on.

Government To Issue Baby Licenses

Not the kind of baby license he has in mind.

Not the kind of baby license he has in mind. (Image source.)

Regular readers will recall that I am a (self-appointed) bioethicist, a post I take on not because I need the work, but because the professionals are making such a hash of it.

Take professional “futurist” Zoltan Istvan’s recent article in Wired, “It’s time to consider restricting human breeding” who poses the question nobody was asking, “In this transhumanist future, should everyone still be allowed to have unlimited children whenever they want?”

He meant it rhetorically, naturally, where all bien pensant would know to answer No. (But then even the best of us cannot have children whenever we want.)

His question has a proviso, relying on “transhumanism”, sometimes abbreviated “H+”. The easy answer is that this is the state of human beings envisioned by those who have watched too much bad science fiction on television. Think Six Million Dollar Man grafted onto an iPad—or maybe its the other way around.

The longer and more tedious definition is exactly the same, except the parts that make up Steve Austin’s limbs have been successfully miniaturized and mass produced and genetically engineered. “Defectives” would not be allowed out of the wome to mix with their betters.

The fallacy underlying transhumanism is not that our body parts won’t be replaced by Apple Corp (or whomever, and for a large fee), which smart money practically guarantees, but that once this happens we become something other than human, something superior, once we reach a “singularity” or pass some “tipping point” or whatever. In other words, transhumanism is yet one more in a long and ever-increasing list of Utopian schemes, and yet more proof that abandoning a classic education leads to a fundamental ignorance of the reality of man’s unchangeable nature.

Transhumanists are far from the first to think of that happy State which awaits us once we work out the technology. In Brave New World, transhumans were born in a factory via “Bokanovsky’s Process”, bred like mushrooms. Istvan thinks this a swell idea—and that the State should be in charge of it.

Enter fallacy number two, appropriately called the Transformation Fallacy. It’s when a common person possessing all the faults, weaknesses, and sins of a common mortal is transformed into a purely altruistic loving caring faultless man of superior-intellect merely by being appointed to a government post. It’s become rare not to see this fallacy. Istvan is a slave to it.

“I cautiously,” Istvan says, inventing a new and opposite meaning for cautiously, “endorse the idea of licensing parents, a process that would be little different than getting a driver’s licence.” State issued, of course. To support this he says,

The philosophical conundrum of controlling human procreation rests mostly on whether all human beings are actually responsible enough to be good parents and can provide properly for their offspring. Clearly, untold numbers of children — for example, those millions that are slaves in the illegal human trafficking industry — are born to unfit parents.

Untold? Millions? Parents who won’t sell their children into slavery stand a higher chance of being licensed. Also, parents “who pass a series of basic tests qualify and get the green light to get pregnant and raise children.” Who will write, score, administrate, and enforce the outcome of these tests? Those who have been changed via the transformation fallacy.

Istvan isn’t alone. He quotes other transhumanists, like Hank Pellissier “founder of the Brighter Brains Institute” (!), Paul “Prognostication Bombed” Ehrlich, an “advocate for government intervention to control human population”, and “bioethics pioneer” (!) Joseph Fletcher whom he quotes as saying “many births are accidental”. Accidental? Accidental? As in, “Honey, I didn’t realize that when we had sex you might have got pregnant“?

Skip it. If you haven’t been convinced that parenting licences are required, Istvan has this up his sleeve: “After all, we don’t allow people to drive cars on crack cocaine.” Devastating, no? Licensing parents would drive down crime rates, too. He says.

How to keep unlicensed people from the unaccidental consequences of doin’ what come naturally? Implanted birth control microchips which “can deliver hormones into the body via an on-off switch on your mobile phone”. We can call it the Lack of Responsibility App.

He closes his article with these words: “As a liberty-loving person, I have always eschewed giving up any freedoms.” Does he, though. For the next words are “However…”

This is the same speech you get from all progressives and statists. “I hate to do it, but it’s for your own good.”

Philosophic Issues in Cosmology VI: Anthropic Coincidences—Guest Post by Bob Kurland

Something (which can be observed) rather than nothing (which cannot).

Something (which can be observed) rather than nothing (which cannot).

Bob Kurland is a retired, cranky, old physicist, and convert to Catholicism. He shows that there is no contradiction between what science tells us about the world and our Catholic faith.

Read Part V.

Scientists are slowly waking up to an inconvenient truth – the universe looks suspiciously like a fix. The issue concerns the very laws of nature themselves. For 40 years, physicists and cosmologists have been quietly collecting examples of all too convenient “coincidences” and special features in the underlying laws of the universe that seem to be necessary in order for life, and hence conscious beings, to exist. —Paul Davies.

The argument (the Anthropic Principle) can be used to explain why the conditions happen to be just right for the existence of (intelligent) life on the earth at the present time. For if they were not just right, then we should not have found ourselves to be here now, but somewhere else, at some other appropriate time. —Roger Penrose.

One doesn’t show that something doesn’t require explanation by pointing out that it is a condition of one’s existence. If I ask for an explanation of the fact that the air pressure in the transcontinental jet is close to that at sea level, it is no answer to point out that if it weren’t, I’d be dead. —Thomas Nagel, Mind and Cosmos.

A common sense interpretation of the facts suggests that a super-intellect has monkeyed with physics, as well as with chemistry and biology, and that there are no blind forces worth speaking about in nature. The numbers one calculates from the facts seem to me so overwhelming as to put this conclusion almost beyond question. —Fred Hoyle

10,000 dials & monkeys

The presence of organic life in the universe (namely us) requires a series of unlikely happenings and restricted values for physical laws and constants. This “fine-tuning” (as it’s been called) has been likened to a room full of 10,000 dials, each of which has to be set to a precise setting in order to achieve action; 10,000 monkeys are let into the room and each adjusts a dial and, lo, action occurs. The set of coincidences was termed “The Anthropic Principle” by Brandon Carter in 1973, when he introduced it in a conference to oppose the “Copernican Principle”, that man has no special place in the universe.

Anthropic Principle

There is a concise summary of the Anthropic Principle by Robert Koons in his philosophy lectures, giving various interpretations, with arguments for and against each. A good collection of articles with different (and opposing views) of the Anthropic Principle is given in God and Design (ed. Neil Manson). There are many versions of the Anthropic Principle ranging from the Weak Anthropic Principle, WAP, which tautologically observes that if the universe weren’t fit for us to be here we would wouldn’t be here discussing the principle, through the Strong Anthropic Principle, SAP, that the universe has been fine-tuned for intelligent life (us), on up to the Completely Ridiculous Anthropic Principle (by Martin Gardner—you complete the acronym).

Can unlikelihood be quantified?

In assessing the improbable nature of the anthropic coincidences, some authors assign a specific probability to the value of some particular physical constant. Such assignment is not always justified, because probability considerations are ill defined, in the usual sense of evidential probability. For example, theoretical calculations have shown that if the strong nuclear force were 2% higher or 2% lower, then the elements as we know them would not have been formed. This does not mean that the probability of having the strong nuclear force at an anthropic value is 4%. In order to give a probability for this range, the population distribution of the parameters for the strong nuclear force would have to be known. Moreover, there is a difficulty in using probability in an after-the-fact, rather than a predictive sense. The way to use probabilities in assessing the anthropic coincidences is via Bayesian probability techniques, with well-defined prior assumptions, and to use the resulting Bayesian probability as a measure of belief.

Ellis’s interpretation

Ellis, in his presentation of the anthropic coincidences, focuses on the special nature of physical laws that allow for the presence of life, rather than on their improbability:

One of the most profound issues in cosmology is the Anthropic question…why does the Universe has the very special nature required in order that life can exist? The point is that a great deal of “fine tuning” is required in order that life be possible. There are many relationships embedded in physical laws that are not explained by physics, but are required for life to be possible; in particular various fundamental constants are highly constrained in their values if life as we know it is to exist…What requires explanation is why the laws of physics are such as to allow this complex functionality (life) to work..We can conceive of universes where the laws of physics (and so of chemistry) were different than in ours. Almost any change in these laws will prevent life as we know it from functioning.

Ellis posits as a first requirement for the laws of physics “the kind of regularities that can underlie the existence of life”: laws that are not based on symmetry and variational principles are unlikely to produce the kind of complexity that would be required for life. He also sets up general conditions that allow for organic life and cosmological boundary/initial conditions. In this respect he cites the following as necessary:

  • “Quantization that stabilizes matter and allows chemistry to exist through the Pauli exclusion principle;
  • The number D of large spatial dimensions must be just 3 for complexity to exist.
  • The seeds in the early universe for fluctuations (quantum fluctuations) that will later grow into galaxies must be of the right size that structures form without collapsing into black holes…
  • The size of the universe and its age must be large enough…we need a sufficiently old universe for second generation stars to come into existence and then for planets to have a stable life for long enough that evolution could lead to the emergence of intelligent life. Thus the universe must be at about 15 billion years old for life to exist.
  • There must be non-interference with local systems. The concept of locality is fundamental, allowing local systems to function effectively independently of the detailed structure of the rest of the Universe. We need the universe and the galaxies in it to be largely empty, and gravitational waves and tidal forces to be weak enough, so that local systems can function in a largely isolated way.
  • The existence of the arrow of time, and of laws like the second law of thermodynamics, are probably necessary for evolution and for consciousness. This depends on boundary conditions at the beginning and end of the Universe.
  • Presumably the emergence of a classical era out of a quantum state is required. The very early universe would be a domain where quantum physics would dominate leading to complete uncertainty and an inability to predict the consequence of any initial situation; we need this to evolve to a state where classical physics leads to the properties of regularity and predictability that allow order to emerge.
  • The fact that the night sky is dark…is a consequence of the expansion of the universe together with the photon (light particle) to baryon (mass particle) ratio. This feature is a necessary condition for the existence of life: the biosphere on Earth functions by disposing of waste energy to the heat sink of the dark night sky. Thus one way of explaining why the sky is observed to be dark at night is that if this were not so, we would not be here to observe it.
  • Physical conditions on planets must be a in a quasi-equilibrium state for long enough to allow the delicate balances that enable our existence, through the very slow process of evolution, to be fulfilled.” (see the Theology of Water.)

There are a number of other constraints, limited values for forces—gravity, electromagnetic, weak nuclear, strong nuclear—and fundamental constants, including that for particle masses and number of particles that are needed for life to evolve. In summary, Ellis puts the Anthropic Principle as the following:

Life is possible because both the laws of physics and the boundary conditions for the universe have a very special nature. only particular laws of physics, and particular initial conditions in the Universe, allow the existence of intelligent life of the kind we know. No evolutionary process whatever is possible for any kind of life if these laws and conditions do not have this restricted form.

Robert Koons summarizes some general objections to invoking the Anthropic Principle for carbon-based life “well isn’t that special” (as the Church Lady might say):

  1. The problem of “old evidence”;
  2. Laws of nature don’t need to be explained;
  3. We had to be here in any event (see Penrose’s quote above);
  4. Exotic life might exist;
  5. The Copernican Principle–rejection of anthropocentricity is fundamental to science;
  6. We’re only one among many universes (see below).

Then:

  1. Objection 1 can be countered by the argument that such evidence is used frequently in science when direct experiments can’t be done. Witness the General Relativity explanation of the advance in the perihelion of Mercury.
  2. Objection 2 would do away with all interpretations of theory, quantum mechanics, and the philosophy of science.
  3. Objection 3 is countered as in Thomas Nagel’s quote above; as information seeking life form we need explanations.
  4. Objection 4 is invalid: we’re talking about conditions for carbon-based life; science-fiction can explore and has explored conditions for exotic life.
  5. Objection 5: the Anthropic Principle was introduced to rebut the Copernican Principle.
  6. Objection 6: the multiverse proposition is not itself proven.

The philosophic/metaphysical context for these Anthropic conditions that Ellis sets forth will be given in the final post. It should be noted that one interpretation of anthropic coincidences is the theory that infinitely many universes with potentially different physical laws and constants exist and so it is not unlikely that in all these one universe with appropriate conditions for life would be present.

The analogy is like that of having a lottery ticket with the numbers 1 1 1 1 1 be the winner. That combination of numbers looks improbable, but since there are a whole host of numbers from 00000 to 99999, it is no less probable than any other number. This brings up the notion of a multiverse, which will be discussed in the next post.

Summary Against Modern Thought: God Is Not Made Of Matter

This may be proved in three ways. The first...

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.

A relatively simple argument today. God is not made of stuff. Who would disagree? Pagans, perhaps. For example, the god of the atheists is a demiurge, a sort of superior created or “evolved” being, and therefore made of matter. But not God. What’s nifty about today’s discussion is the role of “chance”. For that, we turn back (again) to Aristotle.

Chapter 17: That in God there is no matter

1 FROM this it follows that God is not matter.i

2 For matter, such as it is, is in potentiality.ii

3 Again. Matter is not a principle of activity: wherefore, as the Philosopher puts it,[1] efficient and material causes do not coincide. Now, as stated above,[2] it belongs to God to be the first efficient cause of things. Therefore He is not matter.iii

4 Moreover. For those who referred all things to matter as their first cause, it followed that natural things exist by chance: and against these it is argued in 2 Phys.[3] Therefore if God, Who is the first cause, is the material cause of things, it follows that all things exist by chance.iv

5 Further. Matter does not become the cause of an actual thing, except by being altered and changed. Therefore if God is immovable, as proved above,[4] He can nowise be a cause of things as their matter.v

6 The Catholic faith professes this truth, asserting that God created all things not out of His substance, but out of nothing.vi

7 The ravings of David of Dinant are hereby confounded,vii who dared to assert that God is the same as primary matter, because if they were not the same, they would needs differ by certain differences, and thus they would not be simple: since in that which differs from another thing by a difference, the very difference argues composition.

Now this proceeded from his ignorance of the distinction between difference and diversity. For as laid down in 10 Metaph.[5] a thing is said to be different in relation to something, because whatever is different, differs by something, whereas things are said to be diverse absolutely from the fact that they are not the same thing.[6]

Accordingly we must seek for a difference in things which have something in common, for we have to point to something in them whereby they differ: thus two species have a common genus, wherefore they must needs be distinguished by differences. But in those things which have nothing in common, we have not to seek in what they differ, for they are diverse by themselves. For thus are opposite differences distinguished from one another, because they do not participate in a genus as a part of their essence: and consequently we must not ask in what they differ, for they are diversified by their very selves. Thus too, God and primary matter are distinguished, since, the one being pure act and the other pure potentiality, they have nothing in common.

—————————————————————-

iFrom last time, of course.

iiMatter can change, thus it is in potentiality, and we have seen from last time that God is not in potentiality.

iiiThis doesn’t appear controversial, but we have scarcely outlined the nature of cause. There are four kinds of cause: the formal (the form of the thing), material (what the thing is made of), efficient (what brings about the change), and final (the end or direction of the change). The material of the statue, say, is not its efficient cause. Much more on this later.

ivWe are now at Yours Truly’s favorite material. Aristotle (from 2 Phys iv):

Some people even question whether [chance and spontaneity] are real or not. They say that nothing happens by chance, but that everything which we ascribe to chance or spontaneity has some definite cause, e.g. coming ‘by chance’ into the market and finding there a man whom one wanted but did not expect to meet is due to one’s wish to go and buy in the market.

Similarly in other cases of chance it is always possible, they maintain, to find something which is the cause; but not chance, for if chance were real, it would seem strange indeed, and the question might be raised, why on earth none of the wise men of old in speaking of the causes of generation and decay took account of chance; whence it would seem that they too did not believe that anything is by chance…

There are some too who ascribe this heavenly sphere and all the worlds to spontaneity. They say that the vortex arose spontaneously, i.e. the motion that separated and arranged in its present order all that exists. This statement might well cause surprise.

For they are asserting that chance is not responsible for the existence or generation of animals and plants, nature or mind or something of the kind being the cause of them (for it is not any chance thing that comes from a given seed but an olive from one kind and a man from another); and yet at the same time they assert that the heavenly sphere and the divinest of visible things arose spontaneously, having no such cause as is assigned to animals and plants.

Yet if this is so, it is a fact which deserves to be dwelt upon, and something might well have been said about it. For besides the other absurdities of the statement, it is the more absurd that people should make it when they see nothing coming to be spontaneously in the heavens, but much happening by chance among the things which as they say are not due to chance; whereas we should have expected exactly the opposite.

Others there are who, indeed, believe that chance is a cause, but that it is inscrutable to human intelligence, as being a divine thing and full of mystery.

Aristotle says things which are for the sake of something can be caused by chance, and he gives this example (2 Phys v):

A man is engaged in collecting subscriptions for a feast. He would have gone to such and such a place for the purpose of getting the money, if he had known. [But he] actually went there for another purpose and it was only incidentally that he got his money by going there; and this was not due to the fact that he went there as a rule or necessarily, nor is the end effected (getting the money) a cause present in himself — it belongs to the class of things that are intentional and the result of intelligent deliberation. It is when these conditions are satisfied that the man is said to have gone ‘by chance’. If he had gone of deliberate purpose and for the sake of this — if he always or normally went there when he was collecting payments — he would not be said to have gone ‘by chance’.

Notice that chance here is not an ontological (material) thing or force, but a description or a statement of our understanding (of a cause). Aristotle concludes, “It is clear then that chance is an incidental cause in the sphere of those actions for the sake of something which involve purpose. Intelligent reflection, then, and chance are in the same sphere, for purpose implies intelligent reflection.”

And “Things do, in a way, occur by chance, for they occur incidentally and chance is an incidental cause. But strictly it is not the cause — without qualification — of anything; for instance, a housebuilder is the cause of a house; incidentally, a fluteplayer may be so.”

Chance used this way is like the way we use coincidence. But there is also spontaneity, which is similar: “The stone that struck the man did not fall for the purpose of striking him; therefore it fell spontaneously, because it might have fallen by the action of an agent and for the purpose of striking.”

Lastly, “Now since nothing which is incidental is prior to what is per se, it is clear that no incidental cause can be prior to a cause per se. Spontaneity and chance, therefore, are posterior to intelligence and nature. Hence, however true it may be that the heavens are due to spontaneity, it will still be true that intelligence and nature will be prior causes of this All and of many things in it besides.”

vIn short, since God is not movable, he can’t be made of matter, which is always movable.

viSuch a misunderstood word, nothing! It means just what it says. No thing. No fields, no forces, no fields, no equations, no quantum thises or thats, the absence of all entities. Now just you imagine what kind of Being could create something about of this real nothing. Only one: Being itself, I Am That I Am; which is to say, God.

viiZing! More proof that even saints can be contemptuous when the need arises. Notice very carefully that St Thomas does not ask for dialogue with David of Dinant, but is satisfied to destroy his argument.

[1] 2 Phys. vii. 3.
[2] Ch. xiii.
[3] Chs. viii., ix.
[4] Ch. xiii.
[5] D. 9, iii. 6.
[6] Sum. Th. P. I., Q. iii., A. 8, ad 3.

Comments On Dawid’s Prequential Probability

Murray takes the role of a prequential Nature.

Murray takes the role of a prequential Nature.

Phil Dawid is a brilliant mathematical statistician who introduced (in 1984) the theory of prequential probability1 to describe a new-ish way of doing statistics. We ought to understand this theory. I’ll give the philosophy and leave out most of the mathematics, which are not crucial.

We have a series of past data, x = (x1, x2, …, xn) for some observable of interest. This x can be quite a general proposition, but for our purposes suppose its numerical representation can only take the values 0 or 1. Maybe xi = “The maximum temperature on day i exceeds Wo C”, etc. The x can also have “helper” propositions, such as yi = “The amount of cloud cover on day i is Z%”, but we can ignore all these.

Dawid says, “One of the main purposes of statistical analysis is to make forecasts for the future” (emphasis original) using probability. (It’s only other, incidentally, is explanation: see this for the difference.)

The x come at us sequentially, and the probability forecast for time n+1 Dawid writes as Pr(xn+1 | xn). “Prequential” comes from “probability forecasting with sequential prediction.” He cites meteorological forecasts as a typical example.

This notation suffers a small flaw: it doesn’t show the model, i.e. the list of probative premises of x which must be assumed or deduced in order to make a probability forecast. So write pn+1 = Pr(xn+1 | xn, M) instead, where M are these premises. The notation shows that each new piece of data is used to inform future forecasts.

How good is M at predicting x? The “weak prequential principle” is that M should be judged only on the pi and xi, i.e. only how on good the forecasts are. This is not the least controversial. What is “good” sometimes is. There has to be some measure of closeness between the predictions and outcomes. People have invented all manner of scores, but (it can be shown) the only ones that should be used are so-called “proper scores”. These are scores which require pn+1 to be given conditional on just the M and old data and nothing else. This isn’t especially onerous, but it does leave out measures like R^2 and many others.

Part of understanding scoring is calibration. Calibration has more than one dimension, but since we have picked a simple problem, consider only two. Mean calibration is when the average of the pi equaled (past tense) the average of the xi. Frequency calibration is when whenever pi = q, q*100% of the time x = q. Now since x can only equal 0 or 1, frequency calibration is impossible for any M which does produce non-extreme probabilities. That is, the first pi that does not equal 0 or 1 dooms the frequency calibration of M.

Ceteris paribus, fully calibrated models are better than non-calibrated ones (this can be proven; they’ll have better proper scores; see Schervish). Dawid (1984) only considers mean calibration, and in a limiting way; I mean mathematical limits, as the number of forecasts and data head out to infinity. This is where things get sketchy. For our simple problem, calibration is possible finitely. But since the x are given by “Nature” (as Dawid labels the causal force creating the x), we’ll never get to infinity. So it doesn’t help to talk of forecasts that have not yet been made.

And then Dawid appears to believe that, out an infinity, competing mean-calibrated models (he calls them probability forecasting systems) are indistinguishable. “[I]n just those cases where we cannot choose empirically between several forecasting systems, it turns out we have no need to do so!” This isn’t so, finitely or infinitely, because two different models which have the same degree of mean calibration can have different levels of frequency calibration. So there is still room to choose.

Dawid also complicates his analysis by speaking as if Nature is “generating” the x from some probability distribution, and that a good model is one which discovers this Nature’s “true” distribution. (Or, inversely, he says Nature “colludes” in the distribution picked by the forecaster.) This is the “strong prequential principle”, which I believe does not hold. Nature doesn’t “generate” anything. Something causes each xi. And that is true even in the one situation where our best knowledge is only probabilistic, i.e. the very small. In that case, we can actually deduce the probability distributions of quantum x in accord with all our evidence. But, still, Nature is not “generating” x willy nilly by “drawing” values from these distributions. Something we-know-not-what is causing the x. It is our knowledge of the causes that is necessarily incomplete.

For the forecaster, that means, in every instance and for any x, the true “probability distribution” is the one that takes only extreme probabilities, i.e. the best model is one which predicts without error (each pi would be 0 or 1 and the model would automatically be frequency and mean calibrated). In other words, the best model is to discover the cause of each xi.

Dawid also has a technical definition of the “prequential probability” of an “event”, which is a game-theoretic like construction that need not detain us because of our recognition that the true probability of any event is 0 or 1.

Overall

That models should be judged ultimately by the predictions they make, and not exterior criteria (which unfortunately includes political considerations, and even p-values), is surely desirable but rarely implemented (how many sociological models are used to make predictions in the sense above?). But which proper score does one use? Well, that depends on exterior information; or, rather, on evidence which is related to the model and to its use. Calibration, in all its dimensions, is scandalously underused.

Notice that in Pr(xn+1 | xn, M) the model remains fixed and only our knowledge of more data increases. In real modeling, models are tweaked, adjusted, improved, or abandoned and replaced wholesale, meaning the premises (and deductions from the same) which comprise M change in time. So this notation is inadequate. Every time M changes, M is different, a banality which is not always remembered. It means model goodness judgments must begin anew for every change.

A true model is the one that generates extreme probabilities (0 or 1), i.e. the identifies the causes, or the “tightest” probabilities deduced from the given (restricted by nature) premises, as in quantum mechanics. Thus the ultimate comparison is always against perfect (possible) knowledge. Since we are humble, we know perfection is mostly unattainable, thus we reach for simpler comparisons, and gauge model success by it success over simple guesses. This is the idea of skill (see this).

Reminder: probability is a measure of information, an epistemology. It is not the language of causality, or ontology.

—————————————————————————–

Thanks to Stephen Senn for asking me to comment on this.

1The two papers to read are, Dawid, 1984. Present position and potential developments: some personal views: statistical theory: the prequential approach. JRSS A, 147, part 2, 278–292. And Dawid and Vovk, 1999. Prequential probability: principles and properties. Bernoulli, 5(1), 125–162.

Explanation Vs Prediction

The IPCC, hard at work on another forecast.

The IPCC, hard at work on another forecast.

Introduction

There isn’t as much space between explanation and prediction as you’d think; both are had from the same elements of the problem at hand.

Here’s how it all works. I’ll illustrate a statistical (or probability) model, though there really is no such thing; which is to say, there is no difference in meaning or interpretation between a probability and a physical or other kind of mathematical model. There is a practical difference: probability models express uncertainty natively, while (oftentimes) physical models do not mention it, though it is there, lurking below the equations.

Let’s use regression, because it is ubiquitous and easy. But remember, everything said goes for all other models, probability or physical. Plus, I’m discussing how things should work, not how they’re actually done (which is very often badly; not your models, Dear Reader: of course, not yours).

We start by wanting to quantify the uncertainty in some observable y, and believe we have collected some “variables” x which are probative of y. Suppose y is (some operationally defined) global average temperature. The x may be anything we like: CO2 levels, population size, solar insolation, grant dollars awarded, whatever. The choice is entirely up to us.

Now regression, like any model, has a certain form. It says the central parameter of the normal distribution representing uncertainty in y is a linear function of the x (y and x may be plural, i.e. vectors). This model structure is almost never deduced (in the strict sense of the word) but is assumed as a premise. This is not necessarily a bad thing. All models have a list of premises which describe the structure of the model. Indeed, that is what being a model means.

Another set of premises are the data we observe. Premises? Yes, sir: premises. The x we pick and then observe take the form of propositions, e.g. “The CO2 observed at time 1 was c1“, “The CO2 observed at time 2 was c2,” etc.

Observed data are premises because it is we who pick them. Data are not Heaven sent. They are chosen and characterized by us. Yes, the amount of—let us call it—cherishing that takes place over data is astonishing. Skip it. Data are premises, no different in character than other assumptions.

Explanation

Here is what explanation is (read: should be). Given the model building premises (that specified, here, regression) and the observed data (both y and x), we specify some proposition of interest about y and then specify propositions about the (already observed) x. Explanation is how much the probability the proposition about y (call it Y) changes.

That’s too telegraphic, so here’s an example. Pick a level for each of the observed x: “The CO2 observed is c1“, “The population is p”, “The grant dollars is g”, etc. Then compute the probability Y is true given this x and given the model and other observed data premises.

Step two: pick another level for each of the x. This may be exactly the same everywhere, except for just one component, say, “The CO2 observed is c2“. Recompute the probability of Y, given the new x and other premises.

Step three: compare how much the probability of Y (given the stated premises) changed. If not at all, then given the other values of x and the model and data premises, then CO2 has little, and maybe even nothing, to do with y.

Of course, there are other values of the other x that might be important, in conjunction with CO2 and y, so we can’t dismiss CO2 yet. We have a lot of hard work to do to step through how all the other x and how this x (CO2) change this proposition (Y) about y. And then there are other propositions of y that might be of more interest. CO2 might be important for them. Who knows?

Hey, how much change in the probability of any Y is “enough”? I have no idea. It depends. It depends on what you want to use the model for, what decisions you want to make with it, what costs await incorrect decisions, what rewards await correct ones, all of which might be unquantifiable. There is and should be NO preset level which says “Probability changes by at least p are ‘important’ explanations.” Lord forbid it.

A word about causality: none. There is no causality in a regression model. It is a model of how changing CO2 changes our UNCERTAINTY in various propositions of y, and NOT in changes in y itself.1

Explanation is brutal hard labor.

Prediction

Here is what prediction is (should be). Same as explanation. Except we wait to see whether Y is true or false. The (conditional) prediction gave us its probability, and we can compare this probability to the eventual truth or falsity of Y to see how good the model is (using proper scores).

Details. We have the previous observed y and x, and the model premises. We condition on these and then suppose new x (call them w) and ask what is the probability of new propositions of y (call them Z). Notationally, Pr( Z | w,y,x,M), where M are the model form premises. These probabilities are compared against the eventual observations of z.

“Close” predictions means good models. “Distant” ones mean bad models. There are formal ways of defining these terms, of course. But what we’d hate is if any measure of distance became standard. The best scores to use are those tied intimately with the decisions made with the models.

And there is also the idea of skill. The simplest regression is a “null x”, i.e. no x. All that remains is the premises which say the uncertainty in y is represented by some normal distribution (where the central parameter is not a function of anything). Now if your expert model, loaded with x, cannot beat this naive or null model, your model has no skill. Skill is thus a relative measure.

For time series models, like e.g. GCMs, one natural “null” model is the null regression, which is also called “climate” (akin to long-term averages, but taking into account the full uncertainty of these averages). Another is “persistence”, which is the causal-like model yt+1 = yt + fuzz. Again, sophisticated models which cannot “beat” persistence have no skill and should not be used. Like GCMs.

More…

This is only a sketch. Books have been written on these subjects. I’ve compressed them all in 1,100 words.

———————————————————————————-

1Simple causal model: y = x. It says y will be the value of x, that x makes y what it is. But even these models, though written mathematically like causality, are not treated that way. Fuzz is added to them mentally. So that if x = 7 and y = 9, the model won’t abandoned.

Older posts

© 2014 William M. Briggs

Theme by Anders NorenUp ↑