Monday, December 19, 2011

Jabberwocky. Or: Grand Unified Theory of Uncertainty???

Jabberwocky, Lewis Carroll's whimsical nonsense poem, uses made-up words to create an atmosphere and to tell a story. "Billig", "frumious", "vorpal" and "uffish" have no lexical meaning, but they could have. The poem demonstrates that the realm of imagination exceeds the bounds of reality just as the set of possible words and meanings exceeds its real lexical counterpart.

Uncertainty thrives in the realm of imagination, incongruity, and contradiction. Uncertainty falls in the realm of science fiction as much as in the realm of science. People have struggled with uncertainty for ages and many theories of uncertainty have appeared over time. How many uncertainty theories do we need? Lots, and forever. Would we say that of physics? No, at least not forever.

Can you think inconsistent, incoherent, or erroneous thoughts? I can. (I do it quite often, usually without noticing.) For those unaccustomed to thinking incongruous thoughts, and who need a bit of help to get started, I can recommend thinking of "two meanings packed into one word like a portmanteau," like 'fuming' and 'furious' to get 'frumious' or 'snake' and 'shark' to get 'snark'.

Portmanteau words are a start. Our task now is portmanteau thoughts. Take for instance the idea of a 'thingk':

When I think a thing I've thought,
I have often felt I ought
To call this thing I think a "Thingk",
Which ought to save a lot of ink.

The participle is written "thingking",
(Which is where we save on inking,)
Because "thingking" says in just one word:
"Thinking of a thought thing." Absurd!

All this shows high-power abstraction.
(That highly touted human contraption.)
Using symbols with subtle feint,
To stand for something which they ain't.

Now that wasn't difficult: two thoughts at once. Now let those thoughts be contradictory. To use a prosaic example: thinking the unthinkable, which I suppose is 'unthingkable'. There! You did it. You are on your way to a rich and full life of thinking incongruities, fallacies and contradictions. We can hold in our minds thoughts of 4-sided triangles, parallel lines that intersect, and endless other seeming impossibilities from super-girls like Pippi Longstockings to life on Mars (some of which may actually be true, or at least possible).

Scientists, logicians, and saints are in the business of dispelling all such incongruities, errors and contradictions. Banishing inconsistency is possible in science because (or if) there is only one coherent world. Belief in one coherent world and one grand unified theory is the modern secular version of the ancient monotheistic intuition of one universal God (in which saints tend to believe). Uncertainty thrives in the realm in which scientists and saints have not yet completed their tasks (perhaps because they are incompletable). For instance, we must entertain a wide range of conflicting conceptions when we do not yet know how (or whether) quantum mechanics can be reconciled with general relativity, or Pippi's strength reconciled with the limitations of physiology. As Henry Adams wrote:

"Images are not arguments, rarely even lead to proof, but the mind craves them, and, of late more than ever, the keenest experimenters find twenty images better than one, especially if contradictory; since the human mind has already learned to deal in contradictions."

The very idea of a rigorously logical theory of uncertainty is startling and implausible because the realm of the uncertain is inherently incoherent and contradictory. Indeed, the first uncertainty theory - probability - emerged many centuries after the invention of the axiomatic method in mathematics. Today we have many theories of uncertainty: probability, imprecise probability, information theory, generalized information theory, fuzzy logic, Dempster-Shafer theory, info-gap theory, and more (the list is a bit uncertain). Why such a long and diverse list? It seems that in constructing a logically consistent theory of the logically inconsistent domain of uncertainty, one cannot capture the whole beast all at once (though I'm uncertain about this).

A theory, in order to be scientific, must exclude something. A scientific theory makes statements such as "This happens; that doesn't happen." Karl Popper explained that a scientific theory must contain statements that are at risk of being wrong, statements that could be falsified. Deborah Mayo demonstrated how science grows by discovering and recovering from error.

The realm of uncertainty contains contradictions (ostensible or real) such as the pair of statements: "Nine year old girls can lift horses" and "Muscle fiber generates tension through the action of actin and myosin cross-bridge cycling". A logically consistent theory of uncertainty can handle improbabilities, as can scientific theories like quantum mechanics. But a logical theory cannot encompass outright contradictions. Science investigates a domain: the natural and physical worlds. Those worlds, by virtue of their existence, are perhaps coherent in a way that can be reflected in a unified logical theory. Theories of uncertainty are directed at a larger domain: the natural and physical worlds and all imaginable (and unimaginable) other worlds. That larger domain is definitely not coherent, and a unified logical theory would seem to be unattainable. Hence many theories of uncertainty are needed.

Scientific theories are good to have, and we do well to encourage the scientists. But it is a mistake to think that the scientific paradigm is suitable to all domains, in particular, to the study of uncertainty. Logic is a powerful tool and the axiomatic method assures the logical consistency of a theory. For instance, Leonard Savage argued that personal probability is a "code of consistency" for choosing one's behavior. Jim March compares the rigorous logic of mathematical theories of decision to strict religious morality. Consistency between values and actions is commendable says March, but he notes that one sometimes needs to deviate from perfect morality. While "[s]tandard notions of intelligent choice are theories of strict morality ... saints are a luxury to be encouraged only in small numbers." Logical consistency is a merit of any single theory, including a theory of uncertainty. However, insisting that the same logical consistency apply over the entire domain of uncertainty is like asking reality and saintliness to make peace.

Sunday, December 11, 2011

Picking a Theory is Like Building a Boat at Sea

"We are like sailors who on the open sea must reconstruct their ship
 but are never able to start afresh from the bottom." 
Otto Neurath's analogy in the words of Willard V. Quine

Engineers, economists, social planners, security strategists, and others base their plans and decisions on theories. They often argue long and hard over which theory to use. Is it ever right to use a theory that we know is empirically wrong, especially if a true (or truer) theory is available? Why is it so difficult to pick a theory?

Let's consider two introductory examples.

You are an engineer designing a robot. You must calculate the forces needed to achieve specified motions of the robotic arms. You can base these calculations on either of two theories. One theory assumes that an object comes to rest unless a force acts upon it. Let's call this axiom A. The other theory assumes that an object moves at constant speed unless a force acts upon it. Let's call this axiom G. Axiom A agrees with observation: Nothing moves continuously without the exertion of force; an object will come to rest unless you keep pushing it. Axiom G contradicts all observation; no experiment illustrates the perpetual motion postulated by the axiom. If all else is the same, which theory should you choose?

Axiom A is Aristotle's law of inertia, which contributed little to the development of mechanical dynamics. Axiom G is Galileo's law of inertia: one of the most fruitful scientific ideas of all time. Why is an undemonstrable assertion - axiom G - a good starting point for a theory?

Consider another example.

You are an economist designing a market-based policy to induce firms to reduce pollution. You will use an economic theory to choose between policies. One theory assumes that firms face pure competition, meaning that no single firm can influence market prices. Another theory provides agent-based game-theoretic characterization of how firms interact (without colluding) by observing and responding to price behavior of other firms and of consumers.

Pure competition is a stylized idealization (like axiom G). Game theory is much more realistic (like axiom A), but may obscure essential patterns in its massive detail. Which theory should you use?

We will not address the question of how to choose a theory upon which to base a decision. We will focus on the question: why is theory selection so difficult? We will discuss four trade offs.

"Thanks to the negation sign, there are as many truths as falsehoods;
we just can't always be sure which are which." Willard V. Quine

The tension between right and right. The number of possible theories is infinite, and sometimes it's hard to separate the wheat from the chaff, as suggested by the quote from Quine. As an example, I have a book called A Modern Guide to Macroeconomics: An Introduction to Competing Schools of Thought by Snowdon, Vane and Wynarczyk. It's a wonderful overview of about a dozen theories developed by leading economic scholars, many of them Nobel Prize Laureates. The theories are all fundamentally different. They use different axioms and concepts and they compete for adoption by economists. These theories have been studied and tested upside down and backwards. However, economic processes are very complex and variable, and the various theories succeed in different ways or in different situations, so the jury is still out. The choice of a theory is no simple matter because many different theories can all seem right in one way or another.

"The fox knows many things, but the hedgehog knows one big thing." Archilochus

The fox-hedgehog tension. This aphorism by Archilochus metaphorically describes two types of theories (and two types of people). Fox-like theories are comprehensive and include all relevant aspects of the problem. Hedgehog-like theories, in contrast, skip the details and focus on essentials. Axiom A is fox-like because the complications of friction are acknowledged from the start. Axiom G is hedgehog-like because inertial resistance to change is acknowledged but the complications of friction are left for later. It is difficult to choose between these types of theories because it is difficult to balance comprehensiveness against essentialism. On the one hand, all relevant aspects of the problem should be considered. On the other hand, don't get bogged down in endless details. This fox-hedgehog tension can be managed by weighing the context, goals and implications of the decision. We won't expand on this idea since we're not considering how to choose a theory; we're only examining why it's a difficult choice. However, the idea of resolving this tension by goal-directed choice motivates the third tension.

"Beyond this island of meanings which in their own nature are true or false
lies the ocean of meanings to which truth and falsity are irrelevant." John Dewey

The truth-meaning tension. Theories are collections of statements like axioms A and G in our first example. Statements carry meaning, and statements can be either true or false. Truth and meaning are different. For instance, "Archilochus was a Japanese belly dancer" has meaning, but is not true. The quote from Dewey expresses the idea that "meaning" is a broader description of statements than "truth". All true statements mean something, but not all meaningful statements are true. That does not imply, however, that all untrue meaningful statements are false, as we will see.

We know the meanings of words and sentences from experience with language and life. A child learns the meanings of words - chair, mom, love, good, bad - by experience. Meanings are learned by pointing - this is a chair - and also by experiencing what it means to love or to be good or bad.

Truth is a different concept. John Dewey wrote that

"truths are but one class of meanings, namely, those in which a claim to verifiability by their consequences is an intrinsic part of their meaning. Beyond this island of meanings which in their own nature are true or false lies the ocean of meanings to which truth and falsity are irrelevant. We do not inquire whether Greek civilization was true or false, but we are immensely concerned to penetrate its meaning."

A true statement, in Dewey's sense, is one that can be confirmed by experience. Many statements are meaningful, even important and useful, but neither true nor false in this experimental sense. Axiom G is an example.

Our quest is to understand why the selection of a theory is difficult. Part of the challenge derives from the tension between meaning and truth. We select a theory for use in formulating and evaluating a plan or decision. The decision has implications: what would it mean to do this rather than that? Hence it is important that the meaning of the theory fit the context of the decision. Indeed, hedgehogs would say that getting the meaning and implication right is the essence of good decision making.

But what if a relevantly meaningful theory is unprovable or even false? Should we use a theory that is meaningful but not verifiable by experience? Should we use a meaningful theory that is even wrong? This quandary is related to the fox-hedgehog tension because the fox's theory is so full of true statements that its meaning may be obscured, while the hedgehog's bare-bones theory has clear relevance to the decision to be made, but may be either false or too idealized to be tested.

Galileo's axiom of inertia is an idealization that is unsupported by experience because friction can never be avoided. Axiom G assumes conditions that cannot be realized so the axiom can never be tested. Likewise, pure competition is an idealization that is rarely if ever encountered in practice. But these theories capture the essence of many situations. In practical terms, what it means to get the robotic arm from here to there is to apply net forces that overcome Galilean inertia. But actually designing a robot requires considering details of dissipative forces like friction. What it means to be a small business is that the market price of your product is beyond your control. But actually running a business requires following and reacting to prices in the store next door.

It is difficult to choose between a relevantly meaningful but unverifiable theory, and a true theory that is perhaps not quite what we mean.

The knowledge-ignorance tension. Recall that we are discussing theories in the service of decision-making by engineers, social scientists and others. A theory should facilitate the use of our knowledge and understanding. However, in some situations our ignorance is vast and our knowledge will grow. Hence a theory should also account for ignorance and be able to accommodate new knowledge.

Let's take an example from theories of decision. The independence axiom is fundamental in various decision theories, for instance in von Neumann-Morgenstern expected utility theory. It says that one's choices should be independent of irrelevant alternatives. Suppose you are offered the dinner choice between chicken and fish, and you choose chicken. The server returns a few minutes later saying that beef is also available. If you switch your choice from chicken to fish you are violating the independence axiom. You prefer beef less than both chicken and fish, so the beef option shouldn't alter the fish-chicken preference.

But let's suppose that when the server returned and mentioned beef, your physician advised you to reduce your cholesterol intake (so your preference for beef is lowest) which prompted your wife to say that you should eat fish at least twice a week because of vitamins in the oil. So you switch from chicken to fish. Beef is not chosen, but new information that resulted from introducing the irrelevant alternative has altered the chicken-fish preference.

One could argue for the independence axiom by saying that it applies only when all relevant information (like considerations of cholesterol and fish oil) are taken into account. On the other hand, one can argue against the independence axiom by saying that new relevant information quite often surfaces unexpectedly. The difficulty is to judge the extent to which ignorance and the emergence of new knowledge should be central in a decision theory.

Wrapping up. Theories express our knowledge and understanding about the unknown and confusing world. Knowledge begets knowledge. We use knowledge and understanding - that is, theory - in choosing a theory. The process is difficult because it's like building a boat on the open sea as Otto Neurath once said.