raghu wrote: > I think the point is that a much weaker ergodicity assumption is > actually required than commonly assumed. For e.g., it may be necessary > to assume that herding and mob psychology will be pretty much the same > in future as in the past. But it is not necessary to assume that stock > price distributions are stationary (let alone ergodic).
Actually, ergodic is weaker than stationary. Ergodicity only requires that the time joint probability distribution be measure preserving. Stationarity requires that some traits of the time joint probability distribution be somewhat stable over time. But, aside from that, what you say is exactly the point of updating your prior beliefs (probability distribution) in Bayesian inference. I've already written a lot of garbage about this EMH story. It's all in the PEN-L archives. I won't repeat that. Let me end with this instead: Take a portion of the universe and have it evolve in such a way that it reaches a point when it can reflect on the rest of the universe and on itself. Chances are the very nature of this reflection (call it, for the sake of a name, "social practice," of which "human cognition" is a humble subset) is going to run into paradoxes whenever you push things to the corners. When a piece of the universe swims, partially at least, against the stream of such an overarching law of the universe as the second law of thermodynamics, then (if you allow me a functionalist argument here) it will have to have such a quixotic but unstoppable impetus that it won't be satisfied until it takes over the whole universe including itself. Marx's formulation of human social practice as the appropriation of nature (including our own historically shifting human nature), which is the universal curse (or blessing) of human labor, is only a particular instance of this. We are silly if we think we are the first to notice these things. We're simply noticing them in the forms that appear novel to us. The story of thought running into paradoxes already made the name of a number of philosophers throughout human history. So, here's a Platonic conversation mixing amateurs with professionals: Keynes: I just realized that, in practice we tacitly fall back on the convention that today's state of affairs will continue indefinitely into the future, unless we have reasons to expect a change. Socrates: Wise words. Any rules as to how and when to anticipate a change? Rumsfeld: Hmm, not always. It depends. Certainly, there are known unknowns -- in other words, we can expect certain changes on the basis of our prior experience. But, in a sense, those changes are to be expected. They are in our probability space. They are not true surprises. However there are also.... Socrates: Wait... So, can we tacitly fall back on the convention that unanticipated changes will continue to pop up indefinitely into the future? Keynes: You're too smart, Socrates. If we can anticipate that unanticipated changes will happen, then they are not really unanticipated, are they? The problem is that, in the case of the former, we may have an actuarial basis to form our expectations. Not with the latter, because of... Soros: Reflexivity! Rumsfeld: Exactly! And, by the way, I was going to refer to this very thing. I was going to say that there are also unknown unknowns, events that we don't know we may have to face, because there is nothing in available data, known history, etc., cum current tools of historical interpretation, statistical inference, etc., that could prepare us for them. Ramsey: But a mathematical expectation is simply a function of probabilities, which in turn measure our degree of belief on the occurrence of events. If, with the information we currently have, an infinite range of events are implicitly deemed impossible -- are out of the probability space, have zero probability -- that doesn't mean they won't occur. It only means that we believe that they won't occur on the basis of what we now know. Capisci? Expecting human minds to anticipate what is impossible to anticipate is silly! Keynes: I hear echoes of my own 1921 treatise here. But you still seem not to see the point of my beauty contest metaphor? One thing is to predict that a meteorite will hit Mars and another one is to determine the rate of return on a long-term investment project. In the former, human choices don't affect the outcome. In the later, human choices affect the outcome, which is in turn used as an input to make those very choices in an infinite regress... Soros: Right! Reflexivity! Lorenz: And "observing" meteorites doesn't affect their trajectory? Ha! Tiny influences are not necessarily inconsequential. Tiny influences are inconsequential except when they are not. Ramsey: Keynes, I do see the point of your beauty contest. But what do you mean, that we can avoid beliefs about "unexpected" changes? What's the point of all this paraphernalia if not to orient ourselves in practice? Protagoras: Hey, I'm the proto-humanist here. "Humans are the measure of all things," remember? In that sense, I must take due credit. I anticipated Ramsey's (the so-called Bayesian) view of probability. But enough about me. I don't want you to think that I meant to say that "Protagoras is the measure of all things." No. My question to Keynes, Soros, Rumsfeld, or whomever else may wish to take a stab is this: How do you guys know with such certainty that there are unknown unknowns? If they really were unknown unknowns, we wouldn't even be able to name them "unknown unknowns." Naming a thing is the first step in getting to know it. So, do tell, how can you be so sure that there are unknown unknowns? Isn't it because you've extracted that signal from experience, historical evidence, data, etc. with your available tools of inference? If so, then how can you say that there's no basis to anticipate unanticipated changes? Isn't that expectation that unanticipated changes await us some basis to incorporate them into our decision making? [From this point on, the conversation runs in circles... and I must go look at the snow falling.] _______________________________________________ pen-l mailing list [email protected] https://lists.csuchico.edu/mailman/listinfo/pen-l
