Assume you have some system with variable x which is a dynamic system you know follows some law of nature but may be perturbed by some random process. Namely let:
dx(t)/dt = a x(t) + w(t)
and that you have some measurement of y(t) which relates to x(t) but may also have noise:
y(t) = b x(t) + n(t)
Here we assume that w(t) and n(t) are random processes and further we assume they are Wiener processes. Most likely they are not in reality, but who cares, this is an academic problem. Now we want the best possible estimator in a mean square sense of x(t) give that we have y(t) over some interval (s,t).
The answer is a Bayes least mean square estimator gotten by determining the conditional probability density, p(x,t|y; (s,t)). That approach was initially proposed by Kolmogorov and Wiener and then by Kalman. The nonlinear version was done by Stratonovich and, you guessed it, me.
The essence of a Bayesian world is that we are desiring to estimate some variable, say x, and we have a set of observations, say y over some time period, and, this is key, we know both the relationship between y the observation and x the system plus we know how the "noise" disturbs things.
The conditional probability described above is the Bayesian methodology. Now there is another way to consider this problem and also to add complexity, namely it is to allow for the Rowe Conjecture. Now Nick Rowe proposed this conjecture about four years ago as a simple premise. Namely, if one looks at some "Economics Law", in his case the Efficient Market Theory, one knows that this may or may not hold in reality. Thus there is a random process related to the law itself being extant. Also inherent to the EMT or Hypothesis, there is randomness to it being true or not. In addition, for Rowe both nature, namely the Economics law, and the people themselves may be random. This means people may "believe" that the EMT applies or does not apply. The people may be in one state or another and that means the people may act differently based upon their belief no matter which law is in action. Thus the combined system of law and people as a system description is itself totally random. It would be like us saying above that the system, as the EFT, is:
dx(t)/dt = a x(t) + w(t)
or
dx(t)/dt = d x(t) + w(t)
where we may or may not know a and d and further we may have (for the people):
y(t) = b x(t) + n(t)
or
y(t) = e x(t) + n(t)
It actually may be even more complex. But we will not consider that here.
Now we all know gravity does not work that way, nor does thermodynamics, nor even bridge design, but somehow it works in Economics, just look at the Nobel Prizes. Second, as Rowe conjectured, people act either believing this law holds or not believing it holds, and some fraction of the population may hold one view and some another. There is a whole lot of literature on this type of a world as well and even another Nobel Prize in this area.
Thus, using the Rowe Conjecture, we have cycles developed where the law may or may not hold and people may or may not believe in it. This then is one way to explain the Business Cycle, in part. Now there is no Nobel Prize in this statement. But its does provide insight into human behavior and the lack or consistency in economic "theory".
Along comes the Fellow from the South and presents his theory and this is what he says:
It seems to me that xxx position – he only said it was a danger, not that it would happen at any particular time, so it signifies nothing if it doesn’t happen even after four years have passed – is just untenable in its strong form. If saying that something is a danger carries no implications for the likelihood that it will actually occur, what is the point of saying it? You might as well stand up there and say “Nice day for weather” or sing “Mary had a little lamb.”
No, clearly talking about the danger of inflation was some kind of statement about probabilities – in particular, a statement that the probability of inflation is, according to the speaker’s model of the world, higher than it is in other peoples’ models of the world. And that means that actual events do or at least should matter – they may not prove that one model is wrong and another is right, but they should certainly affect your assessment of which model is more likely to be right.
In short, it’s a Bayesian thing.
Well not really. It is a Rowe Conjecture "thing" I believe. A Bayes "thing" is purely probabilistic about a well structured world. A Rowe Conjecture "thing" is a probabilistic structure about a probabilistic world. You see, the theory is just that a theory, and the "theory" has the tendency to change. Furthermore people may or may not believe the theory, especially after the past five years of ranting amongst Economists.
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