Here's Mises:

Case probability means: We know, with regard to a particular event, some of the factors which determine its outcome; but there are other determining factors about which we know nothing.

Case probability has nothing in common with class probability but the incompleteness of our knowledge. In every other regard the two are entirely different...

Case probability is a particular feature of our dealing with problems of human action. Here any reference to frequency is inappropriate, as our statements always deal with unique events which as such — i.e., with regard to the problem in question — are not members of any class. We can form a class "American presidential elections." This class concept may prove useful or even necessary for various kinds of reasoning, as, for instance, for a treatment of the matter from the viewpoint of constitutional law. But if we are dealing with the election of 1944 — either, before the election, with its future outcome or, after the election, with an analysis of the factors which determined the outcome — we are grappling with an individual, unique, and nonrepeatable case. The case is characterized by its unique merits, it is a class by itself. All the marks which make it permissible to subsume it under any class are irrelevant for the problem in question...

On the eve of the 1944 presidential election people could have said:

1. I am ready to bet three dollars against one that Roosevelt will be elected.

2. I guess that out of the total amount of electors 45 million will exercise their franchise, 25 million of whom will vote for Roosevelt.

3. I estimate Roosevelt's chances as 9 to 1.

4.I am certain that Roosevelt will be elected...

[Statement 3] is a proposition about the expected outcome couched in arithmetical terms. It certainly does not mean that out of ten cases of the same type nine are favorable for Roosevelt and one unfavorable. It cannot have any reference to class probability. But what else can it mean?

It is a metaphorical expression. Most of the metaphors used in daily speech imaginatively identify an abstract object with another object that can be apprehended directly by the senses. Yet this is not a necessary feature of metaphorical language, but merely a consequence of the fact that the concrete is as a rule more familiar to us than the abstract. As metaphors aim at an explanation of something which is less well known by comparing it with something better known, they consist for the most part in identifying something abstract with a better-known concrete. The specific mark of our case is that it is an attempt to elucidate a complicated state of affairs by resorting to an analogy borrowed from a branch of higher mathematics, the calculus of probability. As it happens, this mathematical discipline is more popular than the analysis of the epistemological nature of understanding.

Odds are, this is going to be a long day, which defies both class & case probability, as all days are the same 24 hours.

ReplyDeleteSilver and Krugman did successful predict the downfall of Chick-fil-A though. So, I give them credit for that.

ReplyDeleteSomething going on in the various markets...Dow up 160, gold spiking and eliminating its loss.

ReplyDeletekitco.com/market

Mises is not the final word on the nature and utility of probability. "Subjective" probability is an individual, and well, subjective, expression of an individual regarding the likelihood of an outcome whose value is unknown to that individual. It can be useful when complex events are decomposed into simpler events for which one can more readily assign probabilities and there are decisions to be made (my academic field 'decision analysis').

ReplyDeleteA (subjective) probability is a summation of a state of knowledge. I flip and coin and look at it and see it's 'heads', hiding it from you. Our states of knowledge differ, so the probabilities we assign of 'heads' differ. You see that I flipped a weighted coin you had on your desk which comes up 'heads' 70% of many trials, so your probability of heads is 70%.

Then I ask you if you would rather receive $1000 if the coin has come up 'heads' or if Obama is inaugurated for a second term, and you have real trouble deciding, in fact, for this purpose you are virtually indifferent (let's say that means that you'll let me choose which event your prize depends on without getting into further discussion on whether indifference can actually be revealed).

If we are allowed to say that your indifference here means your probabilities based on your states of knowledge of these two events, coin and election, are equal, then we have used a class probability, the coin where heads and tails appear in the long run 70% and 30%, to elicit your case probability, the outcome of a unique election.

One of the wrong turns taken in this field long ago was to equate probability with class probability, where statistics are a guide. Class probability is really frequency, and if the only information or knowledge you have is an applicable frequency, then a probability based on that knowledge must be equal to the frequency. Case or subjective probability is the same process of going from knowledge to likelihood, just without the guide of frequency. In this sense *all* probability is subjective probability.

As an example of this, consider being offered the choice of three curtains, with a desirable prize behind one in a game show produced only once, with a host you don't know and with zero knowledge of how they chose which curtain to place the prize behind. This is not class probability. There are no statistics to guide you. You could try to create classes mentally, running the game many times in your mind, and such a mental construct might be useful, but does not change anything.

There is one thing about the situation that can be used though, and that is its *symmetry*. No curtain is distinguished in any way from the others. Technically we say the curtains are invariant to relabeling (if '1', '2', and '3' were hung on the curtains and you choose '1', you have no rational basis for objecting if they switch '1' and '2'). Therefore none of the three probabilities should be different from the others, and since they must add to 1, each must be 1/3 to accurately describe your state of knowledge.

It's been a while since I have thought about these things, but if anyone would like references I could dig some up. Write me at ruby8333 at gmail dot com.

ScottO, is it inappropriate to say you gave me a boner?

DeleteWow. And all I got was a Ph.D.

DeleteAs we have seen over the years with Krugman, pretty much the opposite of whatever he predicts ends up happening, so him saying Obama will win is the best news in a long time for Romney.

ReplyDeleteWhat about 9:1 odds in a horse race? Surely any given race is a 1-time case. acknowledging that this is a handicapping system, isn't there an implied case probability with real meaning?

ReplyDeleteMy apologies for, and please excuse, any imprecise terminology.

my point is that odds on a case, contrary to answer (3) above, are not necessarily metaphorical but may have direct probability implications. consideralso the implied probability of default based on CDS markets

If the race is under the same conditions, and the same horses, it could fit into a class of "horse races under X conditions".

DeleteEven if it did not, the odds are based on a willingness to bet, and are not necessarily indicative of an underlying probability.