Monday, December 13, 2010

A Treatise on Probability, Chapter 3

Must all probabilities be expressed as quantities?
All this talk about ‘degrees of belief’ might lead one to wonder how to measure such degrees (assuming they can be measured). Are they counted or weighed? Keynes begins by surveying a few other opinions on the matter. One opinion is that a numerical comparison of two probabilities is possible, though in practice it it may be impossible.  Much like counting the number of cells in the human body…we assume that there is such a number even though we don't know it. Some things are weighed and others are counted, but this is merely a limitation on our knowledge.
Another view defines probability numerically and thus assumes that every probability can be represented by a number. This is the popular understanding: probability as the number of favorable cases (the numerator) in relation to the total number of cases (the denominator). Keynes spends quite a few pages surveying some practical views (insurance underwriters and lawyers) and concludes that they “weaken rather than support the contention that all probabilities can be measured and estimated numerically.” He also points to scientific induction (theory confirmation).  Is there a number to represent how much each additional observation confirms a scientific theory?
"We are out for a walk—what is the probability that we shall reach home alive? Has this always a numerical measure? If a thunderstorm bursts upon us, the probability is less than it was before; but is it changed by some definite numerical amount? There might, of course, be data which would make these probabilities numerically comparable; it might be argued that a knowledge of the statistics of death by lightning would make such a comparison possible. But if such information is not included within the k knowledge to which the probability is referred, this fact is not relevant to the probability actually in question and cannot affect its value. In some cases, moreover, where general statistics are available, the numerical probability which might be derived from them is inapplicable because of the presence of additional knowledge with regard to the particular case.”
Can all probabilities be compared?
Having given a satisfactory reason to doubt that all probabilities are quantities, Keynes turns to the question of whether all probabilities can even be arranged in an order of magnitude (said to be greater or less than another). Here we get one of Keyne’s famed examples:
"Is our expectation of rain, when we start out for a walk, always more likely than not, or less likely than not, or as likely as not? I am prepared to argue that on some occasions none of these alternatives hold, and that it will be an arbitrary matter to decide for or against the umbrella. The barometer is high, but the clouds are black, it is not always rational that one should prevail over the other in our minds, or even that we should balance them,--though it will be rational to allow caprice to determine us and to waste no time on the debate."
He boils the whole debate down to 4 alternatives:
1. In some cases there is no probability at all
2. In some cases we can’t measure the difference between two probabilities (ie., we can’t say how our degrees of belief in two conclusions compare with one another)
3. #1 is ontologically false, but we are often unable to know the probability (epistemological limitations)
4. #2 is ontologically false, but we are often unable to know how to compare the probabilities
“We can through stupidity fail to make any estimate of a probability at all, just as we may through the same cause estimate a probability wrongly...if we do not limit it in this way and make it, to this extent, relative human powers, we are altogether adrift in the unknown; for we cannot ever know what degree of probability would be justified by the perception of logical relations which we are, and must always be, incapable of comprehending.”
Regarding #3--that quantifying probabilities is an epistemological problem—Keynes wonders if what we really mean is that some addition to our knowledge might then permit a numerical assessment. He doubts that this position is tenable and offers some counterexamples.  So he seems to reject #3 and #4.

Regarding #1-2, Keynes agrees that not all probabilities can be measured, yet in some cases we can say that one is lesser or greater without any precise comparison of magnitude…I take this to be a statement about ontology as opposed to #3-4.  So he seems to accept #1 and  reject #2.  So's he's going to give us a system that can model numerical, non-numerical, comparable, and non-comparable probabilities.

Quantitative properties of probability
Since Keynes is not in favor of simply defining the problem away, he explores a philosophical theory about the quantitative properties of probability. He says that magnitudes arise [I’m growing tired of all these things arising out of other things!] out of placing things in a an ordered series relative to certainty.


Now we might say there is a second probability that lies somewhere between the certainty and the first probability:


We can therefore say that P2>P1, but only if P1 and P2 can lie on the same ordered series with Certainty. But what if more than one distinct series of probabilities exist? Only those in the same series can be compared. “…neither of two probabilities, which lie on independent paths, bears to the other and to certainty the relation of ‘between’ which is necessary for quantitative comparison.”

Consider comparing two arguments for the same conclusion, but where one has an additional supporting premise that the other doesn’t. Clearly one lies nearer to certainty than the other, but can we numerically represent this?   But no all conclusions can be compared so clearly.

Compare these things:
Red book bound in morocco (RM)
Blue book bound in morocco (BM)
Red book bound in calf (RC)
Blue book bound in calf (BC)

We can say that RM is more similar to BM than RC
BM is more similar to RM than BC, etc.

What about BM, RM, and BC?  Or what can we say about the comparative similarity between {RM,BM} and {RM, RC}? They are equally similar with respect to each other. We can’t put them on the same ordered series and compare between Impossibility and Certainty. Thus “the same probability can belong to more than one distinct series.”

Ok, so probabilities fall between 0 and 1, and they can be placed on more than one series. So far, so good…then Keynes throws us a curve ball. “It is possible therefore, for paths to intersect and cross.”

Do what?! Did we just break into the second dimension?!?!

Intersecting Paths
ABC forms an ordered series, B lying between A and C
BCD forms an ordered series, C lying between B and D
Therefore, ABCD forms an ordered series, B lying between A and D
"It follows…that the probability represented by a given point is greater than that represented by any other point which can be reached by passing along a path with a motion constantly towards the point of impossibility, and less than that represented by any point which can be reached by moving along a path towards the point of certainty. As there are independent paths there will be some pairs of points representing relations of probability such that we cannot reach one by moving from the other along a path always in the same direction.”
Confusing? Have a look at this page.

The basic idea is that you may have numerical and non-numerical probabilities, and among those you may have some that are comparable and others that aren't.  You draw a 1-dimensional line to represent numerical probabilities. You put the numerical ones that can be compared on that line (in between 0 and 1).

Then you take your non-numerical probabilities and represent whatever comparative relationship you are able to determine among those points (based roughly on the x coordinate position, but drawn above the line). These non-numerical probabilities are plotted so that you can follow a line across and see which ones are comparative with respect to each other. If you can’t get from one point to another by any line, then they aren’t comparable. The y coordinate doesn’t seem to mean much of anything, it’s just a visual aid.  This is getting interesting!


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