contact@onbelief.orgTossing Coins and Solitary Events
(Stable Proability vs. Uniqueness)Tossing a Coin
A very lucid description of the distinction between observation as 'objective fact' and the subjectivity of probability of belief is given by C M. Caves and colleagues (see further reading). They suggest we consider the types of observational outcomes, i.e. categories of observational possibilities, of tossing a coin. We can either observe a heads or tails result. From such observation we can formulate logical beliefs that we hold to be true. Logical beliefs by the definition used on this site can be held to be true or false. In the simple situation of coin tossing a belief can easily be constructed from the geometric 2-sided symmetry of the coin alone. This is different from the probability of having a certain number of heads or tails observations after a number of throws, which is the associated probabilistic belief. We can formulate a belief in the probability of a particular outcome frequency given the observational categories and a number of events.
In the case of the unbiased coin there should be an equal or symmetrical probability of both heads and tails. However in this situation Caves and colleagues argue that "even probabilities that follow from symmetry arguments are subjective, because the symmetry argument is applied to the probabilities, not to facts." The word 'fact' could perhaps better be substituted with category of observational possibilities. They go on to state that "a subjectivist interpretation of probability is natural in a deterministic world, where the outcome of any observation can be predicted with certainty given sufficient initial information. Probabilities then simply reflect an agent’s ignorance." Ignorance could be defined in that context as the unknowablity of certainty.
One extremely important concept in Bayes theorem and in relation to all matters of probabilistic belief is our 'previous' knowledge, or in more formal language the posterior probability. In the case of tossing a coin, the posterior probability is the previous 'knowledge' that a 'fair' coin will on average give 50% heads and 50% tails. If a coin looks fair then we will treat it as fair. What is meant by this statement? We can formulate a belief about the out come of tossing a particular coin by
1) memory of previous coin tossing exercises
2) direct observation of tossing the coin for a small sample number of times to observe what can happen (for example 3 times)
3) logical (or mathematical) inference.
Constructing a logical belief from observation of coins
In the case of inference we first of all learn to ignore extremely unlikely events such as the coin falling and coming to rest on the edge. We observe that there are two sides. We therefore logically infer that the coin can either fall on one side or the other. By using geometry, we can argue from the symmetry of the 2-sided coin that there will be one of 2 outcomes. If we were instead rolling a dice using geometry we would infer a 1 in 6 chance of a particular side. In the case of the coin we can construct the logical or mathematical belief that there should be 50% head and 50% tails. In other words before undertaking an exhaustive trial of tossing that particular coin we can formulate a belief. Alternatively after tossing the coin we can formulate a belief. In Bayesian terms these are equivalent. In Bayesian theory this belief is known as the posterior probability although technically it does not refer to posterior in the sense of time. In fact 'posterior' means independent.
Testing the coin
Let us then proceed to test our belief of 50% heads and 50% tails. If after 1,001 tosses we found 500 tails and 501 heads we would then treat our 50% logical belief as true. However what would happen if we had run a different test with 500 tosses and found that on 60% of the outcomes were heads for a coin we suspected might be biased. We are then left with two choices we can say that this was either evidence of bias in the coin or tossing process or conclude that it was just an extremely unlikely outcome. There is really no way to distinguish these possibilities. It is merely a matter of belief, or more precisely strength of belief. In the language of Caves and colleagues we are "ignorant" of the true state of objects or processes in the world.
We then might repeat the 500 tosses on the reasonable expectation of a 60% heads outcome or a 50% outcome. If the second 500 tosses gave a similar result we would be more likely to predict the 60% if we were to try a third time. The process might then continue so that before a 10th set of 500 tosses we might become completely convinced that the 60% outcome was more likely to occur. In other words we would have revised our posterior probability estimate.
Testing Many Coins: Stochastic Behaviour and Stable Probability Versus Uniqueness
What would happen if you were to throw 100 coins in the air. You might reasonably expect that there would be an approximate 50:50 distribution of heads and tails when they landed. If you were to repeat the procedure the expectation would be that the average would converge to that ratio. If you were to repeat the procedure in 10 years time you would expect the same result. In other words there would be a stable probability expectation. In this case the expectation would be based on the belief that each coin having the same geometry will behave in the same way. The average probability for the group of 100 would be the same as that for the individual coins.
Radioactive decay is an example of this kind of stochastic behaviour where a given amount of radioactive nuclei will on average produce an expected number of decays in a fixed time period by a property described as half-life. We cannot know which particular half of the nuclei will decay just that there is a stable probability that only half of the radioactive atoms will exist in the same state after a characteristic period of time for that isotope. For Carbon-14 you might recall it will take 5,730±40 years for half of the nuclei to decay. From a macroscopic point of view there is stochastic behaviour at work. Why one nucleus and not another is a mystery only addressable by quantum theory. However knowledge about the probabilistic behaviour of the population of atoms is nevertheless extremely useful. Indeed it is only because of the random behaviour within a stable probability envelope that Radiocarbon dating is possible.
In the natural world, where individual entities and events vary considerably in their characteristics, a different kind of probabilistic meta-stability can also be observed, at least in the short term. Unlike coins, humans vary considerably in their characteristics. Nevertheless the average human behaviour or characteristic can be meaningful and very real to us. By asking, am I a short fat person or a tall skinny person I am comparing my self to current averages. Am I larger or smaller than the average? In a biological context variability can also have a temporal component for one individual. Am I hotter or colder than I was yesterday?
Solitary Events
Where the solitary event or single unique individual becomes the subject of our analysis the average is less useful. Indeed where the diversity of individual agents or events is of interest to us average behaviour is less relevant than the possible outcome range within which the individual is likely to operate. Certainty is replaced by a probability envelope and a degree of uncertainty. Variation, or more technically variance, is then of itself important.
When unique, or almost unique, events occur a different approach is required to promote understanding. Instead of merely examining frequencies within populations of related individuals or events we begin to consider the meaning of factors contributing to a particular event or to the nature of that individual's makeup. In so doing we form judgments, imprecise beliefs or expectations using observationally based knowledge or logic. We enter the world of subjectivity. This way of understanding the world should not be thought of as inferior to the analysis of stable probabilities, for by definition functionally complex systems with some element of uniqueness, such as the weather or human beings, cannot be completely understood in terms of frequency analysis alone. Frequency analysis is of course very useful and should, wherever possible, be used in conjunction with understanding reached by other means. It this were a January day in Glasgow, Scotland for example it would probably be sensible not to plan a barbecue 2 days from now. In this situation we might know something about the situation not directly related to the probability of movement of weather systems. We might know about the pattern of the seasons and thus know it would be better to hold our outdoor event in summer. We might even know something about the earth's behaviour in space and have a theoretical understanding of why our seasons exist. However if there was, by chance, a very large low pressure weather system in the North East Atlantic a probabilistic analysis of meteorological data might tell us we could expect 2 cm of rain fall and so add to our expectations. Both types of analysis are complementary to each other.
Probability and Human Uniqueness
On a more human scale the individual uniqueness of our lives can seem all important. It is reasonable for me to be sure that you will not reach the age of 111 years before dying. However I might be wrong you might die aged 121. I might then try to analysis your life for a serious of risk factors where some probabilistic risk analysis has been undertaken in the population at large. I might discover that you were wealthier than average and more intelligent. Your personality was such that you were more adverse to risk taking behaviours and so you did not smoke cigarettes, or were not sexually promiscuous and so on. I might discover that you had not lived through periods of war and famine and so discover many factors that were apparently independent of your personally unique characteristics. By carrying out an analysis of your DNA I might find a unique mutation in your immune system that made you resistant to infection and malignancy. However in the end it is likely that I would always have uncertainty about the reason for your relative longevity. Even given an extremely exhaustive analysis of your life and your personal characteristics I might never discover any conclusive set of reasons. I might ultimately be left with a degree of belief that you were in some way different as an individual. On the other hand I might be wrong to search for uniqueness in you. You might be average in many ways but you might have survived to a great age purely by chance. Our understanding of outcome in all complex and many apparently simple systems will remain incomplete. A pretence of certainty in the face of stochastic behaviour at a population or group level and uniqueness at an individual level is both unhelpful and undesirable.
An alternative frequentist view of probability is still based on ignorance
Another way of examining probability is claimed by those who take an exclusively 'frequentist' approach. They consider the idea of an upper and lower probability of a particular frequency of events occurring in a particular sample or 'sample space'. By that method of analysis the analyser wants to know how representative that particular sample of observations is of all possible samples. In that way the observer refers to uncertainty of action (i.e. sampling or coining tossing) rather than uncertainty of belief. For example, the observer cannot be sure how representative the first sample of 500 coin tosses actually is of the first 50,000. Both ways of thinking lead to a conclusion of fundamental uncertainty either through ignorance of action or ignorance of belief.
C. A. Fuchs, and R Schack, in Subjective probability and quantum certainty, 2007 (available as a PDF file also see Resource material for promoting the Bayesian view of everything by Caves)
On the Nature of Belief
www.onbelief.org
Scotland, 12th October 2007 and thereafter
Copyright 2007 onwards
