Randomness
Is randomness a state of mind?
---------------------------------------------------------------------------------
------------------------------------------------
Over the centuries, men of all creeds have had a lot to say about randomness -- an idea also known as chaos -- about which there would seem to be little that one could say.
Either term -- randomness or chaos -- refers to matters that modern scientists, too, have much to say, despite the fact that chaos is non-rational -- that's why random processes are called chaotic, wouldn't you know! One would think there wouldn't be much one could say about chaos. But chaotic or random variations appear to be as common in our world as rational processes, perhaps even more prevalent. And that of course would be justification enough for the expenditure of time and energy on the discussion, whether or not anything rational could be said -- whether there could be any information about it.
Here's what Peak and Frame have to say about the matter, in their book:
...You probably have an intuitive feeling that randomness implies unpredictability. It does, but that's not enough; many processes are unpredictable, yet we wouldn't call them random. For example, the first time we see a movie, we can't predict what the actors will say next. By the hundredth time, though, the dialog contains no surprises at all. What the actors say is determined by rules -- a script -- that we eventually learn. [Here] we take a narrow definition of random: unpredictable even in principle. In the absence of underlying rules, the most compact description of a random sequence of events is the sequence itself; no shorter description is possible.
Peak and Frame note that randomness doesn't mean "anything goes." You have to stay within the context of the randomness. When tossing a coin, for instance, random only refers to the occurrence of heads or tails, not the occurrence of sunspots, say, or whether a ball player will hit the next pitch.
We might say that random variation is an act, process, or result of varying something in a way that has no pattern, purpose, or objective. Or we could say it's a slight difference of something in a group that has the same chance of occurring as another difference in the group.
Is that clear enough for you? Will you now be able to see for yourself these random changes that occur in your world of experience? Will you be able to recognize that a change you witnessed was one of a batch of equally probable changes?
Not likely!
Indeed, is random variation, or randomness, generally, an observable phenomenon, like the colors and sounds you experience?
Or is it something you can only impose on your world, like a model, a presupposition, or a way of seeing things -- a rule for looking? My choice is the latter, that randomness is more in the nature of an invented description of things or events than an empirical fact.
You might note that Roland Omnes, in his Quantum Philosophy, says quantum events occur at random. "No cause is at work to make an excited atom decay at some specific moment." The laws that govern the process "only express the probability of the event taking place at one time rather than another."
-----------------------------------------------
Whether we sink or stay afloat here, depends on how we understand 'random' and 'variable.' We've looked at random and variation, so now let's see what the noun, variable, has to offer.
According to the dictionary, 'variable' expresses having a likelihood for change -- of not being constant, or of being subject to variation. A random variable would therefore be something specific -- a particular property, for instance -- that's subject to change in a non-patterned way, or in a completely arbitrary manner, chaotically. In symbolic form a variable is usually stated in terms of letters from the end of the alphabet, mainly x, y, or z.
Here, again, can you determine whether something in your experience is changing in a way completely without regularity? Does that something qualify for the 'aimless' tag? If so, it's behaving as a random variable, as something having no fixed quantitative or qualitative value and subject to change in a completely non-patterned way. In this connection you might like to take a look at quantum mechanics. Or the book by Edward Beltrami.
---------------------------------------------
A sample, discussed here, is a representation of a population. It's part of a whole, so to speak -- a mirror fragment that reflects the whole image in some way. It's almost like self-referencing, or self-reflecting, but not quite.
Say you identify the salaries of 1000 members of your golf club (Ha!) as the particular class of items you'd like to investigate. Then the salaries of 1000 members make up your population of interest. A sample from the population might then be a group of 10 of the members drawn from the population in some way. The sample could be selected according to specific rules, or the sample could be taken quite arbitrarily, involving no rule or guiding principle -- chaotically but not chaotic. But, if the sample can be drawn in such a way that any other sample of the members drawn in the same way would be equally likely to occur, you would have a random sample -- a carefully randomly selected part of the whole. Does it sound strange?
---------------------------------------------
When you design an experiment to do a random sampling, you need to have some idea what kind of observations you're going to make and what outcomes you could possibly get. This is to say you have to understand your population. You do the experiments to understand your population more fully or to be able to make predictions in given circumstances.
But you can't extend your information or make predictions if you don't know what's possible. In other words, you need to know beforehand all the things that could happen when you take a sample. In this sense, random sampling can't be chaotic. You need to have a sense of what statisticians call the sample space -- the set of possible outcomes. Lacking this rational organization, your results would be pretty haphazard. As an example, you might look at the space of all combinations of values of matrices used to define transformations in neural networks. Or the space of possible outcomes of a measurement in quantum mechanics.
---------------------------------------------
To say that a sample taken from a given population is a random sample is to say that any other sample of the same type from the population has an equal chance of occurring in the sampling.
For instance, in a sample of cards from a regular 52-card deck, and a sample size of one, all of the cards in the deck have the same chance of occurring -- the same probability -- assuming that each card is replaced in the deck after a sample is taken. The frequency distribution of the cards assures you of this outcome, because there is only one of everything in your population. The deuce of hearts, for instance, would be just as likely to occur as, say, the ace of diamonds, or any other card in the deck. That's the basic idea underlying what's known as an honest deck -- no card in the deck has a special advantage.
But what's the chance of actually drawing a specific card? Say the three of spades? Is it the same as drawing a random sample from another population? Like the one-member sample of the numbers from 1 to 100?
Since the samples from each of the two populations have the same chance of occurring, you might think, too hastily, that the chances themselves are the same in the two groups. Should that thought have occurred to you, let me hasten to say they're not the same. The chance of getting the deuce of clubs, for instance, or any of the other 51 cards in the deck, is one in fifty two. But the chance of seeing a specific number in the 1 to 100 set, like 37, is only one in a hundred, as it is for any other one of the hundred numbers. Here, again, it's the flat frequency distribution, plus the assumption of a random selection, that promises the result.
Equal likelihood only assures you that samples are random. It doesn't give you the numerical possibility of getting any one of the samples. For that value you need the probability -- which is about as clear as randomness! And we haven't really said how to generate a random selection process. The model only assumes the randomness.
---------------------------------------------
The problem you have dealing rationally with randomness is that you have to apply rational rules of procedure, or principles. But how can you approach irrationality in a rational manner? It seems impossible, somehow. The prospect itself doesn't seem rational. It's like saying rational things about chaos.
Even so, with the mathematician, Laplace, chaos became subject to mathematical science and thus the laws of games of chance were created. The laws -- the math model -- took the form of patterns of chance behavior -- of rules of behavior -- an example of which is that the toss of a coin leads eventually to the occurrence of an average, like getting heads half the time.
If you toss the coin a number of times, you can compute an average.
This work led to the distinction between the theoretical or mathematical nature of randomness and the empirical or physical nature of randomness. On the one hand there is the problem of defining randomness -- or probability -- in acceptable mathematical terms. And on the other there is the matter of applying the mathematical model or deciding whether it deals adequately with physical events -- i.e., whether the events obey the mathematical criteria for randomness.
The difficulty of defining randomness is that a successful definition may render the randomness non-random. Yet you need a precise definition of randomness in order to establish that a sequence of events is truly random.
--------------------------------------------
What information can you get from random processes? To be of any value, statistics must generate information, but randomness would hardly seem to qualify as such a source. If there is information, where might it reside?
First off, what is information and how is it obtained?
As I developed it in Simulations for Skills Training, information is the meaningful product of rational structuring. Simply put, we get information, or meaning, about the world when we can express something in a normal sentence. In math the information is expressed via equations, which are information forms -- structures through which information is defined. The equations provide what I refer to as repositories of information. These are the collections of sentences that can be derived from interpreting the equations and giving them specific property values.
Statistics, like math, deals with these matters in equation form, so the question comes down to identifying the equations. What you deal with in statistics are populations of properties of the world and the samples that you take from the populations. Your task is to use the sample data to infer something -- to draw a conclusion -- about the population. For instance, you deal with characteristics like the central location of the data or its dispersion. It is the equations that relate the various properties that are the sources of information, and it is the samples that provide the data to fill in the equation blanks -- the variables.
------------------------------------------------
Bart Kosko doesn't like randomness, or at least he doesn't like probability, which for him is the same thing. He says so in his book, Fuzzy Thinking: The New Science of Fuzzy Logic. One reason he doesn't like it is because it covers up the fuzziness that we experience in the world, turning everything into black and white.
The upshot, for me, is that probability is a model, or set of rules, that we use to contend with situations in the real world about which we have scattered knowledge, at best.
-----------------------------------------