Model of induction
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Re: Model of induction
 
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Re: Model of induction
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Re: Model of induction
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Hi Glen, I feel a bit like Nick says he feels when immersed in the stream of such erudite responses to each of your seemingly related, but thread-separated questions. As always, though, when reading the posted responses in this forum, I learn a lot from the various and remarkable ways questions can be interpreted based on individual experiences. Perhaps this props up the idea of social constructivism more than Platonism. So, if you can bear with me, my response here is more of a summary of my takeaways from the variety of responses to your two respective questions, with my own interpretations thrown in and based on my own experiences. Taking each question separately ... Imagine a thousand computers, each generating a list of random numbers. Now imagine that for some small quantity of these computers, the numbers generated are in n a normal (Poisson?) distribution with mean mu and standard deviation s. Now, the problem is how to detect these non-random computers and estimate the values of mu and s. Nick's question seems to be about how to determine non-random event generators from independent streams of reportedly random processes. This is not really difficult to do and doesn't require any assumptions about underlying probability distributions other than that each number in the stream is equally likely as any other number in the stream [i.e., uniformly distributed in probability space] and that the cumulative probability over all possible outcomes sums to unity: the very definition of a random variable ... a non-deterministic event--an observation--mapped to a number line or a categorical bin. A random variable has both mathematical and philosophical properties, as we have heard in this thread. For Nick's question, I think that Roger has provided the most practical answer with Marsaglia's Die Hard battery of tests for randomness. In my professional life, I used these tests to prepare, for example, a QC procedure for ensuring our hashing algorithms remained random allocators after each new build of our software suite. For example, a simple test called the "poker test" using the Chi-squared distribution could be used to satisfy Nick's question with the power of the test (i.e., reducing the probability of rejecting the null hypothesis of randomness when it is true; thus perhaps finding more non-random processes than really exist) increasing with larger sample sizes ... longer runs. So, does anyone here have an opinion on the ontological status of one or both probability and/or statistics? Am I demonstrating my ignorance by suggesting the "events" we study in probability are not (identical to) the events we experience in space & time? At the risk of exposing my own ignorance, I'll also say your question has to do with the ontological status of any random "event" when treated in any estimation experiments or likelihood computation; that is, are proposed probability events or measured statistical events real? For example--examples are always good to help clarify the question--is the likelihood of a lung cancer event given a history of smoking pointing to some reality that will actually occur with a certain amount of uncertainty? In a population of smokers, yes. For an individual smoker, no. In the language of probability and statistics, we say that in a population of smokers we expect this reality to be observed with a certain amount of certainty (probability). To be sure, these tests would likely involve several levels of contingencies to tame troublesome confounding variables (e.g., age, length of time, smoking rate). Don't want to get into multi-variate statistics, though. Obviously, time is involved here but doesn't have to be (e.g., the probability of drawing four aces from a trial of five random draws). An event is an observation in, say, a nonparametric Fisher exact test of significance against the null hypothesis of, say, a person that smokes will contract lung cancer, which we can make contingent on, say, the number of years of smoking. Epidemiological studies can be very complex, so maybe not the best of examples ... So, since probability and statistics both deal with the idea of an event--as your "opponent" insists--events are just observations that the event of interest [e.g., four of a kind] occurred; so I would say epistemologically they are real experiences with a potential (probability) based on either controlled randomized experiments of observational experience. But is a potential ontologically real? 🤔 Asking if those events come with ontologically real probabilistic properties is another, perhaps, different question? This gets into worldview notions of determinism and randomness. We tend to say that if a human cannot predict the event in advance, it is random ... enough. If it can be predicted based, say, on known initial conditions, then using probability theory here is misplaced. Still, there are chaotic non-random events that are not practically predictable ... they seem random ... enough. Santa Fe science writer and book author George Johnson gets into this in his book Fire in the Mind. I would just close with another comment, this time regarding Roger's recounting of Marsaglia's report on the issues with pseudo-random number generators. RANDU was used on mainframes for years but was subsequently found to be seriously flawed. If I remember correctly, the rand() function used in C applications was also found to be deficient. Likely, this is why we need these a battery of randomness tests to be sure. But there has been a great deal of research in this area and things have improved dramatically. There are even so-called true random number generators that "tap" into off-computer and decidedly random-event sources like atmospheric noise [or even quantum-level events]. But even here, some folks who's worldview see the universe as deterministic would say that these generators are not truly random either. Chaotic, yes. But, not random. I say, likely random enough. Finally, I would say that we can use number generators that are random enough for our own purposes. In fact, for running simulation models, say, to compare competing alternatives for decision support, we need to use pseudo-random number generators in order to be able to gain a sizable reduction in the (random) variance of the results. This would tend to sharpen up our test of significance in comparing the resulting output statistics as well. Kind of a fun topic. Hope this adds a little of its own sharpness to the discussion and doesn't just add variance. 🤔 If y' all deem not, I will expect some change from my $0.02. 🤐 Cheers, Robert W. On Tue, Dec 13, 2016 at 3:42 PM, Grant Holland <[hidden email]> wrote:
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove |
Re: Model of induction
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Re: Model of induction
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Re: Model of induction
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Administrator
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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In reply to this post by Grant Holland
Hi Glen, et al, Thanks for cashing mu $0.02 check. :-) When I wrote that "but it doesn't have to be" I wasn't asserting that probability theory is devoid of events. Events are fundamental to probability theory. They are the outcomes to which probability is assigned. In a nutshell, the practice of probability theory is the mapping of the events--outcomes-- from random processes to numbers, thus making the practice purposefully mathematical. And in this regard, we speak of a mathematical entity dubbed a random variable in order to carry out the calculus of probability and statistics. A random variable is like any other variable in mathematics, but with specific properties concerning the values it can take on. A random variable is considered "discrete" if it can take on only countable, distinct, or separate values [e.g., the sum of the values of a roll of seven die]. Otherwise, a random variable can be considered "continuous" if it can take on any value in an interval [e.g., the mass of an animal]. But, a random variable is a real-valued function--a one-to-one mapping of a random process to a number line. This is arguably as long about way of explaining [muck?!] why I said "but it doesn't have to be ... time." Time doesn't have to be involved such that the random variable does not have to distributed in time but often can be, such as in reliability theory--for, example, the probability that a device will survive for at least T cycles or months. Yes to your and Grant's notion that thinking in terms of probability spaces is a good way of thinking of probability and statistic and this mapping, as mathematically we are doing convolutions of distributions [spaces?] when modeling independent, usually identically distributed random trials [activities]. But, let's not confuse the mathematical modeling with the selection process of, say picking four of a kind from a deck of 52 cards. All we are interested in doing is mapping the outcomes--events-- to possibilities over which the probabilities all sum or integrate to no more than unity. The activity gets mapped in the treatment of the random variable in the mapping [e..g., the number of trials]. So, for example, rolling 6s six times in a row is not a function of time, but of six discrete, independent and identically distributed trials. For the computed probability, in this case, it doesn't matter how long it took to roll the dice six times. I am thinking that this is the way your "opponent" is thinking about the problem and suspect that he has been formally trained to see it this way. Not the only way but a classical way. When Eric talks about the historic difference between scientists, mathematicians, and statisticians practicing probability theory and statistics, these differences quickly disappeared when the idea of uncertainty bubbled up into the models found in the fields of physics, economics, measurement theory, decision theory, etc. No longer could the world be completely described by the classical system dynamic models. Maybe before Gauss even (the late-1700s), who was a polymath to be sure, error terms were starting to be added to their equations and had to be estimated. As to my language of "when" an event occurs with some calculated likelihood, it can be a description or a prediction. The researcher may be asking like Nick is [kind of?] asking in the other thread, what is the likelihood of my getting this many 1s in a row if the process is supposedly generating discrete random numbers between, say, one and five? In this case, a psychologically unexpected event has happened. Or in planning his experiment in advance, he may just want to set a halting threshold for determining that any machine that gives him the same N consecutive numbers in a row to be suspect. In that case, the event hasn't happened but has a finite potential for happening and we want to detect that if it happens ... too much. Those "events" don't _happen_. They simply _are_ This bit seems more philosophical than something a statistician would likely [no pun intended] worry about. Admittedly, my choice of words--throughout my post--could have been more precise, but I would not have said that "events simply are." When discussing the nature of time in a "block universe," maybe that could be said, but I would have been in Henri Bergson's corner [to my peril, of course] in the 1922 debate between Bergson and Albert Einstein on the subject of time. :-) Curiously, Bergson's idea of time is coming back--see Time Reborn (2013) by Lee Smolin. But this is likely not what you meant. However, you are an out-of-the-closet Platonist by your own admission. No worries; I have friends who are Platonists, most of them being mathematicians or philosophers or believe the brain to be a computer, but not typically computational scientists and certainly not cognitive scientists. :-) No such thing as computational philosophy ... yet. Hmmm. BTW, a Random Variable--continuous or discrete--does not have to be uniformly distributed, but you want that in a stream of equally-likely numbers input into a Monte Carlo simulation or when computing invertible probability distributions [not all are invertible as you say] to also feed a Monte Carlo simulation. I am pretty sure you have done this, from past discussions. In response to Grant, I would say that we are way beyond the times when we could easily distinguish between mathematicians, statisticians, and scientists. We have computational biology, computational physics, computational economics, computational finance, etc. all of which have elements of computational statistics. Computational statistics--a subset of the field in which I practiced--is a rising and inclusive field. Take a look at the curriculum at George Mason University, for example. I mean, can we say that we need to distinguish between mathematicians and statisticians anymore? To be a statistician these days you need to be able to derive maximum likelihood estimators, for example. To be a mathematician, ... well this is from the Univeristy of Oxford:
Finally and relatedly, I have been trying to follow Nick's evolving query to the forum, but it seems--to me--like he is looking for a way to prove that a generator of numbers is not random. As someone else has already mentioned, one cannot really do this, that is, prove that a sequence is not random ... almost like trying to prove that God does not exist When you think of it, a series of 100 rolls of a dice that is all fives, say, is as equally likely as ANY other specific sequence of rolls or (1/6)^100. So you can't derive anything about randomness by just looking at the numbers. Humans are not good at differentiating between randomness and just chaos. Essentially, anything is possible ... but "how likely?" is the right question. A good--simple way [the die-hard test is not simple]-- to sense the lack of randomness in a stream of numbers is to compare the results with a theoretical distribution that is random using the Chi-squared distribution. The Poker Test fits this criterion of simple, but effective. If the number generator is dealing hands, say--like with four 8s--in a proportion that is not at all likely, then one should be suspicious. But you could not say that it could never happen. The Chi-squared distribution is skewed positively with a tail that goes to infinity. But the thickness of the tails can be decreased with more trials [hands] or so-called degrees of freedom. It's a pretty cool way to do this and is easily accomplished computationally. Hope this clarifies a few things at least. Sorry for the long explanation. I guess I cannot help myself ... :-( Cheers, Robert W. On Wed, Dec 14, 2016 at 1:40 PM, Grant Holland <[hidden email]> wrote:
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove |
Re: Model of induction
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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Re: probability vs. statistics (was Re: Model of induction)
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