tag:blogger.com,1999:blog-3722233.post1814713071082409135..comments2019-12-06T07:15:22.123-05:00Comments on Computational Complexity: ProbabilityLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3722233.post-6818056370972408052011-12-30T13:33:41.359-05:002011-12-30T13:33:41.359-05:00Another way to describe the ideology of belief in ...Another way to describe the ideology of belief in probability is to repeat Suppes' argument in 1956 that, "Because of the many controversies concerning the nature of probability and its measurement, those most concerned with the general foundations of decision theory have abstained from using any unanalyzed numerical probabilities, and have insisted that quantitative probabilities be inferred from a pattern of qualitative decisions."<br /><br />Arnold Koslow paraphrased this to represent Newton's presupposing concepts of magnitude, and of relation in respect to magnitude, when understanding by number the relation in the abstract between any given magnitude and another magnitude of the same kind taken as a unity.<br /><br />"Because of the many controversies concerning the nature of length (mass) and its measurement those most concerned with the general foundations of geometry (mechanics) have abstained from using any unanalyzed numerical lengths (masses), and have insisted that quantitative lengths (masses) be inferred from a pattern of qualitative information."<br /><br />The understanding of mass at that time is like probability now; no one really understands the nature of probability.Doug Mouncehttps://www.blogger.com/profile/10917054134517643425noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1961786268116583942011-12-07T03:46:19.653-05:002011-12-07T03:46:19.653-05:00The question of Fine, is within the context of phi...The question of Fine, is within the context of philosophical theories of probability (Donald Gillies has a book over the subject).<br />Mathematicians (and computer scientists), are not concerned with these questions, because the starting point for them is a "given" probability space, which consists of a set, a \sigma algebra over the set, together with a "given" probability measure, P. <br />Then the mathematical methods are used to find for example the probability P(A) of some event A;<br />and the task of the mathematician is done. <br /><br />After that, P(A) should be interpreted using the same theory that is used to interpret P.Majdodinhttps://www.blogger.com/profile/06638299762577890471noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-7706384573951793492011-12-05T17:45:29.037-05:002011-12-05T17:45:29.037-05:00Isn't your computational point of view just a ...Isn't your computational point of view just a formalization of the frequentist position? I.e., if you have a process $P$ that maps $n$ "random coins" to some space of outcomes, then the probability of outcome $o$ is exactly $|P^{-1}(o)|/2^n$.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-15032123024280004212011-12-05T12:23:44.907-05:002011-12-05T12:23:44.907-05:00You are free to define probability in any way you ...You are free to define probability in any way you like, but the important thing is to show that the concept that you use is roust enough to support a statistical methodology (Savage had it right when he called his book "The Foundations of Statistics" rather than "The Foundations of Probability"). Both the subjectivists and frequentists have methodologies that are derived from their concept of probability, which allow you to draw inferences from data, e.g. what can you say about the bias of a coin after observing several coin flips? Preferably, you should be able to infer, based on your definition of probability, that the statistical methods that we actually use in science lead to at least approximately correct conclusions. Much of the debate in the foundations of probability is concerned with exactly this question.<br /><br />Without a statistical methodology, an interpretation of probability is useless. You might as well say that the moon is made of various kinds of cheese and that the probability of any event, whether or not it involves the moon, is a ratio of the volume of one of the kinds of cheese to the total volume of the moon. That satisfies the Kolmogorov axioms as much as any other definition, but it is useless because it does not allow me to make inferences based on data (and because the moon is not actually made of cheese).<br /><br />Therefore, I would like to ask you how we can understand statistics from your computational point of view?<br /><br />BTW, Terrence Fine's 1973 book on the foundations of probability is a classic, well worth reading.Matt Leiferhttp://mattleifer.infonoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-84918381225896556462011-12-05T10:53:55.190-05:002011-12-05T10:53:55.190-05:00What an odd question. And everyone else picked on...What an odd question. And everyone else picked one or the other? I wouldn't have picked either option either.Anonymousnoreply@blogger.com