{"id":454,"date":"2016-03-04T23:30:52","date_gmt":"2016-03-05T04:30:52","guid":{"rendered":"http:\/\/aristotle2digital.blogwyrm.com\/?p=454"},"modified":"2016-03-05T14:25:17","modified_gmt":"2016-03-05T19:25:17","slug":"a-flip-of-the-coin","status":"publish","type":"post","link":"https:\/\/aristotle2digital.blogwyrm.com\/?p=454","title":{"rendered":"A Flip of the Coin"},"content":{"rendered":"

Ideological conflicts are often the bitterest of arguments that appear in the race of Man.\u00a0 Whether the\u00a0 religious wars of a post-reformation Europe, the polarizing arguments of modern US politics, the simple disputes over which is better – PCs or Macs, or whether somebody should be a dog-person or a cat-person, these conflicts are always passionate and, while important, they are, in some aspect, pointless.\u00a0 Clearly, adherents of both sides always have a good reason for supporting their position; if they didn\u2019t they wouldn\u2019t support it so vehemently.\u00a0 Those bystanders without a strong opinion one way or another are left to just shake their heads.<\/p>\n

One such ideological conflict is the strong, often mean-spirited, argument between the Frequentists and the Bayesians over the right way to characterize or define probabilities.\u00a0 For much of this particular cultural hotspot, I\u2019ve been a bystander.\u00a0 By training and early practice, an outsider would have characterized me as a Frequentist since I am comfortable with and enjoy using sampling techniques, like classical Monte Carlo, to investigate the evolution of a given probability distribution.\u00a0 Over the past 6 or 7 years, I\u2019ve come to a better appreciation of Bayesian methods and find myself in the ever-growing position of seeing the complementary utility of both.<\/p>\n

Nonetheless, finding a simple scenario that captures the essential aspects of these schools of thought and is easy to articulate has been elusive \u2013 that is until recently.\u00a0 I now believe I\u2019ve found a reasonably compact and understandable way to demonstrate the complementary nature of these techniques through the flip of a coin. (Although I am quite sure that there is great room to improve it across the board).<\/p>\n

Before diving into the coin flip model, I would like to summarize the differences between the Frequentist and Bayesian camps.\u00a0 While the coin-flip model is strictly my own \u2013 based on my own thoughts and observations \u2013 the following summary is heavily influenced by the online book entitled Entropic Inference and the Foundations of Physics<\/a><\/em>, by Ariel Caticha.<\/p>\n

The reader may ask why people just can\u2019t agree on a single definition of probability.\u00a0 Why do the two camps even matter.<\/p>\n

On the notion of probability, Caticha has this to say<\/p>\n

The question of the meaning and interpretation of the concept of probability has long been controversial. Needless to say the interpretations offered by various schools are at least partially successful or else they would already have been discarded. But the different interpretations are not equivalent. They lead people to ask different questions and to pursue their research in different directions. Some questions may become essential and urgent under one interpretation while totally irrelevant under another. And perhaps even more important: under different interpretations equations can be used differently and this can lead to different predictions.<\/p>\n
Ariel Caticha<\/div>\n
<\/div>\n<\/div>\n\n

The Frequentist model of probability is based on what I call the gambling notion.\u00a0 Take a probability-based scenario, like throwing a pair of dice, and repeat for a huge number of trials. The probability of a random event occurring, say the rolling of snake eyes (two 1\u2019s)<\/p>\n

\"Snake<\/a><\/p>\n

is empirically determined by how frequently it shows up compared to the total number of trials.\u00a0 The advantage that this method has is that, when applicable, it has a well-defined operational procedure that doesn\u2019t depend on human judgement.\u00a0 It is seen as an objective way of assigning probabilities.\u00a0 It suffers in two regards.\u00a0 First, the notion of what random means is a poorly-defined philosophical concept and different definitions lead to different interpretations.\u00a0 Second, the method fails entirely in those circumstances where trials cannot be practically repeated.\u00a0 This failure manifests itself in two quite distinct ways.\u00a0 First, the question being asked could be entirely ill-suited to repeated trials.\u00a0 For example, Caticha cites the probability of there being life on Mars as one such question.\u00a0 Such a question drives allocation of resources in space programs around the world but is not subject to the creation and analysis of an ensemble of trials.\u00a0 Second, the scenario may naturally lend itself to experimental trials but the cost of such trials may be prohibitively expensive.\u00a0 In these cases, it is common to build a computational model which is subject to a classical Monte Carlo method in which initially assumed distributions are mapped to final distributions by the action of the trial.\u00a0 Both the mechanics of the trial and the assumed initial distribution are subjective and so are the results obtained.<\/p>\n

The Bayesian approach relies heavily on Bayes theorem and, in doing so, allows the user to argue consistently about issues like the life-on-Mars question without needing to define random or determine how to implement trials.\u00a0 This approach eschews the notion of objective determination of probability and, instead, views probability as an epistemological question about knowledge and confidence.\u00a0 Thus two observers can look at the same scenario and quite naturally and honestly assign very different probabilities to the outcome based on each one\u2019s judgement.\u00a0 The drawback of this method is that it is much harder to reach a consensus in a group setting.<\/p>\n

Now to the coin-flip model that embodies both points of view.\u00a0 The basic notion is this:\u00a0 Suppose you and I meet in the street and decide to have lunch.\u00a0 We can\u2019t agree on where to go and decide to pick the restaurant based on a coin flip done in the following fashion.\u00a0 I\u2019ll flip a quarter and catch it against the back of my hand.\u00a0 If you call the outcome correctly we go to your choice; alternatively we go to mine.\u00a0 What is the probability that we end up in your restaurant versus mine?<\/p>\n

Well there are actually two aspects of this problem.\u00a0 The first is the frequentist assignment of probability to the coin flip.\u00a0 Having observed lots of coin flips, we both can agree that prior to the flip, the probability that it be heads or tails is 50-50.<\/p>\n

\"Before<\/a><\/p>\n

But after I\u2019ve flipped the coin and caught it, the probability of it being heads is a meaningless question. \u00a0It is either heads or tails \u2013 it is entirely deterministic.\u00a0 It is just that you don\u2019t know what the outcome is.\u00a0 So the probability of you picking the right answer is not amenable to trials.\u00a0 Sure we could repeat this little ritual every time we go out to lunch, but it won\u2019t be an identical trial.\u00a0 Your selection of heads versus tails will be informed based on all your previous attempts, how you are feeling that day, and so on.<\/p>\n

So it seems that we need both concepts.\u00a0 And after all, we are human and can actually entertain multiple points of view on any given subject, provided we are able to get past our own ideology.<\/p>\n","protected":false},"excerpt":{"rendered":"

Ideological conflicts are often the bitterest of arguments that appear in the race of Man.\u00a0 Whether the\u00a0 religious wars of a post-reformation Europe, the polarizing arguments of modern US politics,… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-454","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/454","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=454"}],"version-history":[{"count":0,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/454\/revisions"}],"wp:attachment":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=454"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=454"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=454"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}