## III.3 Bayesian Analysis and III.4 Braid Groups by F. E. A. Johnson: Page 160.

A conditional probability example is:  suppose n unbiased coins are tossed and three heads are observed.  If the number of coins tossed is between one and ten then we can compute the probability of seeing three heads given that there are n tosses, P(3 Heads | n Tosses), for each value of n from one to ten.   We may instead want to know P( n Tosses | 3 Heads).  By Bayes theorem we have P( n Tosses | 3 Heads) = P( 3 Heads | n Tosses) P( n Tosses) / P( 3 Heads).  A prior distribution is used to guess P( n Tosses) and this is used in the equality.  Using this prior distribution and the equality one obtains the posterior distribution.  If we assume the prior distribution is equal likelihood for each of the number of possible coins from 1 to 10 ( P( n Tosses) = 1/10) then we get the posterior distribution proportional to (1/10) P( 3 Heads | n Tosses).  This example seems very contrived to me.  It makes several assumptions without support, these are: the number of tosses is between 1 and 10, the prior distribution of equally likely tosses and it does not address how to account for the fact that if one has 2 tosses then one cannot observe 3 heads.

A physical model of a mathematical n-braid is that of two parallel planes each with n holes,  a string is run from each hole in the first plane to a hole in the second plane so that no two strings go to the same hole.  The result a physical model of an n-braid.  One can not double back or create knots but can stretch, contract, bend, and otherwise move the strings about in 3-dimensions and still end up with the “same” n-braid.  This notion of “same” is an equivalence relation called a braid isotropy.

Braid composition may proceed as follows.  Consider three parallel planes with with one n-braid between an end plane and the middle plane and a second n-braid between the middle plane and the other end plane.  With the same n-holes in the middle plane.  One may join the two strings, one from each end plane, at the middle plane and remove the middle plane.  The result is the composition of the two n-braids, X and Y, to form the n-braid XY.

With this notion of composition, n-braids, form a group Bn.  One may create an n-braid that acts as X-1.  Then the composition XX-1 = I, the trivial braid where the holes are the same and the strings simply connect to the same hole in the second plane.

Take the trivial n-braid and switch the i th and i+1 st string attachments in the second plane, call this si.  As a group BN, the braid group, is generated by the elements (si)1≤i≤n-1.  There is a similarity between si and the adjacent transpositions that generate the group Sn of permutations of {1,2, … , n}.