Talk:Bernoulli process

message: formal definition is not very formal.  in fact, it doesn't state the independence at all.  bad.
next, bernoulli process are usually not even required to be identically distributed, just independent.
— Preceding unsigned comment added by 140.247.149.65 (talk) 04:57, 25 October 2009‎ (UTC)
: Is the Bernoulli process a sequence of random variables or a single one whose domain is the sequences in a two-element set, canonically {0,1}? The introduction and informal definition clearly say the former but the definition of Bernoulli sequence Zn implies the latter. One element omega in the sample space corresponds to the entire random sequence one sequence <X0, X1, ...>, ie the B process rather than one B trial Xi.
: Either way, the formal definition seems bad to me. Does "for every omega in Omega" belong here, quantifying a probability statement, "the probability of Xi(omega)=1 with probability p ...". For every omega, Xi=1 or Xi=0 without any probability. Right? P64 (talk) 21:55, 18 February 2010 (UTC)
thread_title:  Problems 

message: Is this standard usage, that the Bernoulli sequence is not a sequence of random variates zero and one but a sequence of index numbers (subset of N) where the random variate is one?
I have partly rewritten the article to fit this usage, which is a challenge, because it would be --and it has been for some other editors-- convenient to call the sequence of zero and one a Bernoulli sequence. This needs attention "urgently" inasmuch as I haven't finished rewriting for consistency.
Probably there is a big matter of pedagogical or encyclopedic tactics to resolve, which needs some group decision not simply one person's expertise in one subdiscipline. --P64 (talk) 23:11, 4 March 2010 (UTC)
replies: text: I have added a statement in the main article to this effect. 24.1.53.152 (talk) 10:44, 29 January 2011 (UTC)
thread_title:  Bernoulli sequence 

message: : may be interpreted as the binary digital representation of a real number between zero and one.
First, that isn't unique. The rational numbers whose denominators are powers of 2, aka the multiples of powers of one-half, all have two binary digital representations. If we will gloss over that complication, it should at least be mentioned in a footnote. Many readers do know that 0.9999... = 1. This is a topological stumbling block. Does measure theory handle it without a glitch?
Second, this section needs to be rewritten after we decide the questions of categories, which may be a pedagogical or otherwise tactical rather than matters for deference to standard usage by experts in some subdiscipline.
*When should we use the move that interprets a sequence as a single entity? (eg, a sequence of r.v., aka a discrete-time stochastic process, as a single random variable whose values are sequences of numbers)
*When should we permit trials, and processes and their ilk (experiments, samples, etc) to live equally among the variates and also in the abstract space \Omega? (rather than restrict them as we restrict random variables to functions of Omega into R or Rn)
See also "Bernoulli sequence" above.
See also stochastic process, Markov chain, memoryless.
--P64 (talk) 23:22, 4 March 2010 (UTC)
replies: text: I'm not at all clear about what this complaint is about. This is standard material treated in a variety of textbooks. The real numbers embed in a Cantor space, the Bernoulli process is a Cantor space.  This is not probability 101, but it's not an advanced topic either.linas (talk) 18:16, 29 July 2012 (UTC)
thread_title:  Bernoulli map 

message: The \sigma-algebra defined in the "finite vs infinite" section is not a \sigma-algebra as it does not contain countable intersections. Fix an infinite sequence x = (x_1,x_2,\dots). The countable intersection of the cylinder sets  (which should also be an element of the \sigma-algebra) with the first n digits identical to the first n digits of x equals to the set \{x\}. Therefore, each individual infinite sequence is measurable. One cannot just remove these elements from the \sigma-algebra.
The statement about the open ball in the section "As a metric space" is not correct. The open balls are the cylinder sets. But they are also closed sets, so their complements are open and closed, too.
Patschy (talk) 10:50, 22 August 2012 (UTC)
thread_title:  Two mistakes 

message: I'm not at all sure the simple procedure described in the article matches the referenced paper.
The procedure defined in the article appears to produce wrong answers.  Consider all possible 4-bit input strings, and apply the procedure described in the article:
Initial string	appended	output	probability
0000          	000     	—     	p0
00            	1       	00    	p1
00            	000     	1     	p1
0011          	1       	0     	p2
00            	0       	01    	p1
              	111     	00    	p2
              	0       	011   	p2
11            	111     	0     	p3
00            	000     	1     	p1
              	1       	100   	p2
              	000     	11    	p2
11            	1       	10    	p3
1100          	0       	1     	p2
11            	111     	0     	p3
11            	0       	11    	p3
1111          	111     	—     	p4
Notice how only 2 of the possible 8 3-bit output strings are returned.  If there's a third output bit, it must match the second bit.  That seems wrong.
The paper seems to define the iterated procedure as a recursive process:
# Divide the input into bit pairs.  Discard any trailing odd bit.  (If there are no pairs, stop.)
# Compute the exclusive-or of each pair.
# For each pair where the exclusive or is 1 (the bits differ), output the first bit.
# Take the string of exclusive-or bits (half the length of the input string), and perform the iterated procedure on that.
# For each pair of identical bits (the exclusive or is zero), take one of them, and perform the iterated procedure on that.
(An implementation may also apply a bound to the number of recursion levels, at a cost in output length.)
So given the example bit string 10011011, the procedure would indeed generate 5 output bits, but they would be:
# Output the 3 bits 101
# Perform the iterated procedure on the xor bits 1110:
## Output the bit 1
## Perform the iterated procedure on the xor bits 01:
### Output bit 0
### Perform the iterated procedure on the xor bit 1 (produces nothing)
### Perform the iterated procedure on the empty bit string (produces nothing)
## Perform the iterated procedure on the matching bit 1 (produces nothing)
# Perform the iterated procedure on the matching bit 1 (produces nothing)
I've 
The procedure describe in the paper, tested on all possible 16-bit strings, returns a variety of bit lengths from 0 to 11, but for each output length, each output string is returned an equal number of times.
Initial string	xor	match	output	probability
0000          	00 	00   	—     	p0
00            	   	0    	00    	p1
00            	   	0    	10    	p1
0011          	00 	     	0     	p2
00            	   	0    	01    	p1
              	11 	—    	00    	p2
              	11 	—    	01    	p2
11            	   	1    	01    	p3
00            	   	0    	11    	p1
              	11 	—    	10    	p2
              	11 	—    	11    	p2
11            	   	1    	11    	p3
1100          	00 	     	1     	p2
11            	   	1    	00    	p3
11            	   	1    	10    	p3
1111          	00 	11   	—     	p4
Notice how every possible 1-bit output has probability p2, and every possible 2-bit output has probability p1+p2+p3.
I've done a machine check of longer strings (up to 26 bits), and they also work correctly.
Can someone else confirm the translation from source to Wikipedia is incorrect?  71.41.210.146 (talk) 18:47, 15 January 2014 (UTC)
thread_title:  Iterated Von Neumann extractor