Re: Me and David C. Ullrich

john_ramsden@xxxxxxxxxxxxxx wrote:
> Elmo wrote:
> >
> > [...]
> >
> > 8.I say that "given at least one" is the special case whereas, "at
> > least one is a head" is stated with extreme prejudice toward heads, or,
> > "at least one is a tail" is stated with extreme prejudice toward tails.
> >
> > 9.State "at least one is a head" with extreme prejudice toward tails,
> > and the probability for two heads would be one.
>
> What the _heck_ are you on about with all this "extreme prejudice"
> stuff?! Sounds like you've seen too many Gene Hackman thrillers.
>
John,
Below is an example I found at:
http://en.wikipedia.org/wiki/Bayesian_inference#From_which_bowl_is_the_cookie.3F


Simple examples of Bayesian inference
[edit]
>>From which bowl is the cookie?
To illustrate, suppose there are two bowls full of cookies. Bowl #1 has
10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each.
Our friend Fred picks a bowl at random, and then picks a cookie at
random. We may assume there is no reason to believe Fred treats one
bowl differently from another, likewise for the cookies. The cookie
turns out to be a plain one. How probable is it that Fred picked it out
of bowl #1?

Intuitively, it seems clear that the answer should be more than a half,
since there are more plain cookies in bowl #1. The precise answer is
given by Bayes' theorem. Let H1 correspond to bowl #1, and H2 to bowl
#2. It is given that the bowls are identical from Fred's point of view,
thus P(H1) = P(H2), and the two must add up to 1, so both are equal to
0.5. The datum D is the observation of a plain cookie. From the
contents of the bowls, we know that P(D | H1) = 30/40 = 0.75 and P(D |
H2) = 20/40 = 0.5. Bayes' formula then yields

P(H1|D)=[P(D|H1)P(H1)] / [P(H1)P(D|H1)+P(H2)(D|H2)]

[.5*.75]/[.5*.75+.5*.05]=0.6


Before observing the cookie, the probability that Fred chose bowl #1 is
the prior probability, P(H1), which is 0.5. After observing the cookie,
we revise the probability to P(H1|D), which is 0.6.

<end reference>

The bowls were equally likely. There was a prejudice in the cookies.

When two coins are flipped, there are four equally likely outcomes. At
least one "at least one" statement is always possible. Whether it is
made, or not, is a function of the prejudice. The prejudice can be
toward "heads" and against "tails" or toward "tails" and against
"heads".

When the "at least one is a heads" statement is made every time it's
possible, call this extreme prejudice toward "heads".

When the "at least one is a tails" statement is made every time
possible, call this extreme prejudice toward tails.

With extreme prejudice toward heads and an "at least one is a heads"
statement, we would have:

P(A|B)=P(B|A)P(A) / P(B)

Our numerator will remain 1/4, regardless of the prejudice.

In the denominator, with extreme toward heads, and a heads statement,

1/4+1/4+1/4+0= 3/4 and

1/4 divided by 3/4 = 1/3 (extreme prejudice toward heads, and a heads
statement)


Suppose that there was extreme prejudice toward tails, and we were told
"at least one is a head."

Then: Our numerator is still 1/4.

Our denominator becomes 0+0+0+1/4 = 1/4.

One quarter, divided by one quarter equals one. (extreme prejudice
toward tails, and a heads statement)


When the statement was made without prejudice.

Our denominator is 1/4 + 1/2*1/4 + 1/2*1/4 + 0 = 1/2

and, our answer is 1/2. (no prejudice and an at least one is
statement)

Our answer is a function of the prejudice.

Eldon


> Maybe you have a shadowy subconscious association of the word "given"
> with something benevolent like a "gift", which by contrast makes the
> direct phrase "at least one is a head" sound blunt and final, like
> some assassin calmly plugging their victim with "at least one head"
> shot! Crazy I know, but if you ask me so is this entire discussion.
>
> Mind you, that's about the fourth theory I've put forward - You
> demolish them, one after the other, with each reply! I feel like
> that 19th C British PM Sir William Gladstone, about whom someone
> said "Each time he thinks he's answered the Irish Question, the
> Irish change the question!"
>
>
> Cheers
>
> John R Ramsden (jhnrmsdn@xxxxxxxxxxxx)
>
> * Remove m from com to reply
> * "From" address defunct

.