From: Matt Mahoney (matmahoney@yahoo.com)
Date: Mon Mar 17 2008 - 16:31:13 MDT
--- John K Clark <johnkclark@fastmail.fm> wrote:
> On Sun, 16 Mar 2008 "Matt Mahoney"
> <matmahoney@yahoo.com> said:
>
> > N is the number of trials, not the number of agents.
>
> Then I’m not very interested in N, I don’t really care how often an
> identical being performs an identical experiment and gets an identical
> result.
>
> > A robot zombie with no qualia or consciousness
> > [blah blah] then the robot is killed
>
> That does not compute. Perhaps you mean quantum erasure is used to wipe
> the robot’s memory clean, but then you can’t say in your thought
> experiment that the robot saw this and that because even the universe
> doesn’t know what the robot saw.
Consider a robot whose entire mental state is 2 counters: n_heads and n_tails,
both initialized to 0. The only observation that the robot can make is that
of a coin flip, in which case one of the two counters is incremented depending
on the outcome. The robot's output is a prediction p(heads) =
(n_heads+1)/(n_heads+n_tails+2), i.e. Laplace's derivation of the maximum
entropy prediction of the next coin flip assuming an a-priori bias uniformly
distributed between 0 and 1.
1. A robot flips a coin with unknown bias 100 times and counts 99 heads. What
is its prediction p(heads) of the next coin flip?
2. A robot flips a coin with unknown bias 100 times. If it is heads, it and
the coin are copied 100 times. All coins are actually fair (p(heads) = 0.5).
What is its expected estimate of p(heads) of the next flip for a robot
randomly chosen from the resulting ~ 50^100 robots?
3. 50^100 robots start with 1 coin each with unknown bias (but actually fair).
Whenever a coin comes up tails the robot is killed with probability 0.99.
After 100 flips, only one robot remains. What is its expected estimate of
p(heads)?
4. Start with 50^100 robots. Divide them into groups of 100 robots with 1
coin each. Flip the coin. If it comes up tails, then hold a lottery and kill
99 of the robots in that group. Repeat 100 times, after which one robot is
left. What is its expected estimate of p(heads)?
5. One robot flips a coin with unknown bias (but actually fair) until n_heads
+ n_tails = 100. Whenever a coin comes up tails, both counters are reset to 0
with probability 0.99. What is its expected estimate of p(heads) when
finished (about 50^100 coin flips later)?
Is it possible for a robot to distinguish between these 5 cases? What about a
human?
-- Matt Mahoney, matmahoney@yahoo.com
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