Causation/Causality, Memory, and Convolution 4: Memory vs Complexity
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 19 Feb 2006 00:53:56 -0800
From Osher Doctorow mdoctorow@xxxxxxxxxxx
Which is simpler, being able to reach anywhere back into memory from
some time t_f or having to go in single steps backward from t_f to a
remote past time datum via a sequence t_-1, t_-2, ..., t_-n where -1,
-2, ..., -n are supposed to be subscripts?
Most physicists will arguably say that the former is simpler, and that
is correct in my opinion, but it isn't the opinion of conditional
probability Bayesian Markov chain theorists who prefer the antlike
steps mentioned. Actually, their preference is partly imposed on them
by the fact that conditional probability isn't as intuitive as Probable
Influence/Causation. The latter has no difficulty in going from t_f to
t_-n by 1 + t_f - t_-n. The Markov chain has to do this: it defines
"transition probabilities" which are conditional probabilities such as
being in state S_k at time t_k given that one was in state S_k-1 at
time t_k-1, k = 1, 2, ..., n. Formally:
1) Markov chains specify P(Sk|Sk-1), k = 1 to n
2) Probable Influence/Causation allows any P(Sk --> Sj) = 1 + P(Sj) -
P(Sk), j > k
where k-1, k, j are subscripts and I've dropped the _ subscript
notation.
Some confusion occurs to physicists learning probability-statistics,
and mathematicians too, because Markov chain theory has what is called
"n-step transition functions". These involve the probability of going
from step Si to Step Si+k in n steps (readers can figure out the
conditions on i, k, n as an exercise), but what most people don't
realize is that the n steps have to be n steps of one-step-at-a-time
sequences as in (2) above. Markov chains don't jump from Sk to Sj for
j > k in one step except if j = k+1, since otherwise they would have
new one-step transitions between what is already one-step transitions,
which involves a logical contradiction.
It's sometimes thought that the above is a curiosity that doesn't have
so much significance, but along comes a paper by the UC Berkeley
"contingent", Alexandre J. Chorin and Panagiotis Stinis (Dept. Math.),
"Problem reduction, renormalization, and memory," math.NA/0503612 v1 26
Mar 2005, which shows that in order to reduce complexity of
computational problems involving time dependence, Memory and random
forcing terms are required. Memory in Markov chains is the minimal
type of Memory, one-step-ahead (or one step behind) Memory, while
Probable Influence/Causation Memory is arbitrary many steps. One
could theoretically devise a conditional probability analog of PI
Memory with "given" or "fixed" past times, but it would lose the
properties of Markov chains and would fail at 0 probability events as
usual.
Osher Doctorow
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