Nick,
Re: your queston about stochastic processes....
Yes, your specific description "AND its last value" is what most uses of
"stochastic process" imply. But, technically all that is required to be
a "stochastic process" is that each next step in the process is
unpredictable, whether or not the outcome of one step influences the
outcome of the next. An example of this is the process of flipping a
coin several times in a row. Generally, we assume that the outcomes of
two adjacent flips are stochastically (or statistically) independent,
and that there is no influence between the steps. So, the steps of an
independent stochastic process are not dependent on their previous steps.
On the other hand, selecting dinner tonight probably depends on what you
had last night, because you would get bored with posole too many nights
in a row. And maybe your memory goes back more than just one night, and
your selection of dinner tonite is affected by what you had for 2 or
more nites before. If your memory goes back only one night, then your
"dinner selection process" is a kind of stochastic process called a
"Markov process". Markov processes limit their "memory" to just one
step. (That keeps the math simpler.)
In any event, stochastic processes whose steps depend on the outcomes of
previous steps are "less random" than those that don't, because the
earlier steps "give you extra information" that help you narrow down the
options and to better predict the future steps - some more than others.
So, LEARNING can occur inside of these dependent stochastic processes.
In fact, the mathematics of information theory is all about taking
advantage of these dependent (or "conditional") stochastic processes to
hopefully predict the outcomes of future steps. The whole thing is based
on conditional probability. Info theory uses formulas with names such as
joint entropy, conditional entropy, mutual information and entropy rate.
These formulas can measure /how much /stochastic dependency is at work
in a particular process - i.e how predictable it is. Entropy rate in
particular works with conditional stochastic processes and tries to use
that "extra information" provided by stochastic dependencies to predict
future outcomes.
Re: your "evolution" question... I have been speaking of biological
evolution.
HTH
Grant
On 8/9/17 8:47 AM, Nick Thompson wrote:
Hi everybody,
Thanks for your patience as I emerge (hopefully) from post-surgical fog.
I figured I best start my own thread rather than gum up yours.
First. I had always supposed that a stochastic process was one whose
value was determined by two factors, a random factor AND it’s last
value. So the next step in a random walk is “random” but the current
value (it’s present position on a surface, say) is “the result of a
stochastic process.” From your responses, and from a short rummage in
Wikipedia, I still can’t tell if I am correct or not.
Now remember, you guys, my standard critique of your discourse is that
you confuse your models with the facts of nature. What is this
“evolution” of which you speak? Unless you tell me otherwise, I will
assume you are speaking of the messy biological process of which we
are all a result: -- */The alteration of the design of taxa over
time/*. Hard to see any way in which that actual process is
evidently random. We have to dig deep into the theory that EXPLAINS
evolution to find anything that corresponds to the vernacular notion
of randomness. There is constraint and predictability all over the
place in the evolution I know. Even mutations are predictable. In
other words, the randomness of evolution is a creation of your
imaginations concerning the phenomenon, not an essential feature of
the phenomenon, itself.
So what kind of “evolution” are you guys talking about?
Yes, and forgive me for trolling, a bit. I am trying to wake myself
up, here.
nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/
<http://home.earthlink.net/%7Enickthompson/naturaldesigns/>
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