Back when we were talking about the adequacy of deposition systems as analog to
the lost opportunity updating mechanism for free will, I cross-checked
definitions for the Markov property in 2 of my (old) control textbooks. Here
they are:
from [Estimation Theory and Applications, NE Nahi]
> "A stochastic sequence x(k) is Markov of first order, or simply Markov, if
> for every i the following relationship holds
>
> p{x(i)|x(i–1),…, x(1)} = p{x(i)|x(i–1)} (1.189)
>
> In other words, the conditional probability density function of x(i)
> conditioned on all its past (given) values is the same as using the value in
> the Equation (1.189) is to be satisfied for all i."
from [Applied Optimal Control, Bryson & Ho]
> "A random sequence, x(k), k=0,1,…,N is said to be markovian if
>
> p[x(k+1)/x(k),x(k–1),… ,x(0)] = p[x(k+1)/x(k)] (11.1.1)
>
> for all k; that is, the probability density function of x(k+1) depends only
> on knowledge of x(k) and not on x(k-l), l=1,2,…. The knowledge of x(k) that
> is required can be either deterministic [exact value of x(k) known] or
> probabilistic (p[x(k)] known). In words, the markov property implies that a
> knowledge of the present separates the past and the future."
I think the difference is interesting, particularly the "in other words" and
"in words" parts. The choice of "i-1" vs. "k+1" is also interesting, but much
less than "[future] conditioned on its past" versus "separates past and
future". If we read like a modernist, through the presentations to some
Platonic object behind them, we get the same damned thing, as transformed
through fairly standard transforms (from what you read to what you think). But
if we read it as a postmodernist, we can ask *why* Nahi chose i and i-1 where
B&H chose k+1,k,k-1? And how *might* that choice be related to the more nuanced
phrase "separates past from future"? And there are other differences, like
Nahi's choice of "Markov of first order, or simply Markov" vs. B&H taking
license to avoid allusion to higher order memory.
A skeptical reader simply has to ask what does this *actually* mean? How are
these authors intending to use and reuse this definition later? How will it
compose with other concepts? Etc. Maybe it's all accidental and merely a
function of the authoring/editing processes in each case. Or maybe not.
On 1/5/21 2:28 PM, uǝlƃ ↙↙↙ wrote:
> ... my contrarian nature (and my laziness) forces me to cross-check a
> proposition from one narrative to another,
--
↙↙↙ uǝlƃ
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