Martin Lefley wrote:
> Surely
> accurate Bayesian reasoning needs complete conditional data distributions,
> this is never available and I believe if it is, this applies complete data,
> in which case any learning system could reason consistently, accurately
> perfectly.
I don't think so. Consistency requires a logic which derives exactly the same
result irrespective of the chain of logic/calculation by which it was derived.
As Cox has pointed out, the combination rules of statistics may be unique
in this respect.
However -- even if this is true -- it may not be of any significant practical
import. [And I speak as a statistically inclined lurker.]
Firstly. How big are the inconsistencies for `reasonably' long chains
of derivation.
Secondly. Most floating point systems are formally inconsistent, as
truncation, rounding, under/overflow, etc. may be different for different
ways of ordering the same arithmetic calculation. Mostly this doesn't
matter too much. Sometimes (as numerical analysts know only too well)
it can be important and steps have to be taken to minimise its impact.
[This may be as simple as evaluating (a+b)(a-b) rather than a^2-b^2.
Or as complex as requiring a full analysis of the propagation of truncation
errors.]
Basically `all models are wrong'. So worrying if you have an infinite
number of significant digits in your calculated answer should not be
your biggest concern.
As so often, we have a black and white formal characterisation. With all
sorts of real-world grays in between.
Peter
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