[EM] Bias-Free reply

[Michael Ossipoff]

No, I didn’t assume that the probability-distribution for states’
populations is uniform.

I merely assumed uniformity for that  distribution *within any particular
interval between two whole numbers of quotas*.

…& it wasn’t so much an *assumption*, as much as part of a useful
operational definition for bias.  …a bias easily defined, determined &
avoided.

(…explaining why I’d denied an assumption of special conditions.)

Because in this topic, an interval between two whole numbers of quotas is
often mentioned, it needs an abbreviation. I’ll call it an
“integer-interval” (ii).

Obviously it’s within a lower ii that that distribution varies most.

The 0-1 ii doesn’t count, because , in House-apportionment, each state gets
at least one seat.

It doesn’t count in PR either, because SL specifies .7 (instead of .5) of a
quota as the rounding-point in the 0-1 ii.

That’s to discourage & thwart strategic-splitting. BF’s results are so
close to those of SL, that the .7 rounding-point in the 0-1 ii should be
used in BF too.

…& so, the lowest ii in which the operational-definition makes a difference
is the 1-2 ii.

In that ii, the values of the BF & SL rounding-points differ by only about
2%.

Obviously that % rapidly becomes drastically lower for higher ii s.

…as also must the variation of the probability-distribution within an ii.

As for my useful operational-definition of bias:

1.

One justification for it is that the vagaries & continual variations of the
probability-distribution for states’ populations isn’t the responsibility
of an allocation-rule.

…as is tacitly, performatively, agreed by every allocation-rule that
doesn’t require recalculating an approximation to that distribution at each
census.

2.

But what would an alternative to my useful operational definition look
like? A mess, that’s what.

Obviously the probability distribution within each ii could only be
*approximated* by a formula based on…what? The historical record of
populations in that range?

Why should old records be assumed relevant to today’s probabilities. We
haven’t gotten away from assumptions—a futile goal.

Or maybe an interpolation or least-squares approximation based on the new
populations of the states.

The operative word there is “approximation”.

All that extra work for something that’s still only approximate. Exactitude
hasn’t been gained.

Arguably, in a particular ii, the approximation to the
probability-distribution, a best-guess, is more realistic than a uniform
distribution there.

But the better likely-accuracy of that guess, doesn’t make it more than a
guess.

I didn’t stumble-upon BF as a useful, feasible avoidance of a usefully
operationally defined bias.

That was the purpose.

But sure, any other interpretations or interesting aspects that I didn’t
know about—yeah I stumbled onto that.

Bias has very much been part of the merit evaluation & comparison of PR
methods, & that was always so with House-apportionment as well.

So that was my purpose.

I hadn’t considered the interesting entropy consideration, so yes that was
an accidental result.

Entropy in PR had never occurred to me, but it’s the basis of a number of
useful measures of inequality.

The most relevant 1-number inequality-measure based on summed-aggregation
is a generalized entropy called ge(-1).
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