Elephantine Eighteens via Bayesian Bananas

*PS …Petrolhead Porkies Cowboy Cops… *

*Camera craptrap
conundrum clobbered Sep04*

The binomial distributions generated by tossing
coins and dice suffice to convey the contrivance that is common to so many of
these politically partial projects. Here we’ll use a tetrahedral dice sufficing
for sum simplicity that can be carried over to an any-hedral one including the
hexahedral standard. Binary head / tail or yes / no outcomes from N
realisations with the coin (dihedral dice) are conveyed by expansion of the
elementary expression (½+½)^{N}, whilst for the dice (¼+¾)^{N}
designates the binary distribution when the probability of a yes is 3 times
more than a no in any single realisation ~ if you like, it’s a coin with 3
heads and one tail.

The trick played by epi-demonologists so as to procure the right result desired by their paymasters is ludicrously simple and entails no more than an illusory adjustment of the odds, much the same as has been used for centuries by cheats and charlatans. All we have to do is persuade people that what was always really a tetrahedral dice representing reality should instead have been regarded as a dihedral one defining the reference state and it only transformed into a tetrahedral one as a causal consequence of the “right” policy. This immediately creates the illusion of the “right” outcome because the chance of yes is shifted at a stroke from its arbitrarily assigned false value of ½ to its true value of ¾. By same token the chance of no as “wrong” outcome appears to have been halved from ½ to ¼.

Neat isn’t
it and that’s about the sum of it so far as the rabid report went ~ well it
would have been except that the Darling Minister let slip in his summary
briefing that 1/6^{th} of the test sites returned the “wrong” results
meaning that deaths (KSIs) actually increased so had he been consistent with
claimed causal connection he should then have said cameras also had an adverse
effect on safety. Except that Ali didn’t say that (of course not) ~ he said
“we’ll have to look at those cases again” and by “we” he meant of course his
epi-demonologists and by “look” he surely meant “fudge as far as possible and
then some more”. At any rate at least it was disclosed (it’s for sure that by
no means all wrong results get any airing at all) despite the evident
embarrassment, although quite why it was posed as a conundrum puzzled me until
I realised what was publicised as a glitch actually gave the game away on how
the figures had been fudged.

With
hindsight of course it was obvious but it took a fortnight to dawn on me that
the distributions are interval representations of the records so if a no chance
of ½ in the dihedral misrepresentation is taken to reflect biennial death rate
at a single site then a no chance of ¼ in the tetrahedral one reflects annual
death rate and the comparison should be between dihedral N=1 and tetrahedral
N=2. In fact having N as years fits rather neatly with the evaluation itself if
we suppose that of 1800 cameras or so 600 were on site for 3 years, 1200 for
two years and 600 for one year ~ roughly but adequately for the present purpose
then we might say that 1200 tests were done over two years when the 100 supposedly
saved lives would be a 1/12 reduction claimed as causally connected, always
reminding ourselves that 1/6 of the sites showed an “anomalous” increase.

Okay so two throws with the tetrahedral dice (¼+¾)^{2}=1*(¼)^{2}(¾)^{0}+2*(¼)^{1}(¾)^{1}+1*(¼)^{0}(¾)^{2}
delivers as distribution 1/16 as NN , 6/16 as NY/YN and 9/16 as YY, in total
4/16 as N and 12/16 as Y where N=No and Y=Yes. This is what should have been
compared with one throw of the dihedral dice (½+½)^{1}=1*(½)^{1}(½)^{0}+1*(½)^{0}(½)^{1}
delivering as distribution 8/16 as N and 8/16 as Y. Now we see immediately how
the thought trick not only generates a 1/16 increase in YY (no Ns ~ no deaths)
but also appearance of a 1/16 chance of NN, two Ns (deaths) derived from ¼ of
the throws against never more than one N deliverable from the (false)
reference.

These numbers (1/16 reduction in deaths and
increased deaths from 1/4 of sites) aren’t a million miles from the rabid
report’s 1/12 and 1/6 respectively and it’s for sure that straightforward
tweaking of intervals and facets could provide an improved match. Not
worthwhile though, not given the myriad other uncertainties, and doing it would
take us down the mathsturbation motorway towards elephantine eights though god
forbid anyone in their right mind would go so far as eighteens. Point is that
it was the conjunction of multiple N appearing and accompanying increased Y
that gave their game away ~ and whatever had been politically picked by way of
fudged facets etc, it would always have been betrayed at least within in this
simple framework.

Sad to say this example reflects just one tip of an
enormous iceberg of post-philosophical predilections for provision of
politically proper promotional pap that long since squashed sensible scientists
into conjuring cretins. Sadder still is the attention it attracts from
anonymous apparatchiks with endless appetites for regurgitating this mash into
mindless media machines where it is portrayed as apocalypse always and
everywhere. So poor old percy public is left with only platefuls of fictitious
pulp and nowhere to be seen is any sign of the perspicacity that was his
rightful entitlement due in return for enforced prepayment from his purse.

By popular pressure or rather prompted by punters
puzzled by my claimed connection for fudged figures as originally outlined
(below), I spent a little time tinkering to make their message more meaningful.
How about this then as plausible proposition? Bottom line “benefit” was
promotionally presented in terms of an overall statistic as a lumped value over
all intervals and all sites and on that basis was headlined as 1/12^{th}
reduction of KSI (= Killed or Seriously Injured) in the unpleasantly jarring
jargon of that rabid report. The evaluation of course was not single blind
never mind double ~ it was fully foresighted and it is clear from the report
that the criteria for site inclusion / exclusion were not merely ambiguous but
dubious because getting the right result as “positive” outcome was crucial for
continuation / expansion of the activity and its accompanying advantageous expenditure.

The key to the massage resides in this presumed
reference state as false benchmark being betrayed by the notoriously
unexplained anomaly of increased KSI at 1/6^{th} of the sites included
for evaluation. Okay so we start with an overall index / chance of N for Nasty
meaning the chance of N’ for Nice is simply its complement 1-N ~ or in compact
notation the binomial outcomes are captured in (N+N’)^{1} with exponent
1 denoting the single equivalent (lumped) presumed reference realisation.
Realities of course were very different and as first representation of the
heterogeneities we might separate the sites into two lumped subspecies
respectively resident for (say) one and two intervals of time. Whereas (N+N’)^{1}
denotes representation of the presumed comparator over the entire interval, I
say realities are (taken to be better) reflected by linear addition of interval chances (P,P’) ~ that is, R(P+P’)^{1}+R’(P+P’)^{2}
~ with P’ Pleasant as counterpart of N’ Nice and P+P’=1=R+R’ of course and R as
Ratio reflecting fraction of newer installations in place for only one time
interval, plausibility perhaps demanding 1/3<R<2/3.

To make the present point we are looking for
plausibility of (N,P,R) values consistent with RP’+R’P’^{2}=N’+1/12 as
notional increase in Niceness and same time R’P^{2}=1/6 as concomitant
increase in Nastiness. A couple of simple trial cases suffice to make the point
being pursued, as follows: R=1/3, P’=1/2, N’=1/4 and R=5/8, P’=1/3, N’=1/6,
both plausible on R and roughly spanning the credible range (5/8 is not
meaningfully distinct from 2/3) inside which (1/3<R<5/8), 1/2<P<2/3
and 3/4<N<5/6. Notice just how small are these ranges of N and P as Nasty
and Pernicious chances compared with R ranging from only half as many new
installations as old to almost twice as many. Of course confidence attributes
corresponding to the heavily headlined “correct” outcome weren’t featured in
the media manipulation, nor even mentioned in what I saw, but I bet they
substantially exceeded the ranges deduced here ~ meaning not a lot should have
been made of the findings, certainly not all that self-congratulatory
trumpeting bannered as “Bloodied Hands” for counter-culture campaigners like
me.

The figures speak for themselves so no more really
needs to be said except perhaps to note (coincidentally?) in both of the
illustrative examples the interval chances of Pleasant outcomes (P’=1/2, 1/3) happen
to be exactly twice the lumped chances of Nice outcomes (N’=1/4,1/6) ~ I’ll
leave it to the reader to check the equivalence for tossed coins or dice saying
here simply that looking longer (and harder) inevitably makes things look both
better and worse because extremities (both ways) are rarer than normalities.
That simplicity, indeed tautological truism for monomodal distributions, appears
to have escaped Britain’s Best Brains or at least London’s Limpet Luvvies. Any
rate, it should now be transparent that the trick really has been revealed as
Bayesian Bananas of the sort that sixth formers used to deduce for themselves
as self-evident contrivances but never the conundrums asserted by good old Ali
D during his many manipulative media machinations. That’s the trouble with
leaving things to lawyers / politicians ~ they are undeniably able to balloon
baloney into briefs but that’s got nothing to do with lasting leverage from
logical learning. Enough said ~ indeed a
tad too much for my own good but then I am an incorrigibly cantankerous old
git.