[Humanist] 31.153 on maths for humanists

Humanist Discussion Group willard.mccarty at mccarty.org.uk
Sun Jul 2 08:48:13 CEST 2017


                 Humanist Discussion Group, Vol. 31, No. 153.
            Department of Digital Humanities, King's College London
                       www.digitalhumanities.org/humanist
                Submit to: humanist at lists.digitalhumanities.org



        Date: Sat, 1 Jul 2017 15:50:57 -0400
        From: Henry Schaffer <hes at ncsu.edu>
        Subject: Re:  31.152 pubs: on maths for humanists
        In-Reply-To: <20170701075422.A572067D0 at digitalhumanities.org>


Norman,
  Thanks for a very nice discussion spanning several realms of
investigation. One reason I am certain of your wide experience is your
statement, "The above is a 'frequentist' account, based on probabilities.
The other doctrine is 'bayesian' (who are not to be left alone with
frequentists in the presence of sharp objects)."

--henry

P.S. I've found that many frequentists in the life sciences use the Bonferroni
Correction - which somewhat moves out of the orthodox frequentist territory.

On Sat, Jul 1, 2017 at 3:54 AM, Humanist Discussion Group <
willard.mccarty at mccarty.org.uk> wrote:

>                  Humanist Discussion Group, Vol. 31, No. 152.
>             Department of Digital Humanities, King's College London
>                        www.digitalhumanities.org/humanist
>                 Submit to: humanist at lists.digitalhumanities.org
>
>
>
>         Date: Fri, 30 Jun 2017 23:24:59 +0100
>         From: "Norman Gray" <norman at astro.gla.ac.uk>
>         Subject: Re: [Humanist] 31.149 pubs: on maths for humanists;
> digital logic
>         In-Reply-To: <20170630064201.1BE862F59 at digitalhumanities.org>
>
>
> Greetings.
>
> On 30 Jun 2017, at 7:42, Gabriel Egan wrote:
>
> > In Patrick Juola and Stephen Ramsay's new book _Six Septembers_,
> > announced here, there is a most interesting discussion of
> > the notion of the Null Hypothesis (pp. 246-9).
>
> I may be able to re-explain this (the following is a slightly protracted
> account, but intended to be complementary to Juola and Ramsay's account
> rather than at all disagreeing with it).
>
> > The Null Hypothesis is that this 40-pound cat is a Siamese.
>
> That's right -- the Null Hypothesis is usually the boring hypothesis, or
> the no-new-science-here hypothesis.  You haven't discovered a new breed
> of cat, with Siamese-like markings, just a reeeally fat Siamese.
>
> But 40 lb is surprisingly heavy for a Siamese -- really very surprising.
>   But how surprising, numerically?
>
> The argument on Juola and Ramsay's p248 gives a necessarily rather
> hand-waving estimate that the probability of a Siamese cat being this
> heavy is about 1%.  But this cat (as Gabriel points out) is certainly 40
> lb.  So we have a right to be astonished -- this is a very unlikely
> thing (chance of 1%) to come across.
>
> So at this point we can either (a) decide that today is a weird day, and
> that being accosted by enormous felidae probably won't be the end of it,
> or (b) decide that we don't believe in coincidences, and that something
> is wrong.  Since we do believe (100%) that the cat is that heavy,
> perhaps it's our hypothesis that this is a Siamese that is wrong, so we
> decide to reject that Null Hypothesis.
>
> > << At this point, the test becomes simple logic. If the cat were
> > an ordinary Siamese, it would probably not weigh forty pounds.
> > Therefore, if it does weigh forty pounds, it's probably not an
> > ordinary Siamese. >>
> >
> > This statement seems to me to commit a well-known fallacy. The
> > probability value is a remark on how often the observed data
> > should be expected if the Null Hypothesis is true, not a remark
> > on the truthfulness of the Null Hypothesis.
>
> That's exactly right (except that it's not a fallacy): this figure of 1%
> is just a remark on the unlikeliness of what we've seen, given the Null
> Hypothesis.  It's our choice to take the next step and decide to take a
> closer look at that suddenly-suspicious hypothesis.  The 1% (or
> probability of 0.01, written as p=0.01) is the justification we can
> claim for that decision.
>
> A p-value of p=0.10 (or 10%) is pretty marginal, p=0.05 is publishable,
> p=0.01 is pretty good, as these things go, at least in the social and
> life sciences -- that is, no-one would reproach you for concluding, at
> least provisionally, that this is not a Siamese cat, first appearances
> notwithstanding.  Particle physicists (when discovering Higgs particles)
> like '5-sigma', or about 0.00006%, as a criterion.
>
> One could write a book about the interpretive logic here (and folk have)
> -- this is by no means terminological quibbling -- but I think a key
> point is that the conclusions in statistical logic are not as obligatory
> as in the deductive logic earlier in the book.  The step from 'p=0.01'
> to 'that is not a Siamese' is an inductive leap that we decide to make,
> with a warrant based on the statistical analysis.  I think that Juola
> and Ramsay's account in their Sect. 4.3.1 makes this sound more
> obligatory than it should be, but in contrast their Sect 4.3.2 is really
> saying that the decision is part of a larger very contingent discussion.
>
> The above is a 'frequentist' account, based on probabilities.  The other
> doctrine is 'bayesian' (who are not to be left alone with frequentists
> in the presence of sharp objects).  In the bayesian interpretation, we
> start off with some numerical degree of  'a priori' belief that the cat
> is a Siamese cat, and the discovery that it weighs 40 lb, combined with
> our knowledge of the distribution of cats' weights, allows us (using
> Bayes Theorem) to update our belief that this is a Siamese, specifically
> ending up with a rather _smaller_ 'a posteriori' belief that it is a
> Siamese.  The maths is much the same, but the rationale for our change
> of mind is substantially different.
>
> > I have a personal interest
> > in this that explains why I turned straight to their account
> > of the Null Hypothesis, since such logic has recently been
> > used to much rhetorical effect in my own specialized area,
> > which is authorship attribution by internal evidence. It
> > matters to me whether I'm understanding this topic
> > properly or not, and I'm genuinely asking members of this
> > list to correct me if I'm mistaken.
>
> I suspect the underlying argument (and I'm recapitulating a logic I'm
> sure you already understand) would go something like this:
>
>    1. you calculate some statistic or other from a given text -- say,
> the average word length (though obviously much more sophisticated
> statistics would be more helpful);
>
>    2. by analysing texts known to be by a particular author, Fred, you
> can determine the properties (for example mean and variance) of that
> statistic for Fred's texts;
>
>    3. for a new text X, you calculate the value of the statistic for the
> text X, and then adopting the null hypothesis that 'X is by Fred', you
> ask how unlikely this value is -- how surprised you are that Fred should
> write such a text -- given the known mean and variance obtained in (2).
>
> Given that unlikelihood, you can then have a discussion about how
> defensible it is to ascribe the text X to Fred.  The statistics feed
> into the rhetoric of this discussion; they don't supplant it.
>
> In the real case, I imagine one calculates multiple statistics for
> Fred's texts, calculates the same for broadly comparable texts by all
> authors, and then combines these various distributions together in a
> statistically sophisticated way.  The maths at this point becomes fairly
> hellish, but it remains a more sophisticated version of the basically
> straightforward argument above.  I see that Juola and Ramsay touch on
> this sort of argument in their Sect 4.4.2.
>
> I hope this shines a torch into the gloom.
>
> ----
>
> Just in passing: Juola and Ramsay have written an _ambitious_ book!
> They say near the beginning of Chap. 6 'this is a challenging chapter'.
> Well, it looks to me as if Chap 1--5 are pretty challenging, too.
>
> Enjoy,
>
> Norman
>
> --
> Norman Gray  :  https://nxg.me.uk
> SUPA School of Physics and Astronomy, University of Glasgow, UK




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