Musings

The science fiction writer H G Wells predicted that in modern technological societies statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. How far have we got, a hundred or so years later? A glance at the literature shows a shocking lack of statistical understanding of the outcomes of modern technologies, from standard screening tests for HIV infection to DNA evidence…
— Gigerenzer & Edwards, British Medical Journal 327 (2003), 741–744

1) A woman goes to her doctor’s office for a mammogram. She is told that

1. The incidence of breast cancer is 0.8%.
2. The “sensitivity” of the mammogram is 90%. (If she has breast cancer, the probability is 90% that it will show up on the mammogram.)
3. The “false positive” rate is 7%. (If the woman doesn’t have breast cancer, there’s a 7% chance the mammogram will nonetheless yield a positive result.)

The mammogram comes back “positive”. What is the probability that the woman has breast cancer?

1. 1%
2. 7%
3. 9%
4. 15%
5. 24%
6. 47%
7. 63%
8. 83%
9. 90%

In a recent study, a stunning majority of doctors tested (22/24) could not answer this question correctly.

The abstract to the paper

Bad presentation of medical statistics such as the risks associated with a particular intervention can lead to patients making poor decisions on treatment. Particularly confusing are single event probabilities, conditional probabilities (such as sensitivity and specificity), and relative risks. How can doctors improve the presentation of statistical information so that patients can make well informed decisions?

if, anything, soft-pedals the issue, for it seems that a shocking proportion of doctors themselves don’t understand garden-variety medical statistics.

Matters improved considerably, when the data was presented in terms of “natural”, rather than conditional probabilities (11/24 doctors in the “control group” got the above question right, when it was thus reframed). But that’s not the way this sort of data is conventionally presented to medical practitioners (let alone to patients).

The authors would like to believe that — if one could only “frame” the information in the right way, all would be well with the world. But I don’t see it.

2) Say various factors put the woman in a high risk group for breast cancer. Rather than the 0.8% risk faced by the general population, she faces a 2.4% risk of breast cancer. If the test comes back positive, what is the probability now that she has breast cancer?

More to the point, how do you even decide whether to administer a test or perform a procedure in the first place, without having to grapple with these “icky” statistical considerations. (Positive test results may require invasive/expensive followups, procedures carry their own risks, …)

Maybe patients don’t need to understand this stuff (though that makes a mockery of “informed consent”), but doctors surely do.

And it scares the bejeesus out of me that they, apparently, don’t.

Tip 'o the hat to Chris Bertram at Crooked Timber for making my day.

שנה טובה ומתוקה לכל אורחי הblog הזה.

The Boston Globe reports that the RIAA has dropped its lawsuit against Sarah Seabury Ward, a 66 year old sculptress whom they had accused of sharing 2000 songs on Kazaa. Trouble was that:

1. She’s a self-professed computer neophyte, who’d never installed any file-sharing software on her computer.
2. She seems a rather unlikely aficionado of gangsta rap (among the music she was specifically accused of having traded).
3. Her computer is a Macintosh. So, even if she knew how, and even if she’d wanted to, she couldn’t have shared those songs on Kazaa, which is Windows-only.

You’d think that, after such an obvious screw-up, the RIAA would be gracious, apologize to Mrs. Ward, offer to pay her legal expenses, and give her a gift-certificate to Tower Records. You’d think that, but you’d be wrong.

“Please note, however, that we will continue our review of the issues you raised and we reserve the right to refile the complaint against Mrs. Ward if and when circumstances warrant.”

was what their Law Firm wrote to her lawyer, in agreeing to drop their suit.

One of the signs of Depression is that the individual no longer cares about “keeping up appearances.” The RIAA’s rhetoric may still be that of an aggrieved party seeking redress for injuries sustained. Their body language say differently.

My previous post on Dvali and Kachru’s recent paper stimulated a lot of interest. I got into a really wonderful physics conversation with Konstantin Savvidis of the Perimeter Institute and with Sonia Paban down the hall.

In my previous post, I basically asserted that Dvali and Kachru’s proposal would not work with the sorts of parameters they envisage. But we thought it would be more fruitful to ask what range of parameters would be required to make it work.

Sometimes I get depressed about this whole weblog thing. On the authoring side, it’s just about as good as I could wish for — compose an entry in a LaTeX dialect, click post and bingo! it’s converted to XHTML+MathML and posted to the web.

On the client side, however, it sucks. Never mind that only Gecko-based browsers support MathML. Even their rendering is piss-poor and, at least under MacOSX, getting worse.

Consider the humble minus sign. Up until recently, Mozilla rendered this as a hyphen, “-”. That’s wrong, and looks horrible. The most recent Mozilla builds have, instead, shifted to using the real minus sign glyph from the Symbol font as the default.

There’s one wee problem: Mozilla, under MacOSX doesn’t recognize the Symbol font and so now all the minus signs have disappeared from the equations on my blog. Camino does recognize the Symbol font, but it doesn’t do MathML.

Don’t bother asking if I’ve filed a bug. Using the glyph from the Symbol font is the “right” thing to do, under the circumstances. It would work just fine under MacOSX, if only Mozilla’s font support under MacOSX were not so terrible. But that’s already the subject of enough bug reports.

All I can say is that I really hope this nonsense clears up when the Stix fonts become available. In the meantime, the only viable platform for reading this blog is Mozilla on Linux with the Computer Modern fonts installed — not exactly a mass-market configuration.

I am holding my breath, but anoxia is setting in.

Update (9/24/2003): This particular problem with minus signs can be fixed by including the line

user_pref("font.mathfont-family.\u2212.base", "Math1");

in your user.js file. The general point still remains…

Okounkov, Reshetikhin and Vafa have a paper out today in which they relate the Gopakumar-Vafa conjecture for the Partition Function of the Topological A-Model to the statitstical mechanics of melting crystals.

In the limit of large-volume Calabi-Yau, with Euler characteristic, $\chi$, the genus $g$ vacuum amplitude of the A-Model is

where $c_i$ is the $i^{\mathrm{th}}$ Chern Class of the Hodge bundle, $H\to \overline{\mathcal{M}_g}$ (the bundle whose fiber over $\Sigma_g$ is spanned by the $g$ holomorphic 1-forms on $\Sigma_g$).

The integral is given by

where the $B_i$ are Bernoulli numbers. Gopakumar and Vafa argued that the all-genus result could be evaluated by a 1-loop computation in M-theory. Summing over BPS states, they obtained

where

The present paper notes that this is the partition function for the statistical mechanics of corner-melting of a 3D cubical crystal. They argue for a rather literalist interpretation, where the 3D space is the base of Calabi-Yau, written as a $T^3$ fibration. And they suggest various generalizations, relating the recently-constructed Topological-Vertex of the A-Model to a dimer problem.

Wild stuff!

Anne van Kesteren is horrified to learn that (according to the DTD)

<a id="Grandfather">
  <del id="Father">
    <a id="Son">
      I'm My Own Grandpa
    </a> 
  </del> 
</a>

is legal XHTML.

To which I say, Ray Stevens understood this a long time ago.

No, I’m not talking about Hurricane Isabel. Yesterday, it was OpenSSH. Today, it’s Sendmail.

Now, I have things pretty automated: download the distribution, check the PGP signature, unpack, configure, compile and install — all scripted. Should be a breeze.

But nooo! This time I discover that Sendmail won’t compile with the, recently installed, Apple DevTools Update (gcc 3.3).

So I added

define(`confCC', `gcc2 -traditional-cpp -pipe ${Extra_CC_Flags}')

to my site.config.m4 file — switching the compiler to gcc2 — and it compiled just fine. Whew!

Now all I need, to round out my week, is a security hole in Apache 2.0.47.

Update (9/22/2003): MacOSX 10.2.8 was just released, which deals with various security issues , including this one and the OpenSSH issue mentioned previously. And it “only” took a week!

OpenSSH 3.7.1p1 was released today. It patches a potential security flaw. There are some allegations that there is actually an exploit, but the experts seem to view this as unlikely. No matter, I don’t mess around when it comes to Security.

I downloaded the source (be sure to check the PGP signature!) and compiled. It compiled just fine under MacOSX 10.2.6. But, after installation, the daemon kept dying whenever a client connected, because of some funny business with setuid(). Not wanting to mess around, I grabbed uidswap.c (not a part of the code affected by this vulnerability) from 3.6.1p1 and recompiled. It worked fine after that.

Update (9/17/2003): Despite some back-and-forth with Darren Tucker on openssh-unix-dev, we were unable to resolve this. So, until Apple comes along and releases a binary, MacOSX users should use the previous version of uidswap.c when compiling OpenSSH 3.7.1p1.

Update (9/18/2003): There’s now a patch to configure.ac which fixes the issue for MacOSX.

Update (9/22/2003): MacOSX 10.2.8 was just released, which deals with various security issues , including this one and the Sendmail issue mentioned above. And it “only” took a week! (N.B. the update does not include OpenSSH 3.7.1. Apple just applied a patch to version 3.4. While this fixes the immediate problem, there is a long list of reasons why you shouldn’t be running an outdated version of critical security software.)

Update (9/23/2003): The patch mentioned in the comments is no longer necessary. This, and several other more minor bugs were fixed in OpenSSH 3.7.1p2. Compile and Enjoy!

Just saw the Brazilian Quasar Dance Company, performing “Lend Me Your Eyes” — a really stunning multi-media meditation on the stages of life and the social problems of modern Brazil.

Catch 'em if you can.

Dvali and Kachru have written a paper, in which they try to resuscitate “Old Inflation.” In Old Inflation, the inflaton is caught in a false minimum of the scalar potential, leading to inflation. Eventually, it tunnels through the barrier, and ends up in the true minimum. Unfortunately, the bubbles of true vacuum don’t end up percolating the universe. The alternative, “New Inflation,” dispenses with the false minimum, and simply has the inflaton rolling down the scalar potential towards the true minimum. To achieve enough inflation, it must roll very slowly, which requires the potential to be fine-tuned to be ridiculously flat.

This is a bit distasteful; one would prefer a mechanism which would work for a more robust range of parameters of the potential. Dvali and Kachru’s idea is to have the inflaton dynamically trapped at a saddle-point of the potential. Consider the following potential for two scalar fields $$V(\phi,\psi)= \frac{\lambda_1}{2} \left( \phi^2 (\phi - m_1)^2 +m_2^2 \phi^2\right) +\lambda_2 \phi^2\psi^2 +\frac{\lambda_3}{4} (\psi^2 - m_3^2)^2$$ This potential has a global minimum at $\phi=0$, $\psi=\pm m_3$ and a saddle-point at $\phi=\psi=0$. For a reasonable range of parameters, it has a local minimum at $$\phi = \textstyle{\frac{m_1}{4}}\left(3 + \sqrt{1 - 8m_2^2/m_1^2}\right)\sim m_1,\quad \psi=0$$ We’re particularly interested in the limit $$m_1^2 \gg m_2^2,\quad \lambda_1 m_1^2 \gg \lambda_3 m_3^2$$

The scalar potential, $V(\phi,\psi=0)$. The universe starts in the false vacuum, tunnels through the barrier and starts oscillating about the saddle-point.

We imagine the scalar field starts out in the false vacuum, $\phi\sim m_1,\psi=0$. The universe inflates, but eventually, through bubble nucleation, the scalar field tunnels through the barrier, and $\phi$ starts oscillating about $\phi=0$. So far, this is just like Old Inflation. But $\phi=\psi=0$ is not the true vacuum. It’s actually a saddle-point. $\psi$ would like to roll down the hill towards the true minimum. But, since $\phi$ is oscillating very rapidly (we’ve arranged for the frequency of oscillation to be much greater than the scale given by the negative mass-squared in the $\psi$-direction), $\psi$ is prevented from rolling down the hill by the $\lambda_2 \phi^2 \psi^2$ term in the potential. Thus the universe continues to inflate, and the bubble grows to a size larger than the current horizon.

Eventually, the amplitude of the $\phi$ oscillation is red-shifted away, and $\psi$ rolls down the hill, ending inflation. Or so Dvali and Kachru would have us believe.

My problem with this scenario is that I see no reason to believe that the initial conditions for this second stage of inflation (immediately after the bubble forms) have $\psi=\dot{\psi}=0$. If the initial conditions are not very precisely tuned, you miss the saddle-point, and the scalar field rocks its way down to the bottom of the potential, too rapidly to save you from the evils of Old Inflation. It seems to me that small quantum fluctuations will always ruin the Dvali-Kachru scenario.

Update (9/25/2003): Here’s a more careful (and pessimistic) analysis.

Nothing seems to sow more confusion among markup geeks than the proper usage of the <abbr> and <acronym> elements. Many discussions get sidetracked by the arcana of English grammar (“initialisms” versus “contractions” versus …) rather than focussing on the real issues. HTML is not English, and one should not confuse a discussion of the one with a discussion of the other. In HTML, these elements play two important roles:

1. Provide a definition of the term via the title attribute, as in

   <acronym title="Friend Of A Friend">FOAF</acronym>
   <abbr title="Cold Dark Matter">CDM</abbr>

   In addition to helping out readers who may be unfamiliar with the term, this can be useful to Search Engines (by informing them that a page making frequent reference to “CDM” is actually related to pages talking about “Cold Dark Matter”).
2. Provide a cue to screenreaders (or other assistive technologies) as to whether the term should be pronounced or spelled-out. This might be made explicit in an aural stylesheet

   abbr {speak:spell-out;}
   acronym {speak:normal;}

   or it might be implicit in the rules used by the screenreader.

If the only purpose were to provide a definition, there would be no point in having two distinct element which serve the same function. And, given that there are two elements, it wouldn’t much matter which one you used. On the other hand, if these two elements are to be treated differently by aural browsers, then, on Accessibility grounds, you should be mindful of using the right one in each case.

Which brings us to the next point. For the purpose of providing a definition, it suffices to mark up the first occurrence of an acronym or abbreviation on the page. It’s somewhat annoying to see a page littered with dotted underlines (indicating the presence of a tooltip giving the definition) on each and every occurrence of an abbreviation. Indeed, the WCAG says you only need to provide the definition at the first occurrence on the page.

But that’s not quite the same thing as saying you don’t have to mark up subsequent occurrences at all. Providing cues for screenreaders surely requires marking up all occurrences on the page. (I imagine that this might also help Search Engines, but that’s purely hypothetical.) The best way to achieve these varied goals is, the first time you use the abbreviation, to give its definition, <abbr title="Cold Dark Matter">CDM</abbr>, but on subsequent uses, to merely mark it up as <abbr>CDM</abbr>. This will satisfy the screenreaders, and can be hidden from visual browsers with code like

abbr, acronym {border:none;}
abbr[title], acronym[title] {border-bottom: 1px dotted black;}

Marking up all those abbreviations and acronyms by hand is pretty tedious. Which is why I recently installed the acronym plugin for MovableType. It uses a flat-file database, acronym.db to hold the definitions. It only supports the <acronym> element, but it is trivial to hack it to support <abbr> as well (with two separate databases, acronym.db and abbr.db). If you prefer the old behaviour, you can just put all your definitions in the acronym.db. Since it’s reasonably well-written, getting it to add a title to only the first instance of an acronym/abbreviation in each blog entry was also fairly straightforward. Once it’s installed, you activate it by adding the attribute, acronym="1", to any MT container tag (like <MTEntryBody>) and rebuild.

Anyway, here’s my patch to the acronym plugin (patch < acronym.pl.diff). You need to install the patched plugin and two data files, acronym.db and abbr.db (the latter can be empty, if you wish, but must exist), in your plugins directory.

One note of caution: the plugin is very fast because it doesn’t retokenize after each acronym/abbreviation substitution. This has an unfortunate side-effect. Don’t put other acronyms/abbreviations as part of the definition text in your database!

P.S.: Some people contend that, since Internet Explorer doesn’t understand the <abbr> element, you shouldn’t bother doing the right thing. There are two obvious responses: you could use a Javascript solution, or you could just not worry about it. Since it seemed to me that IE users need all the help they can get, I decided, in this case, to go the Javascript route. Besides, the idea of using crufty Javascript hacks to service a crufty browser seemed … only fitting.

Update (9/12/2003): I should have made an obvious point. If your page is sufficiently complicated and has multiple entry points, a visitor may not enter the page at the top, and hence might miss the definitions found there. In that case, it may not suffice to define each abbreviation once per “page.” It would be more usable to define it once per “section.” (You can see this on my blog’s main page, where each post is a separate section, vis-a-vis marking up abbreviations.) Also, you’ll notice that I haven’t enabled automated abbreviation markup for comments. Some people think that that’s a “service,” but if you leave a comment on one of my posts that says, “Jacques, I think you have succumbed to Steve Jobs’ infamous RDF.” I don’t think you are going to want that to be automatically marked up as “RDF.”

Update (10/29/2003): The patch has been updated to match version 0.5 of the plugin.

Update (12/20/2003): The patch has been updated to match version 0.6 of the plugin.

Update (2/13/2004): The patch has been updated to match version 0.7 of the plugin.

Update (5/28/2004): The patch has been updated to match version 1.0 of the plugin.

One of the topics we will be covering this semester in the Geometry and String Theory seminar, that I run jointly with Dan Freed, is recent work of Caldararu on D-branes in the topological B-model.

Recall the basic setup, which is a twisted version of the $N=2$ $\sigma$-model with target space a Calabi-Yau 3-fold, $X$. The closed-string sector of the topological theory consists of a finite number of states corresponding to infinitesimal deformations of the complex structure of $X$. Naively, a D-brane in this theory consists of a holomorphic submanifold $C\subset X$ with a holomorphic vector bundle, $V$ on it (with certain properties).

If you want to be fancy, you can call that data a (particular example of a) coherent sheaf on $X$ (by extension by zero). In this context, Eric Sharpe and collaborators have written a series of papers recently which prove by explicit calculation that the open-string spectrum stretched between “sheaf $\mathcal{F}$” and “sheaf $\mathcal{G}$” is the ext group $Ext(\mathcal{F},\mathcal{G})$.

A number of years ago, Douglas proposed that the full set of D-branes in the B-model are objects in $D^b(X)$, the Bounded Derived Category of Coherent Sheaves on $X$ and that the open string states are the morphisms of that category. The objects in the derived category are easy to describe. They are (bounded) complexes of coherent sheaves:

and the morphisms are … well, that’s a bit complicated to state. Suffice it to say that, in the special case of one-term complexes, $\mathcal{F}$ and $\mathcal{G}$, the space of morphisms between those two objects in the derived category is again $Ext(\mathcal{F},\mathcal{G})$. So the “familiar” D-branes certainly form a subcategory of $D^b(X)$. But no one has constructed the D-branes corresponding to more general objects in $D^b(X)$, much less computed the corresponding open string spectra.

There are many interesting consequences of Douglas’s conjecture, but there are many puzzling features as well. There’s a $\mathbb{Z}$-grading which doesn’t have an obvious physical interpretation. Physically, one might have a use for a $\mathbb{Z}/2$ grading, corresponding to D-branes and anti-D-branes. Correspondingly, one can construct a category in which we identify the objects $A$ and $A[2]$ (the same complex, $A$, shifted to the left by two units) and whose morphisms are

(with the obvious composition law). Or maybe we should consider a $\mathbb{Z}/6$ grading (since the closed-string sector violates ghost charge by 6 units). Or … There are lots of possibilities. And since we don’t have such a good physical handle on the (existence or properties of) the “exotic” D-branes described by $D^b(X)$, we are not in great shape to decide between them. [To be more precise, Douglas’s conjecture is that the D-branes of the topological B-model correspond to quasi-isomorphism classes of objects in $D^b(X)$. The “exotic” D-branes are the ones not quasi-isomorphic to a (direct sum of) coherent sheaves.]

This is where (I hope) Caldararu’s work comes in. The set of possible D-branes and the corresponding open strings must satisfy a definite set of axioms, the Moore-Segal Axioms of open/closed Topological Field Theory. What Caldararu shows is that $D^b(X)$ satisfies the Moore-Segal axioms.

That’s great! A “physical” explanation for why $D^b(X)$ is the right answer and, perhaps, the alternatives unsatisfactory. After we’ve studied this stuff for a while, maybe I’ll be able to post a cogent explanation for the … ahem! … masses.

Darn it! I was going to post some physics today. Instead, here’s another web-design post.

Anne van Kesteren is trying to add SVG images to his blog, but is confused about how to size them. To rescale properly, they should be sized in relative units (em), rather than absolute ones (px). But the height and width attributes on <object> don’t allow this.

So here’s what I do:

<object type="image/svg+xml" data="/path/to/image.svg"
    width="100%" height="100%" style="width: 18em; height: 7em;">
  <img src="/path/to/image.gif" width= "288" height="112" alt="description" />
</object>

The width and height attributes on the SVG are set to simply fill the allocated box. The actual size of the box is set in relative (em) units using an inline style.

The fallback GIF image is set to have the same size when the text-zoom is set to “normal” (1em = 16px). But the SVG image will rescale along with the text when you use text-zoom to resize the text.

Easy, eh?

Update: And, yes, this does work in Internet Explorer, not that I care.

Update (3/30/2004): In many applications, you need to worry about what’s going to happen to your layout when the fixed-size GIF is is displayed instead of the resizable SVG. Particularly nettlesome is the case where the text size is set to less than the default 16px. Then the GIF is larger than you want. Here are some suggestions for what to do. You can see them in action here.

<rant>

I just happened to fire up IE 5.2 (MacOSX), and looked at this web site. The GIF image of the Asymmetry Data from BELLE in this post was simply missing! Here’s the offending code:

<div style="text-align:center;overflow:auto;margin-bottom:20px;">
  <img src="/~distler/blog/images/asym-highr-wsm.gif"
       title="http://www.kek.jp/press/2003/image/asym-highr-wsm.gif"
       style="border:0;width:567px;height:420px;"
       alt="Asymmetry data in B to phi K_S from BELLE" /><br />
<span class="figurecaption">Asymmetry data from BELLE, from which the value of
 <math xmlns='http://www.w3.org/1998/Math/MathML'>
  <mo lspace="0em" rspace="thinmathspace">sin</mo>
  <mo>(</mo><mn>2</mn><msub><mi>&phi;</mi> <mn>1</mn></msub><mo>)</mo>
 </math>
is determined, plotted as a function of the decay time. The
smooth dotted curve indicates what one would expect from the
Standard Model, whereas the smooth solid curve is a fit to the
observed data.
</span>
</div>

Ignore that IE doesn’t support MathML; many people don’t care about being able to read the formulæ. But not to display the entire <div> because it doesn’t know what to do with overflow:auto?? That’s just brain-dead.

I give up. Life is too short…

Henceforth, InternetExplorer is UNSUPPORTED. If you view these pages in that family of browsers, they may look odd, whole sections may be missing, all sorts of bad stuff may have happened. I don’t know, and I don’t care. And I ain’t gonna “fix” it.

Get yourself a Standards-compliant browser

</rant>

Update (9/5/2003): Having cooled down a bit, I managed to fix this particular problem with IE. But I stand by my “UNSUPPORTED” position, and now have posted a warning for the (mercifully few) IE visitors who frequent this blog.

Howard Dean and his supporters are nothing, if not clever, as demonstrated by this riff on Apple’s Switch campaign.

I hope their “marketshare” proves to be a little higher, though.