A Venn diagram gives a schematic view of joint counts on a set of \(n\) categories, e.g. \(c(S_1^n=s_1^n)\) where \(s_i\in\{0,1\}\). Each “patch” of the diagram corresponds to one of \(2^n\) possible values of \(s_1^n\).

If we have the total count \(C\triangleq \sum_{s_j: j\in \{1,…,n\}} c(S_1^n=s_1^n)\), then we can take the counts as probabilities by normalizing with \(p(S_1^n=s_1^n)=c(S_1^n=s_1^n)/C\).

Suppose we are given only singleton marginals \(p(S_i=1)\triangleq \sum_{s_j: j\in \{1,…,n\}\backslash i} p(S_1^n=s_1^n)\), then we can bound the other probabilities by imposing universal constraints on probabilities to be between 0 and 1.
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Was installing a package on Linux and ran into huge problems.

First, the package — let’s call it ‘somecrap’ — contained a post-installation configuration. Linux has this terminal pseudo-GUI used sometimes for configuration of packages. You may have seen it, it looks like ASCII art. Unfortunately it’s so brittle that when you Ctrl-Z out of it there is no way to get it back. So I had to just Ctrl-C out of it. Turns out the process that runs it (‘whiptail’) is stuck using 100% of CPU. So that’s fine, it too can be killed, but how to get a configuration do-over? The package thinks it’s already configured — and indeed, wrote out a default configuration file. There is no way to get a redo short of removing and reinstalling the package. Right? And that’s where the real problem starts.
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People have proposed using price signals from prediction markets to estimate the odds of certain events. On Intrade right now, you can buy contracts for the two outcomes of the 2012 US Presidential Election. Each contract expires at $10 if the event occurs or $0 if it doesn’t. For example, “Barack Obama wins” contracts are $6.33 a pop right now, while “Mitt Romney wins” contracts go for $3.65. On the page, these are taken directly as probabilities, because it is assumed that the gamble is zero-sum.

Specifically, if \(p\) and \(\bar{p}=1-p\) are respectively the probabilities of two complementary events, and \(a\) and \(b\) are respectively the prices of contracts on them, which can be bought and sold freely, then no-arbitrage imposes that \(-a-b+10 = 0\) and statistical no-arbitrage imposes \(-\bar{p}a +p(10-a) = 0\) and \(-pb +\bar{p}(10-b) = 0\). Solving indeed gives the prices \(a=10p\) and \(b=10\bar{p}\).

However, this isn’t the end of the story.
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This brilliant infinite looping version of Gangnam Style uses machine learning to join the similar points of the song to each other, and allows branching among them. It is slightly annoying, however, because the song does not start on the first beat, so all the bars are misaligned by one. I decided to redo the chart with the correct alignment and labeling, so here it is:

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We would like to prove the following fact:

For any non-negative random variable \(X\) having finite first and second moments, \(\mathbb P(X>0) \ge (\mathbb EX)^2/\mathbb EX^2\).

The proof isn’t difficult. Here are three different ones.
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I swear the concept of compactness was invented to remedy the shortcomings of closedness. Compact sets are closed (in Hausdorff spaces and therefore metric spaces), so compactness is stricter than closedness. It evidently patches some feebleness in the definition of closedness to make it more useful.

Closedness of a set in a metric space (“includes all limit points”), by the sound of it, really wants to be something akin to “has solid boundaries.” But it isn’t. The problem is that the existence of limit points depends on the embedding space. If the embedding space lacks those limit points, then a set in it can be technically closed even though it isn’t really “like” other closed sets. For example, the set \(\mathbb R\) in space \((\mathbb R, d_{\text{Eucl.}})\) is closed, because the space has no point called \(\infty\).
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QE3 was announced today and reactions have been relatively muted. There are some complaints that money is again being redistributed from asset holders to debtors via the mechanism of negative real rates. It seems like a good occasion to put forth two oddities that I’ve always seen as embedded in capital markets as they’re currently constructed. They are: the assumption that money doesn’t spoil, and the assumption of market optimality.
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After more than 1 year — judging by this post — the Alewife bound part of Pythagoras is still broken. Up till a few months ago the graffiti stayed off. But finally somebody caught on that they aren’t actually going to fix this a second time. Blame it on shoddy mechanics — the poor handle broke after only a month in use the first time they fixed it.

Olympic athletes use a huge amount of sponsor money — not to mention legal and illegal performance aids — to reach gold. Soon we will have genetically engineered physiology to reach even greater records. Schools compete for an annual #1 ranking. They spend more and more money to bid for the best professors and build the best facilities, driving up tuition. Coding theorists run massive simulations to find the best code to compete for the one spot in standards. But is the second place athlete, school, and code that much worse? No, usually they are nearly as good as #1.

I’ve often wondered whether many problems in the world are not variations of attempting “exact optimization” — this being the only way to guarantee success in a winner-take-all reward system.
(Read the article)

I finally had enough of two long-standing problems that bugged me in Windows 7, so I fixed them.

1. The ‘System’ process uses a low level of CPU (~6%) when nothing is going on.
2. Chinese characters in the system (e.g. file names) suddenly become boxes.

The first problem caused excessive battery drain along with the CPU fan going on and off all the time. Because Task Manager doesn’t tell you what is under the ‘System’ process, I had to use Process Explorer, which promptly identified the culprit: a revolving series of threads labeled ‘ntoskrnl.exe!KeReleaseInStackQueuedSpinLock+0x1e0′. This would appear to be some issue with a driver stuck on something and aimlessly retrying. It turns out turning off Wifi cured it. An updated driver for the Broadcom 4313 802.11n card fixed the problem permanently. The old version was 5.100.82.82 from 5/2011. The new version that does not have the problem is 5.100.82.112 from 10/2011.

The second problem is caused by a little-known new setting in Windows 7. Under Control Panel -> Fonts -> Font settings, there is an option “Hide fonts based on language settings.” It’s simply broken, so I just turned it off.