De Rerum Natura

I’m sending out a birthday card with a cash gift. It feels strange to give cash. We seem to attach a “tacky” factor to doing so. Leave it to economist Joel Waldfogel to write a book (“Scroogenomics”) on the 20 to 30 percent waste in value that occurs each Christmas season. He argues that cash is actually the best gift we can give someone in most cases.

The gift recipient knows best what they want and giving cash allows them to get the most value for that expenditure. In other words, spending may or may not create value. When I guess what the other person may like, I open the opportunity for destroying the value of the exchange. Unless I now the recipient extremely well or can anticipate their needs perfectly, my gift may have lower value to them than the money spent on the gift. Giving cash allows the recipient to select what they most value as a gift, hence no gift can do better than cash in creating that maximum value. So if we must spend to give gifts, let’s make it most beneficial by giving cash.

I feel better already!

We are awash in data. Computers and cheap memory storage make it possible for us to collect 1/4 of a Gigabyte of information for every woman, man and child in the planet. Most of it is unused or unusable. We may be relatively good at searching this data but we suck at organizing it. Google is much better at finding than at organizing the data it processes. Digital pictures lie in our computers unorganized because we do not have the time to label every picture in which grandma appears.

Jon Stewart’s comments on the usefulness of the Apollo missions:

It was that fateful day in July that we planted the Stars and Stripes in the lunar surface, officially claiming the moon as America’s space Puerto Rico. It was all ours. It was the culmination of a dream. … It took us ten years, astronauts’ lives, billions of dollars, and all we did is hit a f***ing golf ball?

have made me think about how exploration is a good onto itself even when it fails. Indeed, it seems that the unintended consequences or discoveries of such explorations are good enough to justify our urge to carry them out.

I offer as my first example of unintended consequences a single photo — the famous Earthrise captured by the Apollo 8 mission on its fourth orbit around the Moon. As recounted in Richard Poole’s Earthrise: How Man First Saw the Earth it was mission commander Frank Borman that first noticed the Earth rising above the gray moonscape and pointed it out to William Anders, the mission scientific crew member. During the previous three lunar orbits, Anders had been busy photographing the lunar surface for possible landing sites. Anders handed the camera with black and white film he was using to Borman but then proceeded to warn him not take any pictures – “Hey, don’t take that, it’s not scheduled.”. Quickly reversing himself, Anders picked up a roll of color film and took the now iconic photo. The cultural influence of this photo is richly detailed in Poole’s book. I’ll mention one that greatly influenced me: it was used as the front cover image for the Last Whole Earth Catalog. As others have pointed out, it is ironic that not one of the people that envisioned space travel ever thought that its most important result would be the perspective we would gain on our home planet. Norman Cousins stated during a 1975 Congressional hearing on the future of space travel: “what was most significant about the lunar voyage was not that men set foot on the Moon, but that they set eye on the Earth.”

My second example on the unintended consequences of exploration and how failures can sometimes be glorious is Columbus’ Voyage to the Americas. Wishing to reach the East Indies, believing that the Earth was actually half as big as it really is, he hit upon the Americas and to his last days did not realize what he had done. We do not know what we will find when we search out from the safety of our known world, and that is why we should.

I want to apologize to my few readers for the lack of posts since May of last year. This is partly due to personal circumstances that overwhelmed me emotionally and mentally. Since last Fall, however, I have rebounded and have been, since Thanksgiving, been engaged in an immensely satisfying intellectual journey that has greatly generalized the work in the ICML 2008 paper.

Unfortunately, since September 29th, I have been employed as a Research Scientist at the Mobility Division of Nuance Communications. This means that my intellectual output now belongs to my employer and I am obligated to mantain the secrecy of my work until such time as it is patented.

My plans for the immediate future of this blog are  to concentrate on general scientific and “philosophical” matters related to the issues I have explored in previous posts and whatever else strikes my fancy.

Discussions in this blog on precision error estimation via ${\ell }_1$ minimization have made it clear that the technique is only good for recovering precision, not accuracy. This post will argue that accuracy can also be guaranteed if all the detectors have a greater than one-half probability of being correct.

By “guaranteed”, I mean that I have a high probability of being correct and that furthermore, I can estimate this probability with high confidence. The idea is not new to me and follows a well-known result: if all detectors have $P_{\text{true}}>1/2$ then, given enough of them, you can guarantee that simple majority voting will be correct at whatever level you want, for example, ninety-nine per cent of the time.

The idea follows from the following formula

$$P_{\text{voting correct}}=\sum _{i=n/2+1}^n\left(\begin{array}{c}n\\ i\end{array}\right)p^i(1-p{)}^{n-i},$$

where $p$ is the probability of each system detecting the right answer (which the formula assumes is the same for all of them. The formula also assumes that the detectors are uncorrelated. This is an important point and one I will return to in future posts. For the moment, I will assume detector independence to make the discussion simpler. The technique I am proposing, however, does not require it.

The probability of simple voting being correct shown in the formula above comes from counting the sequences where more than half the detectors give you the right answer. My main argument is that if you know the detectors are better than one-half individually, you can do quite well given enough of them.

The following plot shows the probability of being correct for five detectors as a function of their individual ability to figure out the correct answer.

The dashed line shows probability that majority voting gives you the right choice. Note that for $p_{\text{single detector correct}}<1/2$, it pays to ditch voting and just use the output of a single detector. Note also, that there is a point of maximum relative gain. For five detectors this occurs around 0.72 which gets you 0.86 probability of being correct.

Future posts will discuss how we can lever this simple idea in situations where we have no ground truth and are therefore incapable of estimating our correctness rate. Hint: it involves turning the autonomous difference equations used for the mapping application of precision error into majority voting equations.

My introduction of cool pop-out images has created all sorts of bad css entries for this site. The site no longer validates as having well-formed CSS (check out the validate button at the bottom of this page). I apologize to the purists out there. I am proud to say, however, that it validates as correct XML and MathML.

Combining the judgments of different recognizers is always better than using the best one alone. This observation is universal in machine learning realms. But it seldom gets used in practice. Why?

For one, it costs more to implement. Instead of one recognizer, you must deploy several. Computing cycles grow linearly with the number of recognizers if they all have roughly proportional cost. In academic circles, you get rewarded for finding the best recognizer/algoritm so there is little incentive to work in using the work of others along with your own. Companies usually have  proprietary rights to a few recognizers and want to deploy their best one to save development costs, hog less cycles on the customer’s machine, etc.

in certain conditions, however, it actually pays to use different recognizers and to incur the extra development and computing costs. Noisy or uncertain deployment environments can seriously degrade the performance of a well-tuned system. In such situations, the recognizer’s output could be worse than random and it would be impossible to place any certainty on it output.

The situation changes dramatically when we use different recognizers. To see this, consider the case of multiple recognizers, all of which have a probability greater than one half of getting the right answer. One can establish rigorous bounds for the number of recognizers that we would need to guarantee that a simple majority vote would have, say, greater than 99% probability of being right. This is discussed in the article on the Chernof Bound in Wikipedia.

But simple majority voting has its problems. For one, it is incredibly costly. Suppose that your detectors where correct two-thirds of the time. it would take at least 15 detectors to have 90 per cent probability of being correct when you averaged their individual votes

In addition, what if the testing condition starts to differ too much from the training data? The detectors could have degraded performance and 15 detectors may only give you fifty per cent chance of being correct. How would you know that your probability of being correct has dropped?

In future posts, I will explain a scheme using the precision error methodology that I have discussed in previous posts to create a robust voting system that automatically weighs the vote of a collection of systems according to their actual accuracy on the data, and it does this without any ground truth and in a completely data-driven way. Two system may “flip”, one becoming more accurate than the other and the algorithm I propose to combine their decisions would detect this reversal. The practical effect of this scheme is that one can predict, in the field, automatically the probability that one’s collection of recognizers is correct with an efficiency that is higher than simple voting. In compressed sensing, you use less sensors to reconstruct the same signal. I propose something similar, but for decision making: using less decision makers in a robust fashion to make the correct recognition decision.

All the experiments we have been carrying out with precision error have, so far, been with real data. Because of this, we do not have “ground truth” to determine if the reconstruction is correct. That changed today.

Synthetic experiments are a well-known device for studying models or algorithms. By artificially creating data where one knows exactly what is going on, one can then see if the algorithm one is testing is able to reproduce the artificially created “ground truth”.

I did that today by creating an artificial landscape as shown in the left figure at the top. A single example from the ten noisy versions of this landscape is shown in the figure on the right.

The advantage of using precision error is shown in the figures at the bottom. The figure on the left shows what happens after weighting the maps with the discovered precision error covariance matrix. The picture on the right is the result of simple averaging. The difference is clear. The weighted average is better, and precision error estimation was the way to obtain the weights.

Just to show that not all questions behave as nicely as question 9 in the previous post, here is the plot for question 6 in the same exam.The fit is not as good as for question 9. This is expected, there is no reason why the precision error should decay with a perfect exponential behavior. Nonetheless, it still shows a similar decay constant — about six questions. Remember to click on the image to get the larger image in a zoom pane.

To show off the installation of FancyZoom (a trick I learned while visiting the excellent Language Log), I present a graph of the percentage variation in the mean square precision error as a function of the number of questions used to compute it. The image looks small but you can now click on it to obtain a zoomed in version. Try it!

Note how good the fit is to a shifted exponential function of the form:
$$a+b*\exp (-c(n_q–3)).$$
The measurements are the small dots at $n_q=\{\mathrm{4,6,7,10,12}\}$. The fitted values are $a=0.06$, $b=0.2$, and $c=0.43$. The variable $c$ is the decay constant for the variability in the estimate. In particular, if you calculate its inverse $1/c$ you get the number of questions beyond three that will give you less than 33% variability in the estimate. This turns out to be about 2 questions. So ten or twelve questions should be enough for this group of students.

Once again, this suggests that teachers are asking too many questions in their multiple choice exams.