Death penalty and data modeling

This article just appeared in the "International Journal of Law and Information Technology". It purports to show that the death penalty is arbitrary because whether or not somebody who is sentenced to death can be predicted without considering any substantive characteristic of the crime in question. If true, the conclusion certainly would indicate that the death penalty really is applied based more on who was sentenced rather than what they did.

Unfortunately, there are a number of defects in the analysis given in that paper that me somewhat leery of taking the results at face value. Some of these problems represent insufficient detail in the article, others indicate a lack of rigor in the data analysis itself.

First, the authors include without definition the following three variables:

7. Third most serious capital offence
8. Second most serious capital offence
9. First most serious capital offence

These variables sound related to the capital crime for which the prisoner was sentenced. Without definition, we cannot judge. Moreover, without knowing how these variables were encoded, we cannot tell whether the model was able to use this input or not.

Secondly, the authors include the following four time based variables:

15. Month of conviction for capital offense
16. Year of conviction for capital offense
17. Month of sentence for capital offense
18. Year of sentence for capital offense

I understand what these variables represent, but am curious how the authors encoded them. Normally in a model like this, time would be encoded as a continuous linear variable from some relatively recent epoch. Separating a time variable like this often leads to problems if you are looking for recency effects. The other common use of separated month variables is to look for seasonality affects. Typically, though, this would be done using 1 of (n-1) encoding which you clearly did not use given how few inputs you have.

While the encoding of these variables doesn't really bear much on the validity of the results, it does make it unlikely that the model could have used these variables for the purpose intended.

Thirdly, the issue of encoding also comes up the geographical variable:

2. State

Encoded as a single variable, this is almost certainly an integer code. This is a very poor encoding of state (1 of 50 encoding would be much better, several variables with different resolution would be even better).

Regarding the learning results themselves, the authors apparently did not assess which input variables were responsible for their results. They should have used any of the standard techniques such as step-wise variable selection or input randomization. They should have also measured how much value the non-linear classifier that they used mattered to the results. Typically, this would be done by comparing the results available from a linear classifier such as a ridged logistic regression. With the encodings in use by these authors, such a straight-forward comparison is probably not viable.

The authors also do not appear to have done anything to determine whether there is a target leak in the data. This can occur when there is some apparently independent input which is accidentally highly correlated with the desired output. In this case, it is possible that one state is responsible for a large majority of the executions. This might make the output more predictable without providing much ammunition for the original argument.

Finally, the authors do not have any reference in their article about public availability of their data. Without making their data easily available and given how incredibly easy it is for novices to screw up a modeling effort, these results should be considered no more than slightly provocative.

Why the long tail isn't as long as expected

Several bloggers and authors are finding out (belatedly relative to people involved in the industry) that the economic returns in systems that are supposedly governed by long-tail distributions are unexpectedly concentrated in the highly popular items. See here for a blogger's commentary on this article.

In practice, the long tail model does predict certain kinds of consumption very well. I speak from experience analyzing view data at Veoh where, except for the very few top titles, a unit power law described number of views pretty well. This means that the number of views from sparsely watched videos is surprisingly large, adding to the woes of anybody trying to review submissions.

Economically speaking, though, you have to factor in a few kinds of friction. These are reasonably modeled as per unit sale costs and per unit of stock costs. The per unit sale costs don't affect the distribution of revenue and profit, but the per stock unit costs definitely do. If you draw the classic Zipf curve and assume that revenue is proportional to consumption for all items, then a per stock unit cost offsets the curve vertically. The resulting profit curve tells you pretty much immediately how things will fall out.

The graph to the right illustrates. The solid line is the classic long-tail distribution and the dashed lines represent different per stock unit costs. Wherever the solid line is above the dashed line, the model implies positive returns; where it is below, it implies loss. Clearly, the lower the stock unit costs, the deeper into the distribution you can go and still make money.

If we assume that you will only be stocking items that have non-negative return, then the percentage of profit that comes from a particular part of the distribution will vary with the threshold. For high thresholds, almost all of the profit will be from mega-hits but for very low thresholds, a much larger percentage will come from low consumption items.

This sort of situation is shown in this second graph which shows the percentage of total profit achieved for different thresholds and ranks. The concentration of profit in the low rank items for high thresholds is quite clear.

A good example of a per stock unit cost is given in p2p download schemes. The first few downloads of each item have to be paid for by the original source while additional downloads make use of sunk network resources that apply minimal marginal cost back to the source. For systems like bit-torrent, you need more than a hundred downloads before you get to high p2p efficiency which makes the system useful for mainstream content and less useful for body and tail content. The Veoh p2p system has a much lower threshold and is thus useful far down into the tail. Either system, though, leads to an economic return different from the theoretical zero-cost long-tail model.

None of these observations is earth-shaking and, as far as I know these are all common wisdom among anybody trying to make money out of the long tail. Why this is news to the Harvard Business Review is the real mystery to me.

Reading email

And you thought that would be easy.

If you have ever written a program to receive email (not just pick it up from a POP server), you know just how painful that can be. If you wrote the program in Java, you know this even better.

The world just got better:

http://subethasmtp.tigris.org/

This package is just as simple as reading email should be. Your program decides whether to receive the email and then it gets it. The interface is tiny and it looks like they handle enough of the standards to really work.

Whew.

A random walk from eigenvectors to parallel page rank

Power law algorithms are ideal for computing things like page rank. That isn't obvious, however.

The basic outline idea is that hub and authority style algorithms are intimately related to eigenvector or singular value decompositions (depending on whether the links are symmetrical). This also means that there is a close relationship to asymptotic beahavior of random walks on the graph. That probably still isn't obvious, so let's dig in.

If you represent the linkage in the web by a matrix that has columns representing source page and rows representing the target page and with a 1 where-ever the source page has a link pointing to the target page, then if you start with a vector with a single non-zero element equal to 1 as a representation of a single page, then multiplying by the linkage matrix will give you a vector with 1 in the positions corresponding to the pages the original page linked to. If you multiply again, you get all the pages that you can get to in two steps from the original page.

Mathematically, if we call the original vector x and the linkage matrix A, the pages that x links to are just Ax. The pages that are two steps from x are A(Ax) = A² x.

The eigenvector decomposition of A is just a way of writing A as a product of three matrices:

A = U S U'

U' is the transpose of U, and U has the special property that U'U = I (it is called ortho-normal because of this).

S is a diagonal matrix.

There is lots of deep mathematical machinery and beautiful symmetry available here, but for now we can just take this as given.

The set of pages n steps from x are

x_n = Aⁿ x = (U S U')ⁿ x = (U S U')^n-2 (U S U') (U S U') x
= (U S U')^n-2 (U S (U'U) S U') x = (U S U')^n-2 (U S² U') x
= U Sⁿ U' x

This is really cool because Sⁿ can be computed by just taking each diagonal element and raising it to a power.

Eigenvector decompositions have other, really deep connections. For instance, if you take the elements of S (call the i-th one s_i) then

_i s_iⁿ

is the number of paths that are n steps long.

Connected (or nearly connected) clusters of pages can also be derived from the eigenvector decomposition. This is the basis of so-called spectral clustering. For some very impressive examples of spectral clustering see this paper.

So eigenvectors are cool. But how can we compute them? And how can we compute them on BIG graphs in parallel?

First, note that if Aⁿ = U Sⁿ U' and if some of the s_i are bigger than others, the big ones will quickly dominate the others. That is pretty quickly, Aⁿ u₁ s₁ⁿ u₁'. This means that we can compute an approximation of u₁ by just doing Aⁿ x where x is some random vector. Moreover, we can compute u₂ by starting with a different random vector and iterating the same way, but with an additional step where we forbid the result from going towards u₁. With just a few additional wrinkles, this gives us what is called the Lanczos algorithm. Golub and van Loan's excellent book Matrix Computations gives a lot of information on these algorithms.

The cool thing here is that our random vector can represent a single page and we can approximate the final result by following links. Following links is just a (human-readable) way of saying sparse matrix multiplication. If we do this multiplication against lots of different random starting points, we can quickly build parallel algorithms to compute things like page rank.

Words at random, carefully chosen

On comp.ai, Dmitry Kazakov reiterated the lonely cry of a frequentist against statistical natural language. This cry has been repeated many times over the years by many people who cannot abide the treatment of documents and language as if they were random.

Let's examine the situation more carefully.

On Apr 14, 5:28 am, "Dmitry A. Kazakov" wrote:
> ... It cannot be probability because the document is obviously not random ...

The statement "It cannot be probability ..." is essentially a tautology. It should read, "We cannot use the word probability to describe our state of knowledge because we have implicitly accepted the assumption that probability cannot be used to describe our state of knowledge".

The fact that an object has been constructed in its present state by non-random processes outside our ken is no different as far as we can tell than if the object were constructed at random (note that random does not equal uniform). What if the document were, in fact, written using the I Ching (as Philip K Dick is reputed to have written "The Man in the High Castle")? Is it reasonable to describe the text as having been randomly generated now that we know that?

Take the canonical and over-worked example of the coin being flipped. Before the coin is flipped a reasonable observer who knows the physics of the situation and who trusts the flipper would declare the probability of heads to be 100%. After the coin is flipped, but before it is revealed, the situation is actually no different. Yes, the coin now has a state whereas before the coin was only going to have a state, but, in fact, the only real difference is that the physics has become somewhat simpler, the most important factor in our answering the question of the probability has not changed. We still do not know the outcome.

Moreover, if the person flipping the coin looks at the coin, that does not and cannot change our answer.

When WE look at the coin, however, we now suddenly, miraculously declare that the probability is now 100% that the coin has come up heads. Nothing has changed physically, but our estimate has changed dramatically.

Moreover, if we now examine the coin and find that it has two heads, our previous answer of 50% is still valid in the original context. If we were to repeat the experiment, our correct interpretation is to give 100% as the probability before the flip. The only difference is our state of knowledge.

So philosophically speaking, probability is a statement of knowledge.

Moreover, by de Finetti's famous theorem, even if this philosophical argument is bogus, the mathematics all works our AS IF there were an underlying distribution on the parameters of the system. That means that we can profitably use this philosophical argument AS IF it were true.

The upshot is that even if you are a frequentist in your heart of hearts, it will still pay to behave as if you were a Bayesian. And I, as a Bayesian, will be able to behave as if you were rational because I will not know your secret.

How Not to Recommend

A friend of mine pointed out The Netflix Prize: 300 Days Later this morning. The article points out how difficult it has been to make significant improvements on the Netflix rating prediction problem.

This article reinforces a lot of my own feelings. I continue to believe that recommendations based on ratings are inherently flawed. Among other things

• predicting ratings from ratings is hard
• predicting real user preferences from ratings is even harder
• most people don’t rate things very much, if at all (makes #1 even harder)

All of these factors are eased considerably if you just get rid of the idea that you have to predict ratings. Observing real behavior and then predicting that same behavior is vastly easier.

The Power of Meta

Evolutionary algorithms are oohhh so trendy, but they often provide very poor convergence rates. This occurs particularly when the population is in the neighborhood of a good solution, but the mutations being produced are far from that neighborhood because the mutation rate is too high.

Decreasing the mutation rate often doesn't help because then it takes forever to find the right neighborhood.

Changing the mutation rate externally can work, but you will usually get the annealing schedule wrong and have too much or too little mutation at some point.

One of the most effective ways to fix these problems is the technique known as meta-mutation. With meta-mutation, each member of the population has data about how dramatically it should be mutated. Then, when you are far from a good solution, members that take big steps can be big winners and their descendants will also take big steps. When you get close to a good solution, members with large mutation rates will step far from the best solution and will be mostly unable to improve the solution, but some descendants will have smaller mutation rates and these will able to find improved solutions. These calmer descendants will quickly dominate the population because they will be able to produce significant improvements.

Unfortunately, meta-mutation introduces the new problem of how you should mutate the mutation rate. (danger ... recursion alert)

I describe some methods that avoid all of these problems in my 1998 paper Recorded Step Mutation for Faster Convergence. The techniques are very simple and very effective. The paper is available from arxiv.

The real trick is that you can have the mutation rate encode the meta-mutation rate as well. One approach using a fixed distribution that is scaled by the current mutation rate. Another is to use the last step determine a directional component of the new mutation rate. Together, these approaches can improve convergence rate and final accuracy by many orders of magnitude relative to other approaches. Self-similar meta-mutation alone gives results that are nearly identical in symmetric Gaussian bowl problems to analytically determined optimal annealing schedules. When combined with directional mutation, results are near optimal for Gaussian bowls with low rank cross correlations.