Posted on

Multiplication of Learned Probabilities is Dangerous

This is about a design flaw in several learning algorithms such as the Naive Bayes classifier and Hidden Markov Models. A number of people are aware of it, but it seems that not everyone is.

Several learning systems have the property that they estimate some conditional probabilities P(event | other events) either explicitly or implicitly. Then, at prediction time, these learned probabilities are multiplied together according to some formula to produce a final prediction. The Naive Bayes classifier for binary data is the simplest of these, so it seems like a good example.

When Naive Bayes is used, a set of probabilities of the form Pr'(feature i | label) are estimated via counting statistics and some prior. Predictions are made according to the label maximizing:

Pr'(label) * Productfeatures i Pr'(feature i | label)

(The Pr’ notation indicates these are estimated values.)

There is nothing wrong with this method as long as (a) the prior for the sample counts is very strong and (b) the prior (on the conditional independences and the sample counts) is “correct”—the actual problem is drawn from it. However, (a) seems to never be true and (b) is often not true.

At this point, we can think a bit from a estimation perspective. When trying to estimate a coin with bias Pr(feature i | label), after observing n IID samples, the estimate is accurate to (at most) c/m for some constant c. (Actually, it’s c/m0.5 in the general case c/m for coins with bias near 0 or 1.) Given this observation, we should expect the estimates Pr’ to differ by c/m or more when the prior on the sample counts is weak.

The problem to notice is that errors of c/m can quickly accumulate. The final product in the naive bayes classifier is n-way linear in the error terms where n is the number of features. If every features true value happens to be v and we happen to have a 1/2 + 1/n0.5 feature fraction estimate too large and 1/2 – 1/n0.5 fraction estimate too small (as might happen with a reasonable chance), the value of the product might be overestimated by:

(v – c/m)n/2 + n^0.5(v + c/m)n/2 + n^0.5 – vn

When c/m is very small, this approximates as c n0.5 /m, which suggests problems must arise when the number of features n is greater than the number of samples squared n > m2. This can actually happen in the text classification settings where Naive Bayes is often applied.

All of the above is under the assumption that the conditional independences encoded in the Naive Bayes classifier are correct for the problem. When these aren’t correct, as is often true in practice, the estimation errors can be systematic rather than stochastic implying much more brittle behavior.

In all of the above, note that we used Naive bayes as a simple example—this brittleness can be found in a number of other common prediction systems.

An important question is “What can you do about this brittleness?” There are several answers:

  1. Use a different system for prediction (there are many).
  2. Get much more serious about following Bayes law here. (a) The process of integrating over a posterior rather than taking the maximum likelihood element of a posterior tends to reduce the sampling effects. (b) Realize that the conditional independence assumptions producing the multiplication are probably excessively strong and design softer priors which better fit reasonable beliefs.
Posted on

Yahoo’s Learning Problems.

I just visited Yahoo Research which has several fundamental learning problems near to (or beyond) the set of problems we know how to solve well. Here are 3 of them.

  1. Ranking This is the canonical problem of all search engines. It is made extra difficult for several reasons.
    1. There is relatively little “good” supervised learning data and a great deal of data with some signal (such as click through rates).
    2. The learning must occur in a partially adversarial environment. Many people very actively attempt to place themselves at the top of
      rankings.
    3. It is not even quite clear whether the problem should be posed as ‘ranking’ or as ‘regression’ which is then used to produce a
      ranking.
  2. Collaborative filtering Yahoo has a large number of recommendation systems for music, movies, etc… In these sorts of systems, users specify how they liked a set of things, and then the system can (hopefully) find some more examples of things they might like
    by reasoning across multiple such sets.
  3. Exploration with Generalization The cash cow of
    search engines is displaying advertisements which are relevant to search along with search results. Better targeting these advertisements makes money (a small improvement might be worth $millions) and improves the value of the search engine for the user.

    It is natural to predict the set of advertisements which maximize the advertising payoff. This natural idea is stymied by both the extreme
    multiplicity of advertisements under contract (think millions) and a lack of ability to measure hypotheticals like “What would have
    happened if we had displayed a different set of advertisements for this (query,user) pair instead?” This is a combined exploration and
    generalization problem.

Good solutions to any of these problems would be extremely useful (and not just at Yahoo). Even further small improvements on the existing solutions may be very useful.

For those interested, Yahoo (as an organization) knows these are learning problems and is very actively interested in solving them. Yahoo Research is committed to a relatively open method of solving these problems. Dennis DeCoste is one contact point for machine learning research at Yahoo Research.

Posted on

Pittsburgh Mind Reading Competition

Francisco Pereira points out a fun Prediction Competition. Francisco says:

DARPA is sponsoring a competition to analyze data from an unusual functional Magnetic Resonance Imaging experiment. Subjects watch videos inside the scanner while fMRI data are acquired. Unbeknownst to these subjects, the videos have been seen by a panel of other subjects that labeled each instant with labels in categories such as representation (are there tools, body parts, motion, sound), location, presence of actors, emotional content, etc.

The challenge is to predict all of these different labels on an instant-by-instant basis from the fMRI data. A few reasons why this is particularly interesting:

  1. This is beyond the current state of the art, but not inconceivably hard.
  2. This is a new type of experiment design current analysis methods cannot deal with.
  3. This is an opportunity to work with a heavily examined and preprocessed neuroimaging dataset.
  4. DARPA is offering prizes!
Posted on

Research Budget Changes

The announcement of an increase in funding for basic research in the US is encouraging. There is some discussion of this at the Computing Research Policy blog.

One part of this discussion has a graph of NSF funding over time, presumably in dollar budgets. I don’t believe that dollar budgets are the right way to judge the impact of funding changes on researchers. A better way to judge seems to be in terms of dollar budget divided by GDP which provides a measure of the relative emphasis on research.

This graph was assembled by dividing the NSF budget by the US GDP. For 2005 GDP, I used the current estimate and for 2006 and 2007 assumed an increase by a factor of 1.04 per year. The 2007 number also uses the requested 2007 budget which is certain to change.

This graph makes it clear why researchers were upset: research funding emphasis has fallen for 3 years in a row. The reality has been significantly more severe due to DARPA decreasing funding and industrial research labs (ATnT and Lucent for example) laying off large numbers of researchers about when the governments emphasis on basic research started declining.

It is certainly encouraging to see the emphasis on science growing again.

Posted on

Introspectionism as a Disease

In the AI-related parts of machine learning, it is often tempting to examine how you do things in order to imagine how a machine should do things. This is introspection, and it can easily go awry. I will call introspection gone awry introspectionism.

Introspectionism is almost unique to AI (and the AI-related parts of machine learning) and it can lead to huge wasted effort in research. It’s easiest to show how introspectionism arises by an example.

Suppose we want to solve the problem of navigating a robot from point A to point B given a camera. Then, the following research action plan might seem natural when you examine your own capabilities:

  1. Build an edge detector for still images.
  2. Build an object recognition system given the edge detector.
  3. Build a system to predict distance and orientation to objects given the object recognition system.
  4. Build a system to plan a path through the scene you construct from {object identification, distance, orientation} predictions.
  5. As you execute the above, constantly repeat the above steps.

Introspectionism begins when you believe this must be the way that it is done.

Introspectionism arguments are really argument by lack of imagination. It is like saying “This is the only way I can imagine doing things, so it must be the way they should be done.” This is a common weak argument style that can be very difficult to detect. It is particularly difficult to detect here because it is easy to confuse capability with reuse. Humans, via experimental tests, can be shown capable of executing each step above, but this does not imply they reuse these computations in the next step.

There are reasonable evolution-based reasons to believe that brains minimize the amount of computation required to accomplish goals. Computation costs energy, and since a human brain might consume 20% of the energy budget, we can be fairly sure that the evolutionary impetus to minimize computation is significant. This suggests telling a different energy-conservative story.

An energy consevative version of the above example might look similar, but with very loose approximations.

  1. The brain might (by default) use a pathetically weak edge detector that is lazily refined into something more effective using time-sequenced images (since edges in moving scenes tend to stand out more).
  2. The puny edge detector might be used to fill a rough “obstacle-or-not” fill map that coarsens dramatically with distance.
  3. Given this, a decision about which direction to go next (rather than a full path) might be made.

This strategy avoids the need to build a good edge detector for still scenes, avoids the need to recognize objects, avoids the need to place them with high precision in a scene, and avoids the need to make a full path plan. All of these avoidances might result in more tractable computation or learning problems. Note that we can’t (and shouldn’t) say that the energy conservative path “must” be right because that would also be introspectionism. However, it does exhibit an alternative showing the failure of imagination in introspectionism on the first approach.

It is reasonable to take introspection derived ideas as suggestions for how to go about building a (learning) system. But if the suggestions don’t work, it’s entirely reasonable to try something else.