Akinator.com and Naive Bayes classifier

Question

Context: I am a programmer with some (half-forgotten) experience in statistics from uni courses. Recently I stumbled upon http://akinator.com and spent some time trying to make it fail. And who wasn't? :)

I've decided to find out how it could work. After googling and reading related blog posts and adding some of my (limited) knowledge into resulting mix I come up with the following model (I'm sure that I'll use the wrong notation, please don't kill me for that):

There are Subjects (S) and Questions (Q). Goal of the predictor is to select the subject S which has the greatest aposterior probability of being the subject that user is thinking about, given questions and answers collected so far.

Let game G be a set of questions asked and answers given: $\{q_1, a_1\}, \{q_2, a_2\} ... \{q_n, a_n\}$.

Then predictor is looking for $P(S|G) = \frac{P(G|S) * P(S)}{P(G)}$.

Prior for subjects ($P(S)$) could be just the number of times subject has been guessed divided by total number of games.

Making an assumption that all answers are independent, we could compute the likelihood of subject S given the game G like so:

$P(G|S) = \prod_{i=1..n} P(\{q_i, a_i\} | S)$

We could calculate the $P(\{q_i, a_i\} | S)$ if we keep track of which questions and answers were given when the used have though of given subject:

$P({q, a} | S) = \frac{answer\ a\ was\ given\ to\ question\ q\ in\ the\ game\ when\ S\ was\ the\ subject}{number\ of\ times\ q\ was\ asked\ in\ the\ games\ involving\ S}$

Now, $P(S|G)$ defines a probability distribution over subjects and when we need to select the next question we have to select the one for which the expected change in the entropy of this distribution is maximal:

$argmax_j (H[P(S|G)] - \sum_{a=yes,no,maybe...} H[P(S|G \vee \{q_j, a\})]$

I've tried to implement this and it works. But, obviously, as the number of subjects goes up, performance degrades due to the need to recalculate the $P(S|G)$ after each move and calculate updated distribution $P(S|G \vee \{q_j, a\})$ for question selection.

I suspect that I simply have chosen the wrong model, being constrained by the limits of my knowledge. Or, maybe, there is an error in the math. Please enlighten me: what should I make myself familiar with, or how to change the predictor so that it could cope with millions of subjects and thousands of questions?

Answer 1

This game looks similar to 20 questions at http://20q.net, which the creator reports is based on a neural network. Here's one way to structure such network, similar to the neural network described in Concept description vectors and the 20 question game.

You'd have

1. A fixed number of questions, with some questions marked as "final" questions.
2. One input unit per question, where 0/1 represents no/yes answer. Initially set to 0.5
3. One output unit per question, sigmoid squished into 0..1 range
4. Hidden layer connecting all input units to all output units.

Input units for questions that have been answered are set to 0 or 1, and the assumption is that neural network has been trained to make output units output values close to 1 for questions that have "Yes" answer given a set of existing answers.

At each stage you would pick the question which NN is the least sure about, ie, corresponding output unit is close to 0.5, ask the question, and set corresponding input unit to the answer. At the last stage you pick an output unit from the "final question" list with value closest to 1.

Each game of 20 questions gives 20 datapoints which you can use to update NN's weights with back-propagation, ie, you update the weights to make the outputs of current neural network match the true answer given all the previous questions asked.

Answer 2

I don’t think it is really a classification problem. 20 questions is often characterized as a compression problem. This actually matches better with the last part of your question where you talk about entropy.

See Chapter 5.7 (Google books) of

Cover, T.M. and Joy, A.T. (2006) Elements of Information Theory

and also Huffman coding. This paper I found on arXiv may be of interest as well.

Gill, J.T. and Wu, W. (2010) "Twenty Questions Games Always End With Yes” http://arxiv.org/abs/1002.4907

For simplicity assume yes/no questions (whereas akinator.com allows allows maybe, don’t know). Assume that every possible subject (what akinator.com knows) can be uniquely identified by a sequence of yes/no questions and answers — essentially a binary vector.

The questions that are (and their answers) asked define a recursive partitioning of the space of subjects. This partitioning also corresponds to a tree structure. The interior vertices of the tree correspond to questions, and the leaves correspond to subjects. The depth of the leaves is exactly the number of questions required to uniquely identify the subject. You can (trivially) identify every known subject by asking every possible question. That’s not interesting because there are potentially hundreds of questions.

The connection with Huffman coding is that it gives an optimal way (under a certain probabilistic model) to construct the tree so that the average depth is (nearly) minimal. In other words, it tells you how to arrange the sequence of questions (construct a tree) so that the number of questions you need to ask is on average small. It uses a greedy approach.

There is, of course, more to akinator.com than this, but the basic idea is that you can think of the problem in terms of a tree and trying to minimize the average depth of its leaves.