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Context: I am a programmer with some (half-forgotten) experience in statistics from uni courses. Recently I stumbled upon 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?

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I doubt it is a Naive Bayes, rather some decision tree extended each time it fails to recognize someone. – mbq Jan 7 '11 at 23:39
Wouldn't such decision tree be too unwieldy to update? Plus, I see no easy way to account for accidentally-wrong/honest-mistake answers and still get it right in the end with decision tree – ADEpt Jan 7 '11 at 23:55
Looks like a reincarnation of the twenty year old 20-questions guesser, now at Here's a popular explanation how it works – Yaroslav Bulatov Jan 8 '11 at 1:01
Excuse me, but I think that using "artificial intelligence" and "neural networks" without any context hardy counts as explanation. And I cannot see how one could use neural net for this kind of thing - what would be the output function, for example? – ADEpt Jan 8 '11 at 7:03
Hi @ADEpt, It has been a while since the question was asked, but can you share the source code for the implementation you had back there? – prikha Jan 23 '14 at 8:17

2 Answers 2

up vote 10 down vote

This game looks similar to 20 questions at, 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.

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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”

For simplicity assume yes/no questions (whereas allows allows maybe, don’t know). Assume that every possible subject (what 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 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.

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That's a good start, but I think there is more to the problem. For example: how do they get the answers to the questions? Presumably they use the answers given by previous players (a reinforcement learning problem), in which case we face the tradeoff of choosing a question that (a) solves the problem for the current player, and (b) provides information for future players. – Simon Byrne Jan 11 '11 at 11:10
At the first glance, ability to draw analogy between 20 questions and Huffman coding hinges on the ability to ask "range questions". That is, instead of "Have you character ever been to space?" we are asking "blanket" questions like "Has he ever been to space, or is a male, or is bald, or was in a movie, or ... (100500 other options)?" Am I right? If so, then I should probably edit my question to make it clear that I am interested in "ask one by one" variety – ADEpt Jan 11 '11 at 19:41
Plus, most of the articles that use Huffman codes as model for 20 questions, imply that questioner is free to make up his own questions, which in essence boil down to "Does bit $i$ in codeword for the object is $0$"? However, in my case (or, rather,'s case) the set of questions is predefined, and it (obviously) has nothing to with Huffman codewords. Right now I can't see how to make transition from my question to Huffman codes. Perhaps you could elaborate? – ADEpt Jan 11 '11 at 19:52
@vqv: Re: "I don’t think it is really a classification problem. 20 questions is often characterized as a compression problem." Aren't statistical inference and information compression directly related / the same problem? – Yang Jan 14 '11 at 8:42
@Yang Are you referring to Jorma Rissannen’s argument? Statistical inference and information compression both make use of probabilistic models to describe uncertainty, however their perspectives and those if the researchers in those areas are generally very different. What I mean to say above is that 20 questions can be more naturally formulated as a compression (specifically, source coding) problem rather than a classification problem. … continued below – vqv Jan 14 '11 at 19:48

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