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Let as assume that we have a binary classification problem. We also have several classifiers. Instead of assigning a vector to a class (0 or 1) each classifier returns a probability that a given vector belongs to class 1. It means that for each input vector, that has to be classified we get a vector of real number between 0 and 1. For example:

(0.81, 0.67, 0.43, 0.99, 0.53)

where the number of components (probabilities) is equal to the number of classifiers. Now we want to "combine" these "weak" classifiers to get on "strong" classifier. In other words we need to find a way to map a given vector of probabilities into one number (probability).

So, my questions is: What is the "correct" way to do it? Of course I can train another classifier that uses the vector of probabilities and returns one probability. In other words we can find out how to combine the "weak" probabilities in an empirical way. However, I assume, that we can use the fact that the components of the vector are not just "some numbers" (or features) they are probabilities, they are already predictions and, as a consequence, they have to be combined in a corresponding appropriate way.

ADDED

In comments it has been proposed to average the "weak" probabilities. But what if it is possible to estimate quality (power) of each "weak" classifier (and it should be possible), doesn't it make sense to suppress "bad" classifier (for example by using their predictions (probabilities) with smaller weights or by ignoring them completely)? Does it makes sense to use just one (the best) weak classifier? Does it make sense to check correlation between the weak classifiers. For example what should we do if two "weak" classifiers always give the same result. Shouldn't we through one of them out as not having any additional value?

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    $\begingroup$ Ensemble learning is what I would google. $\endgroup$ – Berkan May 6 '15 at 8:03
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    $\begingroup$ The default is averaging (bagging), which is known to be very powerful. $\endgroup$ – Marc Claesen May 6 '15 at 8:50
  • $\begingroup$ @MarcClaesen, I made an extension of my question, that is inspired by your comment. $\endgroup$ – Roman May 6 '15 at 9:34
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    $\begingroup$ Rather then removing "bad" classifiers you could rather use weighted mean where weights are based on inverse classification errors, see: stats.stackexchange.com/questions/147881/… $\endgroup$ – Tim May 6 '15 at 10:09
  • $\begingroup$ @Roman I don't see any reason not to treat 'predictions' from the previous classifier as new features to a new classifier. For example, logistic regression takes probabilities as input and weight them with respect to their relative importance. Hence, the last classifier's weights show how important each previous classifier's predictions is. If weight for the first classifier is higher than the second classifier, that means the first classifier is more reliable. $\endgroup$ – yasin.yazici May 7 '15 at 22:21
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Practically speaking bagging, boosting, and stacking all constitute reasonable ways to combine the weak predictions, as others have mentioned. Taking that one step further though, trying as many of them as possible and seeing what performs best is common the context of competitive machine learning too (e.g. You might try a stacked classifier as well as well as a simple average ensembler and choose the better of the two as per performance on some hold-out data).

In fact, if accuracy is your only goal then you're likely better off having a few different ensembling techniques in mind like this and focusing on the diversity of the classifiers being ensembled rather than how they're combined. Since the best ensemble technique will depend on your problem and and data and there is no approach that's always best, your time is better spent just making the ensemble choice another part of the training process rather than worrying about making a single correct choice.

Theoretically speaking though, and as an attempt to answer the question more directly, I think there are some probabilistic arguments that could be used to justify the best way to "suppress" the weak classifiers. IMO the reasoning behind Bayesian Model Averaging and Information-Criteria-Based Averaging is pretty enlightening and has ties to some of the approaches in Machine Learning like weighting classifiers via binomial deviance. For example, here's a process for combining classifiers through the use of akaike weights (as an example of information-criteria based model averaging):

** Note this is all assuming the classifiers are pretty well calibrated and that actual out-of-sample deviance is used in cross-validation rather than an estimate of it like AIC

Suppose you have $K$ classifiers fit to training data and each of those classifiers then makes $N$ predictions on test data. You could then compute the "likelihood" of each models predictions as follows:

$$ L_k = \Pi_{i=1}^n{ P_{M_k}(y_i)}$$

where:

  • $L_k$ = Likelihood of predictions from model $k$
  • $y_i$ = Response $i$ in the test data
  • $P_{M_k}(y_i)$ = The probability that classifier $k$ attributes to $y_i$

Given the likelihood of the predicted data, the weight or relative likelihood of each classifier, $w_{M_k}$ could then be defined as: $$L_{max} = \max{L_k} $$ $$w_{M_k} = \frac{e^{2log(L_{max}/L_k)}}{\sum_{j=1}^{k}{e^{2log(L_{max}/L_k)}}}$$

The weights for each model, $w_{M_k}$, can then be interpreted as the probability that classifier $M_k$ is the true model and an expected outcome coming from an ensemble of all $k$ classifiers would have an expected value equal to the sum of the probability of each classifier times its prediction:

$$ y_{new} = \sum_{j=1}^k { w_{M_k} \cdot M_k(X_{new}) }.$$

I hope the notation doesn't bog you down but the point is that there are theoretical ways (the Bayesian ones are especially interesting) to determine what probability each model in an ensemble should have and then use that to make predictions, rather than some more heuristic weighting or equal-voting scheme. These more intuitive averaging strategies don't generally perform better in empirical studies (or so it seems), but I thought throwing the notion of them into the mix might help you like they helped me.

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As you might surmise from the diversity of comments, combining weak learners into strong ones isn't a task with a single, "correct" approach, but a field of approaches with varying levels of known strengths and drawbacks. For a brief introduction, the wikipedia entry for ensemble learning is a reasonable place to start. There you'll find reference to many of the methods discussed in comments, such as averaging predictions or combining them using logistic regression. In particular, here's an excerpt from the section on stacking:

Stacking (sometimes called stacked generalization) involves training a learning algorithm to combine the predictions of several other learning algorithms. First, all of the other algorithms are trained using the available data, then a combiner algorithm is trained to make a final prediction using all the predictions of the other algorithms as additional inputs. If an arbitrary combiner algorithm is used, then stacking can theoretically represent any of the ensemble techniques described in this article, although in practice, a single-layer logistic regression model is often used as the combiner.

The of questions in your edit seem to me best answered in individual cases using cross-validation. Whether @Tim's suggestion of inverse-error-weighted mean performs better than @yasin.yazici's logistic regression will, I suspect, vary depending on the algorithms and dataset. Will a single-layer neural net using the classifications features produce good results? Could be, as neural nets can learn non-linear functions. Maybe it will learn that when two of the probabilities agree, they're nearly always right, and produce a hidden layer node that weights these two very highly.

But until you try those and compare them, who knows? Thankfully, cross-validation provides a means of comparing each of those approaches for a given, practical problem.

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  • $\begingroup$ I will buy it if you compare those techiques like bagging, forest as well as stacking. And point some reference at the end. $\endgroup$ – Henry.L Sep 8 '15 at 3:28
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Combining weak learners into a strong one is exactly what boosting is designed for.

Boosting is a specific example of ensemble method.

"Improved Boosting Algorithms Using Confidence-rated Predictions" is a popular algorithm (over 3000 citations) that uses the confidence of the weak learner.

Note that there is a big difference between the regular use of boosting and your problem. In regular boosting, you have the ability to train a weak learner on the data set of your choice. The boosting algorithm deals with building the proper data sets and combining the data set.

Here, you are already given the weak learners. If you can train new learners, you can ignore those you already have and use boosting in the regular way. Otherwise, since the weak learners are known, you might be able to aggregate them in a better way by treating them as regular features and build a classifier on top of them.

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