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Effective Estimation of Posterior Probabilities: - Explaining the Accuracy of Randomized Decision Tree Approaches Wei Fan Ed Greengrass Joe McCloskey Philip S. Yu Kevin Drummey

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Example Simple Decision Tree Method Construction: at each node, a feature is chosen randomly Discrete: only if it has never been chosen previously on a given decision path starting from the root of the tree. every example on the same path has the same discrete feature value. Continuous feature: can be chosen multiple times on the same decision path. each time a random threshold value is chosen

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continues… Stop when: Number of examples in the leaf node is too small. The total height of the tree exceeds some limits. Each node of the tree keeps the number of examples belonging to each class. For example, 10 + and 5 – Construct at least 10 trees but no need to be more than 30.

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Classification Each tree output estimated posterior probability: A node with 10 + and 5 - outputs P(+|x,t) = 0.67 Multiple trees average their probability estimates as the final output. Use the estimated probability and given loss function to choose label that minimize expected loss. 0-1 loss or traditional accuracy: choose the most probable label Cost-sensitive: choose the label that minimize risk.

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Difference from Traditional No Gain function. Info gain Gini index Kearn-Mansour criteria others No Feature Selection. Dont choose feature with highest gain Multiple trees. Relies on probability estimates.

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How well it works? Credit card fraud detection: Each transaction has a transaction amount. There is an overhead $90 to challenge a fraud. Predict fraud iif P(fraud|x) $1000 > $90 P(fraud|x) $1000 is expected loss When expected loss is more than overhead, do sth. Three models: Traditional Unpruned decision tree Traditional Pruned decision tree RDT

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Results

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Randomization Feature selection randomization: RDT: completely random. Random Forest: consider random subset at each node. etc Feature subset randomization. Fixed random subset. Data randomization: Bootstrap sample. Bagging and Random Forest Data Partitioning Feature Combination.

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Methods Included RDT: Choose feature randomly. Choose threshold for continuous randomly. RF and RF+ (variation of Random Forest): Chooses k features randomly. Choose the one among k with highest infogain Variation I: use original dataset. Variation II: output probability instead of voting.

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More Methods Bagged Probabilistic Tree: Bootstrap Compute probability. Traditional Tree Disjoint Subset Trees: Shuffle the data. Equal-sized subsets. Traditional Tree

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Some concepts True posterior probability P(y|x) Probability of an example to be a class y as a condition of its feature vector x Generated from some unknown function F Given a loss function, the optimal decision y* is the class label that minimizes the expected loss. 0-1 loss: the most probable label. Binary problem: class +, class – P(+|x) = 0.7 and P(-|x) = 0.3 Predict + Cost-sensitive loss: choose the class label that reduces expected risk. P(fraud|x) * $1000 > $90 Optimal label *y may not always be the true label. For example, 0-1 loss, P(+|x) = 0.6, the true label may be – with 0.4 probability

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Estimated Probability We use M to approximate true function F. We almost never know F. Estimated probability by a model M, P(y|x,M). The dependency on M is none-trivial: Decision tree uses tree structure and parameters within the structure to approximate P(y|x) Mixture model uses basis functions such as naïve Bayes and Gaussian. Relation between P(y|x,M) and P(y|x)?

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Important Observation If P(y|x,M) = P(y|x), the expected loss for any loss function will be the smallest. Interesting cases: P(y|x, M) = P(y|x) and 0-1 loss,100% accuracy? Yes, only if the problem is deterministic or P(y|x) =1 for the true label and 0 for all others! Otherwise, you can only choose the most likely label, but it can still be wrong for some examples. Can M beat the accuracy of P(y|x), even if P(y|x, M) =! P(y|x)? Yes, for some specific example or specific test set. But not in general or not expected loss

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Reality Class labels are given, however P(y|x) is not given in any dataset unless the dataset is synthesized. Next Question: how to set the true P(y|x) for a realistic dataset?

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Choosing P(y|x) Naïve Approach Assume that P(y|x) is 1 for the true class label of x and 0 for all class labels. For example, two class problem + and – If xs label is +, assume P(+|x) = 1 and P(-1|x) = 0 Only true if the problem is determinisitic and noise free. Rather strong assumption and may cause problems. X has true class label: + M1: P(+|x,M1) = 1, P(-|x,M1) =0 M2: P(+|x,M2) = 0.8, P(-|x,M2) = 0.2 Both M2 and M1 are correct. But Penalize M2

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Utility-based Choice of P(y|x) Definition: v is the probability threshold for model M to correctly predict the optimal label y* of x. If P(y*|x,M) > v, predict y* Assume *y to be the true class label of an example. Example, binary class, 0-1 loss v=0.5 or If P(y|x,M) > 0.5, predict y Example, credit card fraud cost-sensitive loss P(y|x,M) * $1000 > $90 v = 90/1000 = 0.09 In summary, we use [v, +1] as the range of true probability P(y|x) This is weaker than assuming P(y|x) = 1 for the true class label.

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Example Two class problem: Naïve assumption: P(y|x) = 1 for the correct label. 0 for all others. We assume P(y|x) (0.5, 1] It includes naïve assumption P(y|x) = 1. We re-define some measurements to fix the problem of penalty.

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Desiderata If P(y|x,M) [v, 1], the exact value is trivial, since we already predict the true label. When P(y|x,M) < v(x,M), the difference is important. Measures how far off we are from making the right decision. Take into account the loss function, since the goal is to minimize its expected value.

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Evaluating P(y|x,M) Improved MSE Square Error: Where [[a]] = min(a, 1) Cross-entropy: Undefined either when P(y|x.M) = 0 or true probability P(y|x) = 0 No relation to loss function. Reliability plots previously proposed and used such as Zadrozny and Elkan02 (explain later)

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Synthetic Dataset True probability P(y|x) is known and can be used to measure the exact MSE. Standard Bias and Variance Decomposition of MSE

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Charts

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Binary Dataset Donation Dataset: Send a letter to solicit donation. Costs 68c to send a letter Cost-sensitive loss: P(donate|x) * amt(x) > 68c Used MLR to estimate amt(x). Better results could be obtained by Heckmans two-step procedure (Zadrozny and Elkan02)

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How much money we got

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Reliability Plot Divide score or output probability into bins Either equal size such as 10 or 100 bins. Or equal number of examples. For those examples in the same bin: Average the predicted probability of these examples, and call it bin_x Divide the number of examples with label y by the total number of examples in the bin, call it bin_y Plot (bin_x, bin_y)

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Reliability Plot Unpruned Tree

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Multi-Class Dataset Artificial Character Dataset from UCI Class labels: 10 letters Three loss functions: Top 1: the true label is the most probable letter. Top 2: the true label is among the two most probable letters. Top 3: the true label is among the top three.

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Losses

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MSE

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Detailed Probability

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What we learned On studies of probability approximation: Assuming P(y|x)=1 is a very strong assumption and cause problems. Suggested a relaxed choice of P(y|x). Improved definition of MSE that takes into loss. Methodology part: Proposed a variation of Random Forest.

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Summary of Experiments Various experiments Synthetic with true probability P(y|x) Binary and multi-class problems Reliability plots and MSE show that randomized approaches approximate P(y|x) significantly closer. Bias and Variance Decomp of Probability as compared to loss function. Reduction comes mainly from variance Bias is reduced as well

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What next We traditionally think that probability estimation is a harder problem than class labels: Simplified approach: naïve Bayes. Uncorrelated assumption. Finite mixture models: still based on assumption of basis function. Logistic regression: sensitive to example layout, and subjective use to categorical features. Bayes network: need knowledge about causal relations. NP-hard to find the optimal one.

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continued We show that rather simple randomized approaches approximate probability very well. Next step: is it time for us to re-design some better and simpler algorithms to approximate probability better?

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