Effective Estimation of Posterior Probabilities:

-Explaining the Accuracy of Randomized Decision Tree Approaches

Wei FanEd GreengrassJoe McCloskey

Philip S. YuKevin Drummey

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

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.

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.

Difference from Traditional

No Gain function. Info gain Gini index Kearn-Mansour criteria others

No Feature Selection. Don’t choose feature with highest “gain”

Multiple trees. Relies on probability estimates.

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

Results

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.

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.

More Methods

Bagged Probabilistic Tree: Bootstrap Compute probability. Traditional Tree

Disjoint Subset Trees: Shuffle the data. Equal-sized subsets. Traditional Tree

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

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)?

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’’

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?

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 x’s 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

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.

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”.

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.

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 Elkan’02 (explain later)

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

Charts

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Unp Pruned RD RF+

BiasVariace

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 Heckman’s two-step procedure (Zadrozny and Elkan’02)

How much money we got

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)

Reliability Plot Unpruned Tree

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.

Losses

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Unp Pruned RD RF+

top1

top2

top3

MSE

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Unp Pruned RD RF+

P(y|x)=1

top1

top2

top3

Detailed Probability

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.

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

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.

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?