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Uncertainty in Multiclass Classification

Dr. Rishabh Pathak
1. What is Uncertainty in Classification? Uncertainty refers to the model’s confidence or doubt in its predictions. Quantifying uncertainty is important to understand how reliable each prediction is. In multiclass classification , uncertainty estimates provide probabilities over multiple classes, reflecting how sure the model is about each possible class. 2. Methods to Estimate Uncertainty in Multiclass Classification Most multiclass classifiers provide methods such as: predict_proba: Returns a probability distribution across all classes. decision_function: Returns scores or margins for each class (sometimes called raw or uncalibrated confidence scores). The probability distribution from predict_proba captures the uncertainty by assigning a probability to each class. 3. Shape and Interpretation of predict_proba in Multiclass Output shape: (n_samples, n_classes) Each row corresponds to the probabilities of ...
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Multi-class classification

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Dr. Rishabh Pathak

1. Problem Setup

·         Definition: In multi-class classification, the goal is to assign an input x∈Rd to one out of k classes or categories. The label y can take values in the set: y∈{1,2,...,k}

·         Examples:

·         Email classification into three classes: spam, personal, and work-related.

·         Handwritten digit recognition where k=10.

2. Modeling Multi-class Classification

·         Output Representation: Unlike binary classification, where the output is a scalar probability, in multi-class classification we model a probability distribution over k discrete classes: p(y=j∣x;θ)forj=1,…,k where θ represents model parameters.

·         Multinomial Distribution: The output distribution for a given x is modeled as a multinomial distribution over k classes: p(y∣x;θ)=Multinomial(ϕ1​,ϕ2​,…,ϕk​) with parameters (probabilities) ϕj​=p(y=j∣x;θ) satisfying: ϕj​≥0and∑j=1k​ϕj​=1

3. Parameterization of the Model

·         Parameter Vectors: We have k parameter vectors: θ1​, θ2​,…,θk ​with θj​∈Rd

·         Scores for each class: For input x, compute the score for each class j as: sj​ = θjT​x

These scores represent a measure of confidence that x belongs to class j.

4. The Softmax Function

·         To convert these scores sj​ into probabilities ϕj​, we use the softmax function: ϕj​=∑l=1k​esl​esj​​

·         Properties of Softmax:

·         Outputs a valid probability distribution.

·         Emphasizes the highest scoring classes exponentially, making them more likely.

5. Loss Function: Cross-Entropy Loss

·         Given training examples {(x(i), y(i))}i=1n​, the loss function is: L(θ)=−∑i=1n​logp(y(i)∣x(i);θ)

·         Plugging in the softmax probabilities: L(θ)=−∑i=1n​log∑j=1k​eθjT​x(i)eθy(i)T​x(i)​

·         Goal: Minimize this negative log-likelihood (or equivalently maximize the likelihood) over θ1​,…,θk​.

6. Training via Gradient Descent

·         Gradient Computation: The gradient of the loss with respect to each parameter vector θj​ is: ∇θj​​L=−∑i=1n​x(i)(1{y(i)=j}​−p(y=j∣x(i);θ)) where 1{.}​ is the indicator function.

·         Update Rule: Parameters are updated in the direction opposite to the gradient by an amount proportional to the learning rate η: θj​←θj​−η∇θj​​L

7. Making Predictions

  • Given a new input x, predict the class y^​ as: y^​=argmaxj∈{1,…,k}​θjT​x
  • This corresponds to selecting the class with the highest linear score.

8. Relationship to Binary Classification

  • The softmax regression (multiclass generalization) reduces to logistic regression for k=2, where the softmax converts to the sigmoid function: p(y=1∣x)=eθ1T​x+eθ2T​xeθ1T​x​=1+e−(θ1​−θ2​)Tx1​

9. Summary Points 

  • The multinomial logistic regression model classifies inputs into one of k classes.
  • Each class gets its own parameter vector θj​.
  • The softmax function converts linear scores into probabilities.
  • Training optimizes the cross-entropy loss via gradient methods.
  • The decision boundary between classes is linear (or piecewise linear), as it depends on linear functions θjT​x.
  • This approach generalizes the binary logistic regression model in an intuitive way.

10. Additional Notes

  • Multi-class perceptrons can be implemented similarly by learning separate weight vectors and picking the max scoring class.
  • More complex multi-class classifiers can involve neural networks that learn non-linear functions before the softmax output layer.

 

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