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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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Classification and Logistic Regression

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Dr. Rishabh Pathak

1. Classification Problem

  • Definition: Classification is a supervised learning task where the output variable y is discrete-valued rather than continuous.
  • In particular, consider binary classification where y∈ {0,1} (e.g., spam detection: spam =1, not spam =0).
  • Each training example is a pair (x(i), y(i)), where x(i)∈Rd is a feature vector, and y(i) is the label.

2. Why Not Use Linear Regression for Classification?

  • Linear regression tries to predict continuous values, which is problematic for classification as the prediction can be outside [0,1].
  • For example, predicting y≈1.5 or −0.2 is meaningless when y is binary.
  • Instead, we want the output hθ​(x) to be interpreted as the probability that y=1 given x.

3. Logistic Regression Model

Hypothesis:

hθ​(x)=g(θTx)=1+e−θTx1​,

where:

  • g(z)=1+e−z1​ is the sigmoid function, which maps any real value to the interval (0, 1).
  • θ∈Rd+1 are parameters (including intercept term).
  • hθ​(x) can be interpreted as the estimated probability P(y=1∣x;θ).

Decision Boundary:

  • Predict y=1 if hθ​(x)≥0.5; otherwise, predict y=0.
  • The decision boundary corresponds to θTx=0, which is a linear boundary in input space.

4. Loss Function and Cost Function

Probability Model:

  • Logistic regression models conditional probability directly:

P(y=1∣x;θ)=hθ​(x),P(y=0∣x;θ)=1−hθ​(x).

  • Equivalently, likelihood for data point (x(i),y(i)):

p(y(i)∣x(i);θ)=(hθ​(x(i)))y(i)(1−hθ​(x(i)))1−y(i).

Cost (Loss) Function:

  • Use negative log-likelihood (cross-entropy loss) as cost per example:

J(i)(θ)=−[y(i)loghθ​(x(i))+(1−y(i))log(1−hθ​(x(i)))].

  • Overall cost function (average over n examples):

J(θ)=n1​∑i=1n​J(i)(θ).

  • This loss is convex in θ, enabling efficient optimization.

5. Training Logistic Regression

·         Use methods such as gradient descent or more advanced optimization (Newton's method, quasi-Newton) to minimize cost J(θ).

·         The gradient of the cost function is:

∇θ​J(θ)=n1​∑i=1n​(hθ​(x(i))−y(i))x(i).

  • Update rule in gradient descent:

θ:=θ−α∇θ​J(θ),

where α is the learning rate.

6. Multi-class Classification

·         When y∈{1,2,...,k} for k>2, logistic regression generalizes to multinomial logistic regression or Softmax regression.

·         Model outputs hˉθ​(x)∈Rk called logits.

·         The Softmax function converts logits into probabilities:

P(y=j∣x;θ)=∑s=1k​exp(hˉθ​(x)s​)exp(hˉθ​(x)j​)​.

  • Loss for example (x(i),y(i)) is the negative log-likelihood:

J(i)(θ)=−logP(y(i)∣x(i);θ).

7. Discriminative vs. Generative Classification Algorithms

  • Discriminative algorithms (like logistic regression) model p(y∣x) directly or learn a direct mapping from x to y.
  • Generative algorithms model the joint distribution p(x,y)=p(x∣y)p(y).
  • Example: Gaussian Discriminant Analysis (GDA).
  • Logistic regression is an example of a discriminative approach focusing purely on p(y∣x).

8. Linear Hypothesis Class and Decision Boundaries

  • Logistic regression hypothesis class:

H={hθ​:hθ​(x)=1{θTx≥0}​},

which are classifiers with linear decision boundaries.

  • More generally, hypothesis classes can be extended to neural networks or other complex architectures.

9. Perceptron Learning as Contrast to Logistic Regression

·         Perceptron also uses a linear classifier but with a different loss and update rule.

·         Logistic regression provides probabilistic outputs and optimizes a convex cost function, generally yielding better statistical properties.

10. Practical Considerations

  • Feature scaling often improves numerical stability.
  • Regularization (e.g., L2) is frequently added to cost to prevent overfitting.
  • Logistic regression handles input features linearly; non-linear boundaries require feature engineering or kernel methods.

Summary:

Logistic regression is a fundamental classification algorithm that models the conditional probability of the positive class using a sigmoid of a linear function of input features. It is trained via maximizing likelihood (or minimizing cross-entropy loss) and extends naturally to multi-class problems via Softmax. It is a discriminative model focusing directly on p(y∣x) and yields linear decision boundaries. It contrasts with generative models by its direct approach to classification.

 

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