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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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Locally Weighted Linear Regression (LWR)

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Dr. Rishabh Pathak

Locally Weighted Linear Regression (LWR) is a non-parametric regression technique designed to address the limitations of traditional linear regression, especially when the data relationship is not well modeled by a simple global linear function.

  • Basic Idea: Instead of fitting a single global linear model, LWR fits a linear model locally around the query point x. It places more weight on training examples close to x, and less weight on examples farther away.
  • Weighting Scheme: Each training example (x(i), y(i)) is assigned a weight w(i) based on its distance from the query point x, typically using a kernel function like the Gaussian:

w(i)=exp(−2τ2∥x(i)−x∥2​),

where τ is a bandwidth parameter controlling how quickly the weight decreases with distance.

  • Fitting and Prediction: To predict y at x, LWR:
  1. Solves a weighted least squares problem minimizing:

∑i​w(i)(y(i)−θTx(i))2,

where each data point's contribution is scaled by its weight.

  1. Uses the fitted parameters θ to output the prediction:

y^​=θTx.

  • Non-Parametric Nature: Unlike standard linear regression that produces a single set of parameters θ, LWR adapts parameters locally for each query point. It requires retaining the entire training set for prediction, making it a non-parametric method.

Advantages:

  • Can handle complex, non-linear relationships without explicitly defining a global model.
  • Makes prediction sensitive to the local structure of data.
  • Reduces the dependency on carefully selecting features.

Considerations:

  • Choosing the bandwidth τ is critical; too small leads to high variance (overfitting), too large leads to high bias (underfitting).
  • Computationally expensive for large datasets since it fits a model for each query point.

This method smooths between fitting the data globally and simply using nearest neighbor predictions, providing a flexible approach to regression when data relationships vary locally

 

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