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Uncertainty in Multiclass Classification

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 ...

Probabilistic interpretation

The probabilistic interpretation in machine learning and statistics refers to understanding algorithms and models in terms of probability theory, which offers a principled framework for reasoning under uncertainty and making predictions.

Overview of Probabilistic Interpretation

  1. Connection Between Likelihood and Cost Function:

A common example is the interpretation of least-squares regression. Under specific probabilistic assumptions about the data, minimizing the least-squares cost corresponds exactly to maximizing the likelihood of the observed data.

  • Suppose the data (x(i),y(i)) are generated according to the model:

y(i)=θTx(i)+ϵ(i),

where ϵ(i)N(0,σ2) are independent Gaussian noise terms.

  • The likelihood of the dataset under parameters θ is:

p(y(i)x(i);θ)=σ1exp(−2σ2(y(i)θTx(i))2).

  • Maximizing this likelihood (or equivalently, the log-likelihood) over all data points is:

θMLE=argmaxθi=1nlogp(y(i)x(i);θ),

which simplifies to minimizing the squared error loss:

argminθ21i=1n(y(i)θTx(i))2.

Hence, least-squares regression corresponds to the Maximum Likelihood Estimation (MLE) under Gaussian noise assumptions.

  1. Bayesian Perspective:

Going beyond MLE, Bayesian statistics treats model parameters θ as random variables with a prior distribution p(θ). This leads to the posterior distribution over parameters given data S={(x(i),y(i))}i=1n:

p(θS)=p(S)p(Sθ)p(θ)=θ(i=1np(y(i)x(i),θ)p(θ))(i=1np(y(i)x(i),θ))p(θ)

where the posterior incorporates both the likelihood and prior belief about θ. This approach regularizes the problem, reduces overfitting, and allows for uncertainty quantification.

  1. Summary :
  • The equivalence between least squares and MLE is shown under Gaussian noise assumptions.
  • Bayesian view treats parameters as random variables with priors, leading to posterior inference that naturally incorporates regularization effects.
  • Such probabilistic interpretations provide principled foundations for many classical and modern machine learning methods, like logistic regression and generative models.

Additional Notes:

  • Probabilistic interpretations enable the design of new models and cost functions by modeling p(yx;θ) in flexible ways.
  • They facilitate working with uncertainty, making predictions that include confidence estimates.
  • Extensions include generalized linear models and generative learning algorithms, which build on these ideas. 

 

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