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Parametric Methods in Machine Learning & Maximum Likelihood Estimation (MLE)

Parametric Methods in Machine Learning

Parametric methods are a class of machine learning models that assume a fixed number of parameters to describe the data distribution. These models make strong assumptions about the underlying structure of the data and do not grow in complexity with more data.

Key Characteristics:

1. Fixed Parameters: The model structure is predetermined (e.g., linear regression, logistic regression).

2. Simpler and Faster: Requires fewer resources compared to non-parametric methods.

3. Less Flexible: Assumes a specific distribution (e.g., Gaussian distribution).

Examples of Parametric Models:

• Linear Regression: Assumes a linear relationship between features and target.

• Logistic Regression: Uses a sigmoid function for classification.

• Naïve Bayes: Assumes feature independence.

Maximum Likelihood Estimation (MLE)

MLE is a method used in parametric models to estimate parameters that maximize the likelihood of observed data.

Concept: Given a dataset $X = \{x_1, x_2, ..., x_n\}$, and assuming the data follows a probability distribution $P(X | \theta)$ with parameters $\theta$, MLE finds $\theta$ that maximizes:

$$L(\theta) = P(X | \theta)$$

Taking the log-likelihood for computational ease:

$$\theta^* = \arg\max_{\theta} \sum_{i=1}^{n} \log P(x_i | \theta)$$

Applications of MLE:

• Gaussian Distribution Estimation: Finding mean and variance.

• Logistic Regression: Optimizing weights for classification.

• Hidden Markov Models (HMMs): Learning transition probabilities.

Advantages of Parametric Methods & MLE

• Fast and Efficient for small datasets.

• Interpretable due to fixed assumptions.

However, they may fail with complex data, making non-parametric approaches (e.g., neural networks) preferable in some cases.