Machine learning and artificial intelligence

  Chapter 28: Parameter Estimation II - Priors & Maximum A Posteriori (MAP) In the context of machine learning and statistics, parameter estimation is a crucial process. After exploring Maximum Likelihood Estimation (MLE) in previous chapters, we now delve into a more sophisticated technique: Maximum A Posteriori (MAP) Estimation . This method incorporates prior knowledge about the parameters, making it a more general approach than MLE. 1. Introduction to Priors In Bayesian statistics, a prior represents our beliefs about the parameters before observing any data. It's a probability distribution that reflects our knowledge or assumptions about the parameter's values. Incorporating priors allows us to update our beliefs in light of new evidence, leading to the posterior distribution . Given a parameter θ \theta θ and data X X X , the prior distribution is denoted as P ( θ ) P(\theta) P ( θ ) . The likelihood, P ( X ∣ θ ) P(X|\theta) P ( X ∣ θ ) , represents the probability ...

  Machine Learning (Chapter 27): Maximum Likelihood Estimate (MLE) Introduction to Maximum Likelihood Estimate (MLE) Maximum Likelihood Estimation (MLE) is a fundamental statistical method used in machine learning and statistics to estimate the parameters of a probability distribution or statistical model. The core idea behind MLE is to find the parameter values that maximize the likelihood function, which measures how likely it is to observe the given data under different parameter values. Likelihood Function Given a dataset D = { x 1 , x 2 , … , x n } \mathcal{D} = \{x_1, x_2, \dots, x_n\} D = { x 1 ​ , x 2 ​ , … , x n ​ } and a probabilistic model with a parameter θ \theta θ , the likelihood function L ( θ ) \mathcal{L}(\theta) L ( θ ) is defined as the probability of observing the data given the parameter θ \theta θ . Mathematically, it is expressed as: L ( θ ) = P ( D ∣ θ ) = ∏ i = 1 n P ( x i ∣ θ ) \mathcal{L}(\theta) = P(\mathcal{D}|\theta) = \prod_{i=1}^{n} P(x_i|\theta) ...