Last Updated on November 5, 2019 Density estimation is the problem of estimating the probability distribution for a sample of observations from a problem domain. There are many techniques for solving density estimation, although a common framework used throughout the field of machine learning is maximum likelihood estimation. Maximum likelihood estimation involves defining a likelihood function for calculating the conditional probability of observing the data sample given a probability distribution and distribution parameters. This approach can be used to search […]

Last Updated on November 1, 2019 Linear regression is a classical model for predicting a numerical quantity. The parameters of a linear regression model can be estimated using a least squares procedure or by a maximum likelihood estimation procedure. Maximum likelihood estimation is a probabilistic framework for automatically finding the probability distribution and parameters that best describe the observed data. Supervised learning can be framed as a conditional probability problem, and maximum likelihood estimation can be used to fit the […]

Last Updated on October 28, 2019 Logistic regression is a model for binary classification predictive modeling. The parameters of a logistic regression model can be estimated by the probabilistic framework called maximum likelihood estimation. Under this framework, a probability distribution for the target variable (class label) must be assumed and then a likelihood function defined that calculates the probability of observing the outcome given the input data and the model. This function can then be optimized to find the set […]

Last Updated on August 28, 2020 Model selection is the problem of choosing one from among a set of candidate models. It is common to choose a model that performs the best on a hold-out test dataset or to estimate model performance using a resampling technique, such as k-fold cross-validation. An alternative approach to model selection involves using probabilistic statistical measures that attempt to quantify both the model performance on the training dataset and the complexity of the model. Examples […]

Last Updated on August 28, 2020 Maximum likelihood estimation is an approach to density estimation for a dataset by searching across probability distributions and their parameters. It is a general and effective approach that underlies many machine learning algorithms, although it requires that the training dataset is complete, e.g. all relevant interacting random variables are present. Maximum likelihood becomes intractable if there are variables that interact with those in the dataset but were hidden or not observed, so-called latent variables. […]

Monte Carlo methods are a class of techniques for randomly sampling a probability distribution. There are many problem domains where describing or estimating the probability distribution is relatively straightforward, but calculating a desired quantity is intractable. This may be due to many reasons, such as the stochastic nature of the domain or an exponential number of random variables. Instead, a desired quantity can be approximated by using random sampling, referred to as Monte Carlo methods. These methods were initially used […]

Probabilistic inference involves estimating an expected value or density using a probabilistic model. Often, directly inferring values is not tractable with probabilistic models, and instead, approximation methods must be used. Markov Chain Monte Carlo sampling provides a class of algorithms for systematic random sampling from high-dimensional probability distributions. Unlike Monte Carlo sampling methods that are able to draw independent samples from the distribution, Markov Chain Monte Carlo methods draw samples where the next sample is dependent on the existing sample, […]

Density estimation is the problem of estimating the probability distribution for a sample of observations from a problem domain. Typically, estimating the entire distribution is intractable, and instead, we are happy to have the expected value of the distribution, such as the mean or mode. Maximum a Posteriori or MAP for short is a Bayesian-based approach to estimating a distribution and model parameters that best explain an observed dataset. This flexible probabilistic framework can be used to provide a Bayesian […]