Bayesian Nonparametric Methods Assignment Help

A Bayesian nonparametric model is a Bayesian model with infinite parameters. Typically, the parameter space is chosen as the set of all possible solutions to a given learning problem. In a regression problem, for example, the parameter space can be the set of continuous functions, whereas in a density estimation problem, the space can be all densities. A Bayesian nonparametric model explains a finite sample of observations using only a finite subset of the available parameter dimensions, with the set of dimensions chosen depending on the sample so that the effective complexity of the model (as measured by the number of dimensions used) adapts to the data. Nonparametric estimation and model selection, for example, can thus be formulated as Bayesian inference problems. Gaussian process is a popular example of a Bayesian nonparametric model.

KEY FEATURE

It explains a finite sample of observations using only a finite subset of the available parameter dimensions so that the effective complexity of the model adapts to the data.

Bayesian inference is a statistical inference method that uses Bayes’ theorem to update the probability of a hypothesis as new evidence or information becomes available.

It is a crucial technique in mathematical statistics. Gaussian process regression and Dirichlet process mixture models for clustering are two examples of Bayesian nonparametric models.

Recent applications of Bayesian nonparametric models include regression, classification, clustering, latent variable modeling, sequential modeling, image segmentation, source separation.

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    The majority of machine learning is concerned with learning an appropriate set of parameters from training data within a model class. Model selection or model adaptation are the meta-level problems of determining appropriate model classes. These are critical concerns for machine learning practitioners, not only for avoiding over- and under-fitting, but also for discovering the causes and structures underlying data. Selecting the number of clusters in a clustering problem, the number of hidden states in a hidden Markov model, the number of latent variables in a latent variable model, or the complexity of features used in nonlinear regression are all examples of model selection and adaptation.

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    Nonparametric models are a model selection and adaptation approach in which model sizes are allowed to grow with data size. In contrast, parametric models have a fixed number of parameters. A parametric approach to density estimation, for example, would be to use maximum likelihood to fit a Gaussian or a mixture of a fixed number of Gaussians. A Parzen window estimator, which centres a Gaussian at each observation, is a nonparametric approach (and hence uses one mean parameter per observation). A support vector machine with a Gaussian kernel is another example.
    The representer theorem demonstrates that the decision function is a linear combination of Gaussian radial basis functions centred at each input vector, with increasing complexity as the number of observations increases. Nonparametric methods have long been popular in non-Bayesian (classical) statistics. They frequently outperform parametric models in applications, and while theoretical results for such models are typically more difficult to prove than theoretical results for parametric models, appealing theoretical properties have been established for a wide range of models. Bayesian nonparametric methods provide a Bayesian framework for nonparametric model selection and adaptation. A Bayesian formulation of nonparametric problems is difficult because a Bayesian model defines prior and posterior distributions on a single fixed parameter space, whereas the dimension of the parameter space in a nonparametric approach should vary with sample size. The Bayesian nonparametric solution to this problem involves employing an infinite-dimensional parameter space and invoking only a finite subset of the available parameters on any given finite data set. This subset expands with the data set. “Infinite-dimensional” can thus be interpreted as “of finite but unbounded dimension” in the context of Bayesian nonparametric models. A Bayesian nonparametric model is one that is a Bayesian model on an infinite-dimensional parameter space and can be evaluated on a finite sample using only a finite subset of the available parameters to explain the sample.
    In the following section, we will provide more concrete examples of standard machine learning problems and the corresponding Bayesian nonparametric solutions. As we’ll see, the parameter space in is typically made up of functions or measures, whereas is typically achieved by marginalising out surplus dimensions over the prior. Stochastic processes are random functions and measures, and more broadly probability distributions on infinite-dimensional random objects; examples include Gaussian processes, Dirichlet processes, and beta processes. Nonparametric Bayesian models are frequently named after the stochastic processes they contain. Following the examples are theoretical considerations, such as formal constructions and representations of the stochastic processes used in Bayesian nonparametric models, exchangeability, and issues of consistency and convergence rate.

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