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Bayesian Mixture Posteriors
Contents:
  1. Mixture models
  2. Clustering with Gaussian Mixture Models – Python Machine Learning
  3. Neural networks: representation.

The density functions are estimated by the Gaussian mixture model GMM and the t-student mixture model. The model parameters are estimated by algorithms based on the expectation-maximization EM method.


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The estimated densities calculated for a sequence of feature vectors are inputs to analyzed classification rules. These rules are derived from Bayes decision theory with some heuristic modifications. The performance of the proposed rules was tested in an automatic, text independent, speaker identification task. Achieved results are presented. The above update can also be applied to updating a Poisson measurement noise intensity.

Similarly, for a first-order auto-regressive process, an updated process noise variance estimate can be calculated by. The updated model coefficient estimate is obtained via. The convergence of parameter estimates such as those above are well studied. A number of methods have been proposed to accelerate the sometimes slow convergence of the EM algorithm, such as those using conjugate gradient and modified Newton's methods Newton—Raphson. This idea is further extended in generalized expectation maximization GEM algorithm, in which is sought only an increase in the objective function F for both the E step and M step as described in the As a maximization-maximization procedure section.

The Q-function used in the EM algorithm is based on the log likelihood. Therefore, it is regarded as the log-EM algorithm. Obtaining this Q-function is a generalized E step. Its maximization is a generalized M step. No computation of gradient or Hessian matrix is needed. EM is a partially non-Bayesian, maximum likelihood method.

In this paradigm, the distinction between the E and M steps disappears. Now, k steps per iteration are needed, where k is the number of latent variables. For graphical models this is easy to do as each variable's new Q depends only on its Markov blanket , so local message passing can be used for efficient inference.

Mixture models

In information geometry , the E step and the M step are interpreted as projections under dual affine connections , called the e-connection and the m-connection; the Kullback—Leibler divergence can also be understood in these terms. The aim is to estimate the unknown parameters representing the mixing value between the Gaussians and the means and covariances of each:.

The inner sum thus reduces to one term. These are called the "membership probabilities" which are normally considered the output of the E step although this is not the Q function of below.


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  • This has the same form as the MLE for the binomial distribution , so. The algorithm illustrated above can be generalized for mixtures of more than two multivariate normal distributions. The EM algorithm has been implemented in the case where an underlying linear regression model exists explaining the variation of some quantity, but where the values actually observed are censored or truncated versions of those represented in the model.

    EM typically converges to a local optimum, not necessarily the global optimum, with no bound on the convergence rate in general. It is possible that it can be arbitrarily poor in high dimensions and there can be an exponential number of local optima. Hence, a need exists for alternative methods for guaranteed learning, especially in the high-dimensional setting.

    Alternatives to EM exist with better guarantees for consistency, which are termed moment-based approaches [34] or the so-called spectral techniques [35] [36] [ citation needed ].

    Moment-based approaches to learning the parameters of a probabilistic model are of increasing interest recently since they enjoy guarantees such as global convergence under certain conditions unlike EM which is often plagued by the issue of getting stuck in local optima. Algorithms with guarantees for learning can be derived for a number of important models such as mixture models, HMMs etc.

    For these spectral methods, no spurious local optima occur, and the true parameters can be consistently estimated under some regularity conditions [ citation needed ]. From Wikipedia, the free encyclopedia. Machine learning and data mining Problems. Dimensionality reduction. Structured prediction. Graphical models Bayes net Conditional random field Hidden Markov. Anomaly detection. Artificial neural networks.

    Reinforcement learning.

    Clustering with Gaussian Mixture Models – Python Machine Learning

    Machine-learning venues. Glossary of artificial intelligence. Related articles. List of datasets for machine-learning research Outline of machine learning. EM clustering of Old Faithful eruption data. The random initial model which, due to the different scales of the axes, appears to be two very flat and wide spheres is fit to the observed data. In the first iterations, the model changes substantially, but then converges to the two modes of the geyser.

    Visualized using ELKI. Further information: Information geometry. Scandinavian Journal of Statistics. Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable. Mixture models can be extended within the Bayesian network framework to create models with additional structure. It is interesting to point out, that even with additional structure, the joint distribution of the model will still be a Mixture Model, however the additional structure allows us to use a compact probability distribution leading to increased performance, reduced memory consumption and greater interpretability when viewing the model graphically.

    If we add temporal links to a mixture model, we can create a number of useful models. The link labeled 1 in Image 5, indicates that the link has order 1, which means that the Cluster Node is linked to itself in the next time step. This is easier to see if we unroll the model for a few time steps, as shown in Image 6. In the same way that we can extend a mixture model with additional structure, we can extend temporal models with additional structure. Mixture models are a very popular statistical technique.

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    We have shown how a simple Bayesian network can represent a mixture model, and discussed the type of tasks they can perform. We have also suggested ways in which mixture models can be extended within the Bayesian network paradigm, including time series models. Mixture models This article describes how mixture models can be represented using a Bayesian network.

    Neural networks: representation.

    What are mixture models? Clustering - a term often used, because each group of similar data is called a cluster. Segmentation - a term often used when the groups are used to separate entities such as customers for the purposes of marketing. Density estimation - a term often used, because a probabilistic model such as a mixture model, estimates a probability density function pdf. Image 1 - plot of a Gaussian mixture model with training data Usage Mixture models have a wide range of uses. Data exploration Mixture models are useful for identifying key characteristics of your data, such as the most common relationships between variables, and also unusual relationships.

    Segmentation Because clustering detects similar groups, we can identify a group that has certain qualities and then determine segments of our data that have a high probability of belonging to that group. Anomaly detection Unseen data can be compared against a model, to determine how unusual anomalous that data is.

    Prediction Although Mixture models are an unsupervised learning technique, we can use them for prediction if during learning, we include variables we wish to predict output variables. Mixture models as Bayesian networks Mixture models are simple Bayesian networks, and therefore we can represent them graphically as shown in Image 2. Image 2 - Bayesian network mixture model The node named cluster is a discrete variable, with a number of discrete states, each representing an individual cluster. Image 3 - Alternative Bayesian network mixture model Image 4 shows a Mixture model in which the probability of each continuous variable is independent of the other continuous variables given the cluster.

    This model has fewer parameters, however cannot represent the rotations of ellipses shown in Image A model such as this is termed a diagonal model, because if you constructed a multivariate Gaussian over the continuous variables, all values of the covariance matrix would be zero, except for the diagonal variance entries.

    Gaussian Mixture Model

    Image 4 - Diagonal Bayesian network mixture model Unlike some clustering techniques where a data point only belongs to a single cluster hard clustering , probabilistic mixture models use what is known as soft clustering, i. Learning Usually the parameters of a model are learned from data.