| | 76 | <hr/><a name="_details"></a><h2>Detailed Description</h2> |
| | 77 | <p>Mixture of Exponential Family Densities. </p> |
| | 78 | <p>An approximate estimation method for models with latent discrete variable, such as mixture models of the following kind: </p> |
| | 79 | <p class="formulaDsp"> |
| | 80 | <img class="formulaDsp" alt="\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \]" src="form_40.png"/> |
| | 81 | </p> |
| | 82 | <p> where <img class="formulaInl" alt="$\psi$" src="form_41.png"/> is a known function of past outputs, <img class="formulaInl" alt="$w=[w_1,\ldots,w_n]$" src="form_42.png"/> are component weights, and component parameters <img class="formulaInl" alt="$\theta_i$" src="form_43.png"/> are assumed to be mutually independent. <img class="formulaInl" alt="$\Theta$" src="form_44.png"/> is an aggregation af all component parameters and weights, i.e. <img class="formulaInl" alt="$\Theta = [\theta_1,\ldots,\theta_n,w]$" src="form_45.png"/>.</p> |
| | 83 | <p>The characteristic feature of this model is that if the exact values of the latent variable were known, estimation of the parameters can be handled by a single model. For example, for the case of mixture models, posterior density for each component parameters would be a BayesianModel from Exponential Family.</p> |
| | 84 | <p>This class uses EM-style type algorithms for estimation of its parameters. Under this simplification, the posterior density is a product of exponential family members, hence under EM-style approximate estimation this class itself belongs to the exponential family.</p> |
| | 85 | <p>TODO: Extend <a class="el" href="classbdm_1_1BM.html" title="Bayesian Model of a system, i.e. all uncertainty is modeled by probabilities.">BM</a> to use rvc. </p> |
| 227 | | <hr/><a name="_details"></a><h2>Detailed Description</h2> |
| 228 | | <p>Mixture of Exponential Family Densities. </p> |
| 229 | | <p>An approximate estimation method for models with latent discrete variable, such as mixture models of the following kind: </p> |
| 230 | | <p class="formulaDsp"> |
| 231 | | <img class="formulaDsp" alt="\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \]" src="form_40.png"/> |
| 232 | | </p> |
| 233 | | <p> where <img class="formulaInl" alt="$\psi$" src="form_41.png"/> is a known function of past outputs, <img class="formulaInl" alt="$w=[w_1,\ldots,w_n]$" src="form_42.png"/> are component weights, and component parameters <img class="formulaInl" alt="$\theta_i$" src="form_43.png"/> are assumed to be mutually independent. <img class="formulaInl" alt="$\Theta$" src="form_44.png"/> is an aggregation af all component parameters and weights, i.e. <img class="formulaInl" alt="$\Theta = [\theta_1,\ldots,\theta_n,w]$" src="form_45.png"/>.</p> |
| 234 | | <p>The characteristic feature of this model is that if the exact values of the latent variable were known, estimation of the parameters can be handled by a single model. For example, for the case of mixture models, posterior density for each component parameters would be a BayesianModel from Exponential Family.</p> |
| 235 | | <p>This class uses EM-style type algorithms for estimation of its parameters. Under this simplification, the posterior density is a product of exponential family members, hence under EM-style approximate estimation this class itself belongs to the exponential family.</p> |
| 236 | | <p>TODO: Extend <a class="el" href="classbdm_1_1BM.html" title="Bayesian Model of a system, i.e. all uncertainty is modeled by probabilities.">BM</a> to use rvc. </p> |
