Changes between Initial Version and Version 1 of OfficialTolArchiveNetworkBysPrior

Timestamp:

Dec 24, 2010, 12:54:36 PM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkBysPrior

@@ +1 @@
+[[PageOutline]]
+
+= Package BysPrior =
+
+BysPriorInf stands for Bayesian Prior Information and allows to define prior 
+information handlers to be used in estimation systems (max-likelihood and 
+bayesian ones).
+
+A prior is a distribution function over a subset of the total set of variables 
+of a model that expresses the knowledge about the phenomena behind the model.
+
+The effect of a prior is to add the logarithm of its likelihood to the 
+logarithm of the likelihood of the global model. So it can be two or 
+more priors over some variables. For example, in order to stablish a 
+truncated normal we can define a uniform over the feasible region and 
+an unconstrainined normal.
+
+In order to be estimated with NonLinGloOpt (max-likelihood) and BysSampler 
+(Bayesian sampler), each prior must define methods to calculate the logarithm
+of the likelihood (except an additive constant), its gradient and its hessian,
+and an optional set of constraining inequations, in order to define the feasible 
+region. Each inequation can be linear or not and the gradient and hessian must 
+be also calculated. Note that this implies that priors should be continuous and 
+two times differentiable but this an admisible restricion in almost all cases.
+
+== Non informative priors ==
+
+Let [[LatexEquation( \beta )]] a uniform random variable in a region 
+[[LatexEquation(\Omega\in\mathbb{R}^{n} )]] which likelihood function is [[BR]]
+
+[[LatexEquation(lk\left(\beta\right) \propto 1 )]]
+
+Since the logarithm of the likelihood but a constant is zero, when 
+log-likelihood is not defined for a prior, the default assumed will be the 
+uniform distribution, also called non informative prior.
+
+=== Domain prior ===
+The easiest way, but one of the most important, to define non informative 
+prior information is to stablish a domain interval for one or more variables. 
+
+In this cases, you mustn't to define the log-logarithm nor the constraining 
+inequation functions, but simply it's needed to fix the lower and upper 
+bounds:[[BR]][[BR]]
+
+[[LatexEquation( \beta\in\Omega\Longleftrightarrow l_{k}\leq\beta\leq u_{k}\wedge-\infty\leq l_{k}<u_{k}\leq\infty )]]
+
+=== Polytope prior ===
+A polytope is defined by a system of arbitrary linear inequalities
+
+[[LatexEquation( A\beta\leq a\wedge A\in\mathbb{R}^{r\times n}\wedge a\in\mathbb{R}^{r} )]]