This is a distribution that is used in Monte Carlo sampling, so it (with all statistics) will be approximated by sample points from Markov chain obtained by MCMC algorithms. There is only one requirement which must be satisfied: it must be possible to calculate the density of such distribution at any point.
Tabela 9.13. SamplingDistribution methods
| Method name and syntax | Description of parameters | Output | Output type |
|---|---|---|---|
pdf(x) | x - the point in which to calculate the density function (double[]). | the value of probability density function at the specified point | double |
dim() | dimension of the distribution | int |
StandardMultivariateDistribution is an adaptor for any MultivariateDistribution object, which allows any multiavariate distribution from DistributionLibrary to be sampled by MCMC algorithms.
The constructor has the following syntax:
StandardMultivariateDistribution(distribution)
where distribution is a MultivariateDistribution object.
The methods are the same as in case of MultivariateDistribution.
The likelihood function
is the conditional probability function considered as a
function of parameters
and
,
where
is the observed outcome of an experiment. In other words, when
is viewed as a function of
with fixed
,
it is a probability density function.
On the other hand, when it viewed as a function of
with fixed
(i.e. it refers to past events with known outcomes),
it is a likelihood function.
LogLikelihood is the ancestor of likelihood classes. It is defined in the likelihood module of the AdvancedMiner system.
Tabela 9.14. LogLikelihood methods
| Method name and syntax | Description of parameters | Output | Output type |
|---|---|---|---|
getLog() | the likelihood value in logarithmic form | double | |
getParameters() |
the values of
parameters for which the likelihood is calculated
| double[] | |
| setParameters(theta) | theta - an array of
parameters for which the likelihood should be calculated (double[])
| void | |
getDim() | the dimension of the likelihood, i.e. the number of parameters. | int |
The cosntructor has the following syntax:
LogitLogLikelihood(x, y, theta)
Tabela 9.15. LogitLogLikelihood constructor settings
| Parameter name | Parameter type | Description |
|---|---|---|
| x | double[][] | a two dimesional array of observations: the 1st. dimension is the number of obervation, the 2nd. dimenion is the number of attribute |
| y | double[] | an array of targets, containing either the values of 1 or 0 |
| theta | double[] |
an array containing the
likelihood coefficients.
|
The method for LogitLogLikelihood are the same as for LogLikelihood. In particular the method getLog() works as a logit function and the value it returns is calculated as
SamplingLikelihoodDistribution is an adapter which allows any likelihood function to be treated as a sampling distributions and used in Monte Carlo algorithms. It is used in BayesianInterference for likelihood sampling to obtain the posterior distribution.
The constructor has the following syntax:
SamplingLikelihoodDistribution(likelihood[, log])
Tabela 9.16. SamplingLikelihoodDistribution constructor settings
| Parameter name | Parameter type | Description |
|---|---|---|
| likelihood | LogLikelihood object | the likelihood function which has to be sampled |
| log | boolean | specifies whether the pdf() should return likelihood (false, default) or loglikelihood (true) values. |
SamplingLikelihoodDistribution methods are the same as in the case of SamplingDistribution.
The density of the posterior distribution is calculated as the product of likelihood and prior density:
The constructor hast the following syntax:
SamplingPosteriorDistribution(likelihood, prior[, log])
Tabela 9.17. SamplingPosteriorDistribution constructor settings
| Parameter name | Parameter type | Description |
|---|---|---|
| likelihood | a LogLikelihood object | the likelihood function |
| prior | a MultivariateDistribution object | the a-priori distribution |
| log | boolean | specifies whether the pdf() should return the a-posteriori density in normal (false, default) or logarithmic (true) form. |
SamplingPosteriorDistribution methods are the same as in the case of SamplingDistribution.
This class transforms the samples from a Markov chain (or a number of Markov chains) into a regular multivariate distribution.
The constructor hast the following syntax:
DistributionFromMarkovChain(chains)
where chains is a MarkovChain object or an array of MarkovChain objects (i.e. MarkovChain[]).
On addition to methods inherited from MultivariateDistribution, DistributionFromMarkovChain hast the following methods: