Likelihood by Reference

A Bayesian Reference problem is for data that originate from a computational model \(f\):

\[\begin{split}d = (x_j, y_j)_{j=1...N}\; with \\ y_j = f(x_j) + \epsilon\end{split}\]

The distribution of noise \(\epsilon\) defines the likelihood model of the data. You can choose between three types of noise likelihood models: Normal, Negative Binomial and Positive Normal.

The following likelihood functions are available in Korali:

Normal

\[p(d | \vartheta) = {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left((x-\mu )/\sigma \right)^{2}}\]

where \(\mu\) is the mean and \(\sigma\) is the Standard Deviation.

Positive Normal

The Normal likelihood truncated at 0.

StudentT

\[p(d | \vartheta) = {\frac {\Gamma((n+1)/2)}{{\sqrt {n\pi} \Gamma(n/2)}}}(1+d^2/n)^{-(n+1)/2}\]

where \(n\) is refered to as Degrees Of Freedom.

Positive StudentT

The StudentT likelihood truncated at 0.

Poisson

\[p(d | \vartheta) = {\frac {\lambda^d e^{-\lambda} }{d!}}\]

where \(\lambda\) is the mean.

Geometric

\[p(d | \vartheta) = \lambda(1-\lambda)^{d-1}\]

where \(\lambda\) is the mean.

Negative Binomial

\[p(d | \vartheta) = {d+r-1\choose d} p^r (1-p)^d\]

where \(p\) is the success probability and \(r\) is the dispersion parameter.

Usage

e["Problem"]["Type"] = "Bayesian/Reference"

Compatible Solvers

This problem can be solved using the following modules:

Variable-Specific Settings

These are settings required by this module that are added to each of the experiment’s variables when this module is selected.

Prior Distribution

  • Usage: e[“Variables”][index][“Prior Distribution”] = string

  • Description: Indicates the name of the distribution to use as prior distribution.

Distribution Index

  • Usage: e[“Variables”][index][“Distribution Index”] = unsigned integer

  • Description: Stores the the index number of the selected prior distribution.

Name

  • Usage: e[“Variables”][index][“Name”] = string

  • Description: Defines the name of the variable.

Configuration

These are settings required by this module.

Computational Model

  • Usage: e[“Problem”][“Computational Model”] = Computational Model

  • Description: Stores the computational model. It should the evaluation of the model at the given reference data points.

Reference Data

  • Usage: e[“Problem”][“Reference Data”] = List of real number

  • Description: Reference data required to calculate likelihood. Model evaluations are compared against these data.

Likelihood Model

  • Usage: e[“Problem”][“Likelihood Model”] = string

  • Description: Specifies the likelihood model to approximate the reference data to.

  • Options:

    • “Normal”: The user specifies the mean and the standard deviation of the normal likelihood.

    • “Positive Normal”: The user specifies the mean and the standard deviations of the truncated normal on [0,infty]

    • “StudentT”: The user specifies the degrees of freedom (>0) of the Student’s t-distribution.

    • “Positive StudentT”: The user specifies the degrees of freedom (>0) of the half Student’s t-distribution.

    • “Poisson”: The user specifies the mean (>0) of the Poisson distribution.

    • “Geometric”: The user specifies the inverse mean (>0) of the Geometric distribution.

    • “Negative Binomial”: The user specifies the mean and the dispersion parameter of the Negative Binomial distribution.

Variable Defaults

These following configuration will be assigned to each of the experiment variables by default. Any settings defined by the user will override the given settings specified in these defaults.

{
"Distribution Index": 0
}