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
}