# Custom Likelihood

While a Bayesian Reference type problem is for data that originate from a functional dependency, $$d = (x_j, y_j)_{j=1...N}\;$$ with $$y_j = f(x_j) + \epsilon$$, a Custom Likelihood model makes no such assumption.

With a Custom Likelihood, the function $$p(d|\vartheta)$$ is given directly by a user-defined model of the form $$f:\; \mathbb{R}^N\rightarrow\mathbb{R}$$, where $$N$$ is the number of variables.

## Likelihood Models

Whereas with an Additive Normal Likelihood, the computational model is assumed to be of the form $$f(x;\vartheta)$$, where $$d$$ is a set of M given data points. The output of the model represents the values of the function at the given points for which Korali can build a likelihood function $$p(d|\vartheta)$$, and a prior probability density $$p(\vartheta)$$.

Currently, Korali uses a Normal estimator for the error component of the likelihood calculation, using a statistical-type variable, sigma:

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

With a Custom Likelihood, the function $$p(d|\vartheta)$$ is given directly by a user-defined model of the form $$f:\mathbb{R}^N\rightarrow\mathbb{R}$$, where $$N$$ is the number of variables.

## Usage

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

## 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.

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

• Description: Stores the user-defined likelihood model. It should return the value of the Log Likelihood of the given sample.

## 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
}