In this tutorial we show how to directly sample from a function using Metropolis Hastings (MCMC).

Problem Description

We are given the function \(g(\vartheta)=\exp(-\vartheta^2)\) for \(\vartheta\in[-10,10]\).

We assume that \(f\) represents the unnormalized density of a distribution. We want to draw samples from this distribution.

For the rest of the tutorial we will work with the function \(f(\vartheta)=\log g(\vartheta) = -\vartheta^2\) for numerical reasons. In general we advise users of Korali to work in log space.

The Objective Function

Create a folder named model. Inside, create a file with name and paste the following code,

\[ \begin{align}\begin{aligned}#!/usr/bin/env python\\def evaluateModel( x ): v = x["Parameters"][0] x["Evaluation"] = -v*v\end{aligned}\end{align} \]

This is the computational model that represents our objective function.

Sampling with MCMC

First, open a file and import the korali module

#!/usr/bin/env python3
import korali

Import the computational model,

import sys
from directModel import *

The Korali Experiment Object

Next we construct a korali.Experiment object and set the computational model,

e = korali.Experiment()
e["Problem"]["Objective Function"] = model

The Problem Type

Then, we set the type of the problem to Direct Evaluation

e["Problem"]["Type"] = "Evaluation/Direct/Basic"

The Variables

In this problem there is only one variable,

e["Variables"][0]["Name"] = "X"

The Solver

We choose the solver MCMC and set the initial mean and standard deviation of the parameter X.

e["Solver"]["Type"]  = "MCMC"
e["Variables"][0]["Initial Mean"] = 0.0
e["Variables"][0]["Initial Standard Deviation"] = 1.0

e["Solver"]["Burn In"] = 500
e["Solver"]["Termination Criteria"]["Max Samples"] = 5000

We also set some settings for MCMC. For a detailed description of the MCMC settings, see MCMC

Configuring the output

To reduce the output frequency we write

e["File Output"]["Frequency"]    = 500
e["Console Output"]["Frequency"] = 500
e["Console Output"]["Verbosity"] = "Detailed"


Finally, we are ready to run the simulation,

k = korali.Engine()

The results are saved in the folder _korali_result/.


You can see a histogram of the results by running the command python3 -m korali.plot