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# Sampling¶

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 directModel.py 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
sys.path.append('./model')
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"


## Running¶

Finally, we are ready to run the simulation,

k = korali.Engine()
k.run(e)


The results are saved in the folder _korali_result/.

## Plotting¶

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