Documentation

Table of Contents

1. Effects
   1. Sample
   2. Observe
   3. Random
   4. Weight
   5. Emit
2. Metropolis-Hasting
3. Slice Sampling
4. Wrapper
5. Examples
   1. Linear Regression
   2. 2D-Robot
   3. SIR model

Effects

Sample

The default handler for the effect Sample draws a sample from a given distribution using the function draw. draw uses the Box-Muller method to draw two random samples from a Normal/Gaussian distribution using the effekt Random to generate two random doubles for input of Box-Muller.
The effect Sample also is used for the tracing and trace handling for Metropolis-Hastings and gets handled in handleTracing and handleReusingTrace. The first handler constructs the trace while the second handler takes the first element of an existing trace.

Observe

The default observe handler handleObserve uses the effect Weight to weigh the density of the observed value and distibution.

Random

With the bulit in function random, this effect generates a random value as a Double.

Weight (Factor)

The first handler handleRejection performs rejection sampling and the second handler handleWeight multiplies the factors by which the program gets weight together.

Emit

Emit is defined as an interface because of its type polimorphic properties but it acts just like any other effect in this program. The default handler performes a println of the current element, often used in loops. This effect is also used to limit possibly indefinetly running loops usign the handleLimitEmit. This handler counts the steps or loops of a given function it runs, and only resumes if the given number of steps is not jet surpased. Like the default handler, the limiter prints the current element in each cycle of the loop. But possibly contrary to intuition, handleLimitEmit does not perform n consecutive steps of the algorithm but performes the first step from the initial configuration n times.

Metropolis-Hasting

The algorithm got split into two seperate functions. The helper function metropolisStep, which performes one iteration of the algorithm, and the main algorithm metropolisHastingsAlgo which performs (possibly indefinete) cycles of the algorithm using metropolisStep.
metropolisStep also makes use of the function propose which recursively adds Gaussian noise to an existing trace and uses the resulting trace as a proposal for the new trace. This is also called independent Metropolis-Hastings. Another version of propose called proposeSingleSite proposes only one new sample to a random element of the trace per iteration and resuses the other samples of the trace. Using this version of propose we get the Single-Site Metropolis-Hastings algorithm, MetropolisHastingsSingleSiteAlgo which is sometimes also called Random Walk Metropolis-Hastings.

Slice Sampling

sliceSamplingAlgo is another algorithm in this library with which we can performe inference on programs. This algorithm can be used to draw random samples from a distribution, it is a Markov chain Monte Carlo type algorithm.

Wrapper

All inference algorithms constructed in this library have a wrapper with which it is possible to use the algorithms without having to use the effect handler since they are built into the wrappers. Because of handleLimitEmit it is necessary to give the wrapper an integer n that represents the number of steps the algorithm should perform. Following algorithms are included in this library rejectionSampling sliceSampling metropolisHastings metropolisHastingsSingleSite.

metropolisHastingsSingleSite[R](n){...}

This is an example on how calling the algorithm metropolisHastingsSingleSite works, where R is the return Type of the example/program one wants to apply the algorithm to. E.g. in the robot example this is State a user defined type.

Examples

Linear Regression

This example uses linear regression which is an analysis method to predict values based on values that are already given. Here we want to approximate the slope and the y-intercept of a linear graph given points the graph crosses.

2D-Robot

In this example we trace/approximate the path of a robot that moves over a cartesian plane. At the center of the plane (0,0) is a radar station that measures the distance to the robot with noise. During one unit of time the robot changes its position based on the current trajectory and also changes its velocity in both x and y direction by adding gaussian noise. This is done via the move function. With meassure we can approximate the new position of the robot based on the new meassurement we get, but also considering the velocity of the robot.

SIR model

Simulates a population with susceptible, infected and recovered patients. The function step imitates the transition of patients from e.g. susceptible to infected with the given transition probabilities. The transition probabilities do not correlate to any real live example but rather are just a way of visualization. With test we can test the population for the diseas/virus but we also have to take into consideration that the tests can yield false positives(FP) or false negatives(FN).