hbayes-0.5: Bayesian Networks

Bayes.Examples.ContinuousSampling

Description

Sampling example with continuous distributions

Continuous networks can't be handled by any of the functions defined for the discrete networks. So, instead of using exact inference algorithms like the junction trees, sampling method have to be used.

In this example, we want to estimate a parameter which is measured by noisy sensors.

There are `nbSensors` available. They are described with a `normal` distribution centered on the value of the unknown parameters and with a standard deviation of 0.1.

The unknown parameter is described with a `uniform` distribution bounded by 1.0 and 2.0.

First, we describe the sensor:

```sensor :: `DN` -> `CNMonad` `DN`
sensor p = do
`normal` () p 0.1
```

It is just a `normal` distribution. The mean of this distribution is the parameters p. This parameter has special type `DN`. All expressions used to build the continuous bayesian network are using values of type `DN`. A value of type `DN` can either represent a constant, a variable or an expression.

If the sensor was biased, we could write:

```    `normal` ()  (p + 0.2) 0.1
```

The Bayesian network describing the measurement process is given by:

```    test = `runCN` \$ do
a <- `uniform` "a" 1.0 2.0 -- Unknown parameter
sensors <- sequence (replicate `nbSensors` (sensor a))
return (a:sensors)
```

We are connecting `nbSensors` nodes corresponding to the `nbSensors` measurements. In real life it can either be different sensors or the same one used several times (assuming the value of the parameter is not dependent on time).

Now, as usual in all the examples of this package, we get the bayesian graph and a list of variables used to compute some posterior or define some evidences

```    debugcn = do
let ((a:sensors), testG) = test
```

Then, we generate some random measurements and create the evidences

```    g <- create
measurements <- sequence . replicate  nbSensors \$ (MWC.normal 1.5 0.1 g)
let evidence = zipWith (=:) sensors measurements
```

Evidence has type `CVI` and is created with the assigment operator `=:` .

Now, we generate some samples to estimate the posterior distributions.

```    n <- `runSampling` 10000 200 (`continuousMCMCSampler` testG evidence)
```

This function is generating a sequence of graphs ! We are not interested in the sensor values. They are known and fixed since they have been measured. So, we extract the value of the parameter.

```    let samples = map (g -> `instantiationValue` . fromJust . `vertexValue` g \$ (`vertex` a)) n
```

And with the samples for the parameters we can compute an histogram and get an approximation of the posterior.

```    let samples = map (g -> `instantiationValue` . fromJust . `vertexValue` g \$ (`vertex` a)) n
h = `histogram` 6 samples
print h
```

We see in the histogram that the estimated value is around 1.5.