Refactoring Probability Distributions: part 1, part 2, part 3, part 4, part 5

Welcome to the 5th (and final) installment of Refactoring Probability Distributions! Today, let's begin with an example from Bayesian Filters for Location Estimation (PDF), an excellent paper by Fox and colleagues.

In their example, we have a robot in a hallway with 3 doors. Unfortunately, we don't know where in the hallway the robot is located:

The vertical black lines are "particles." Each particle represents a possible location of our robot, chosen at random along the hallway. At first, our particles are spread along the entire hallway (the top row of black lines). Each particle begins life with a weight of 100%, represented by the height of the black line.

Now imagine that our robot has a "door sensor," which currently tells us that we're in front of a door. This allows us to rule out any particle which is located between doors.

So we multiply the weight of each particle by 100% (if it's in front of a door) or 0% (if it's between doors), which gives us the lower row of particles. If our sensor was less accurate, we might use 90% and 10%, respectively.

What would this example look like in Haskell? We could build a giant list of particles (with weights), but that would require us to do a lot of bookkeeping by hand. Instead, we use a monad to hide all the details, allowing us to work with a single particle at a time.

localizeRobot :: WPS Int
localizeRobot = do
  -- Pick a random starting location.
  -- This will be our first particle.
  pos1 <- uniform [0..299]
  -- We know we're at a door.  Particles
  -- in front of a door get a weight of
  -- 100%, others get 0%.
  if doorAtPosition pos1
    then weight 1
    else weight 0
  -- ...

What happens if our robot drives forward?

Driving down the hall

When our robot drives forward, we need to move our particles the same amount:

This time, we don't detect a door, so we need to invert our blue line. Once again, this allows us to filter our some particles.

In Haskell, we would write:

localizeRobot = do
  -- ...
  -- Drive forward a bit.
  let pos2 = pos1 + 28
  -- We know we're not at a door.
  if not (doorAtPosition pos2)
    then weight 1
    else weight 0
  -- ...

The next time we move, things start to get really interesting:

We can rule out all but one clump of particles! In Haskell:

localizeRobot = do
  -- ...
  -- Drive forward some more.
  let pos3 = pos2 + 28
  if doorAtPosition pos3
    then weight 1
    else weight 0
  -- Our final hypothesis.
  return pos3

And that's it! We can run our particle system as follows:

> runWPS' localizeRobot 10

In this example, we run our particle system with 10 particles, 3 of which survive the whole way to the end. These are clustered about 1/3rd of the way down the hallway, giving us a pretty good idea of where we are.

Unweighted Particle System Monad

OK, so how does our monad work? This gets pretty technical, so you may want to review part 1 and part 2 before continuing. Alternatively, you could just scroll down to the last section.

To begin with, let's ignore the weights. We can represent a particle system as function that maps an integer to a random list of particles:

newtype PS a = PS { runPS :: Int -> Rand [a] }

The integer tells us how many particles to return:

> runRand (runPS (uniform [1..10]) 5)

If we have a random value of type Rand a, we can build a particle system of type PS a by sampling that random value repeatedly. Building on the code from part 2, we get:

liftRand :: Rand a -> PS a
liftRand r = PS (sample r)

Given a particle system of type PS a, and a function of type a -> b, we can construct a particle system of type PS b by running the particle system and applying our function to each particle:

instance Functor PS where
  fmap f ps = PS mapped
    where mapped n =
            liftM (map f) (runPS ps n)

We also need to define the monadic return and >>= operators. The former can be built from return operator for Rand:

instance Monad PS where
  return = liftRand . return
  ps >>= f = joinPS (fmap f ps)

As for the latter, it's usually easiest to define >>= in terms of a join operation. And join is where the magic happens---we need to collapse a value of type PS (PS a) into a value of type PS a:

joinPS :: PS (PS a) -> PS a
joinPS psps = PS (joinPS' psps)

joinPS' :: PS (PS a) -> Int -> Rand [a]
joinPS' psps n = do
    pss <- (runPS psps n)
    xs <- sequence (map sample1 pss)
    return (concat xs)
  where sample1 ps = runPS ps 1

The code is a little tricky, but the basic idea is simple: We run our outer particle system, producing n particles. But each of these particles is itself a particle system, which we run to produce a single particle. We then combine the n single-particle results into one result with n particles.

We also need to make our particle system a probability monad. Again, we can do this by reusing code from Rand:

instance Dist PS where
  weighted = liftRand . weighted

Note that this monad isn't ready for production use---it's still pretty slow, and it doesn't give us any way to access our sensors.

Weighted Particle System Monad

The weighted version of the particle system is a bit easier. We can just reuse PerhapsT, which transforms an existing monad into one with a per-element weight:

type WPS = PerhapsT PS

We also need to make WPS into a probability distribution:

instance Dist (PerhapsT PS) where
  weighted = PerhapsT . weighted . map liftWeighted
    where liftWeighted (x,w) = (Perhaps x 1,w)

Next, we need a way to update the weight of a given particle. Note that this is closely related to applyProb from part 4.

weight :: Prob -> WPS ()
weight p = PerhapsT (return (Perhaps () p))

Finally, we need a way run a weighted particle system:

runWPS wps n = runPS (runPerhapsT wps) n

In our example, we also use a wrapper function to clean up the output a bit:

runWPS' wps n =
  (runRand . liftM catPossible) (runWPS wps n)

catPossible (ph:phs) | impossible ph =
  catPossible phs
catPossible (Perhaps x p:phs) =
  x:(catPossible phs)
catPossible [] = []

You can download the latest version of this code using Darcs:

darcs get

In a more sophisticated implementation, we'd want to periodically resample our particles, discarding the ones with weights near zero, and increasing the number of particles near likely locations.

More cool stuff

Bayesian Filters for Location Estimation (PDF) is an excellent (and very accessible) overview of belief functions. If you're into robotics or probability, it's definitely worth checking out.

As sigfpe points out, PerhapsT is closely related m-sets, which turn up a lot whenever you're working with matrices. In fact, you can build a monad from any measure space, including possibility and other models of beliefs.

Don't miss sigfpe's excellent articles on quantum mechanics. He shows how to generalize the probability monad to support quantum amplitudes and mixed states.

Given all this cool stuff, I strongly suspect that monads are the Right Way to represent probability and beliefs in programming languages.