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CHAPTER 9 INTRODUCING LANGUAGE-ORIENTED PROGRAMMING
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The simplest distribution is always x; this is a distribution that always samples to the same value. Its expectation and support are easy to calculate. The expectation of a function H is just H applied to the value, and the support is a set containing the single value x. The next distribution defined is coinFlip, which is a distribution that models the ability to choose between two outcomes. Listing 9-8 shows how you can define a workflow builder for distribution objects. Listing 9-8. Defining a Builder for Probabilistic Modeling Using Computation Expressions let bind (dist:Distribution<'T>) (k: 'T -> Distribution<'U>) = { new Distribution<'U> with member d.Sample = (k dist.Sample).Sample member d.Support = Set.unionMany (dist.Support |> Set.map (fun d -> (k d).Support)) member d.Expectation H = dist.Expectation(fun x -> (k x).Expectation H) } type DistributionBuilder() = member x.Delay f = bind (always ()) f member x.Bind (d, f) = bind d f member x.Return v = always v member x.ReturnFrom vs = vs let dist = new DistributionBuilder() The types of these primitives are as follows: val bind: Distribution<'T> -> ('T -> Distribution<'U>) -> Distribution<'U> val dist: DistributionBuilder Listing 9-8 shows the all-important bind primitive; it combines two distributions, using the sample from the first to guide the sample from the second. The support and expectation are calculated by taking the support from the first and splaying it over the support of the second. The expectation is computed by using the first distribution to compute the expectation of a function derived from the second. These are standard results in probability theory and are the basic machinery you need to get going with some interesting modeling. Before you begin using workflow syntax, you define two derived functions to compute distributions. Listing 9-9 shows the additional derived operations for distribution objects that you use later in this example. Listing 9-9. Defining the Derived Operations for Probabilistic Modeling Using Computation Expressions let weightedCases (inp: ('T * float) list) = let rec coinFlips w l = match l with | [] -> failwith "no coinFlips" | [(d,_)] -> always d | (d,p)::rest -> coinFlip (p/(1.0-w)) (always d) (coinFlips (w+p) rest) coinFlips 0.0 inp let countedCases inp = let total = Seq.sumBy (fun (_,v) -> v) inp weightedCases (inp |> List.map (fun (x,v) -> (x, float v / float total)))
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CHAPTER 9 INTRODUCING LANGUAGE-ORIENTED PROGRAMMING
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The two functions weightedCases and countedCases build distributions from the weighted selection of a finite number of cases. The types are as follows: val weightedCases : ('T * float) list -> Distribution<'T> val countedCases : ('T * int) list -> Distribution<'T> For example, here is the distribution of outcomes on a fair European roulette wheel: type Outcome = Even | Odd | Zero let roulette = countedCases [ Even,18; Odd,18; Zero,1] You can now use sampling to draw from this distribution: > roulette.Sample;; val it: Outcome = Even > roulette.Sample;; val it: Outcome = Odd And you can compute the expected payout of a $5 bet on Even, where you would get a $10 return: > roulette.Expectation (function Even -> 10.0 | Odd -> 0.0 | Zero -> 0.0);; val it: float = 4.864864865 Now, let s model another scenario. Let s say you have a traffic light with the following probability distribution for showing red/yellow/green: type Light = | Red | Green | Yellow let trafficLightD = weightedCases [ Red,0.50; Yellow,0.10; Green, 0.40 ] Drivers are defined by their behavior with respect to a traffic light. For example, a cautious driver is highly likely to brake on a yellow light and always stops on a red: type Action = Stop | Drive let cautiousDriver light = dist { match light with | Red -> return Stop | Yellow -> return! weightedCases [ Stop, 0.9; Drive, 0.1 ] | Green -> return Drive } An aggressive driver is unlikely to brake on yellow and may even go through a red light:
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