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Listing 9-7. Implementing Probabilistic Modeling Using Computation Expressions type Distribution<'a> = abstract Sample : 'a abstract Support : Set<'a> abstract Expectation: ('a -> float) -> float let always x = { new Distribution<'a> with member d.Sample = x member d.Support = Set.singleton x member d.Expectation(H) = H(x) } let rnd = System.Random() let coinFlip (p:float) (d1:Distribution<'a>) (d2:Distribution<'a>) = if p < 0.0 || p > 1.0 then failwith "invalid probability in coinFlip" { new Distribution<'a> with member d.Sample = if rnd.NextDouble() < p then d1.Sample else d2.Sample member d.Support = Set.Union(d1.Support,d2.Support) member d.Expectation(H) = p * d1.Expectation(H) + (1.0-p) * d2.Expectation(H) } The types of these primitives are as follows: type Distribution<'a> = abstract Expectation: ('a -> float) -> float abstract Sample : 'a abstract Support : Set<'a> val always: 'a -> Distribution<'a> val coinFlip : float -> Distribution<'a> -> Distribution<'a> -> Distribution<'a> 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 just 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.

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Caution Use of instead of triggers on object views can get fairly complex, especially when enabling updates on the view. You may want to read this section with either a fresh mind or a fresh cup of coffee or both!

Here is the first solution worth serious consideration In this case, when an update occurs, instead of finding the row to update by querying only on the columns that comprise the primary key, you query on all columns of the table that may have been changed since you did the read In the case where a row has changed, the update will fail What the application does in that circumstance is where the specific requirements of your application come in, but generally it involves informing the user that the data has changed since the time that she read it, and asking her if she d like to overwrite the changes, cancel her update, or examine the differences between her data and the new data in the database The downside to this approach is that it can be extremely expensive for the database to perform these operations.

Listing 9-8. Defining a Builder for Probabilistic Modeling Using Computation Expressions let bind (dist:Distribution<'a>) (k: 'a -> Distribution<'b>) = { new Distribution<'b> with member d.Sample = (k(dist.Sample)).Sample member d.Support = Set.Union(dist.Support.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.Let(v,f) = bind (always v) f member x.Bind(d,f) = bind d f member x.Return(x) = always x let dist = new DistributionBuilder() The types of these primitives are as follows: val bind: Distribution<'a> -> ('a -> Distribution<'b>) -> Distribution<'b> val dist: DistributionBuilder Listing 9-8 shows the all-important bind primitive; this 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 we begin using workflow syntax, we define two derived functions to compute distributions. Listing 9-9 shows the additional derived operations for distribution objects that we will use later in this example. Listing 9-9. Defining the Derived Operations for Probabilistic Modeling Using Computation Expressions let weightedCases (inp: ('a * 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 = List.sumByInt (fun (_,v) -> v) inp weightedCases (inp.Map (fun (x,v) -> (x,(float v/float total))))