A monad tutorial for Clojure programmers (part 4)

In this fourth and last part of my monad tutorial, I will write about monad transformers. I will deal with only one of them, but it’s a start. I will also cover the probability monad, and how it can be extended using a monad transformer.

Basically, a monad transformer is a function that takes a monad argument and returns another monad. The returned monad is a variant of the one passed in to which some functionality has been added. The monad transformer defines that added functionality. Many of the common monads that I have presented before have monad transformer analogs that add the monad's functionality to another monad. This makes monads modular by permitting client code to assemble monad building blocks into a customized monad that is just right for the task at hand.

Consider two monads that I have discussed before: the maybe monad and the sequence monad. The maybe monad is for computations that can fail to produce a valid value, and return nil in that case. The sequence monad is for computations that return multiple results, in the form of monadic values that are sequences. A monad combining the two can take two forms: 1) computations yielding multiple results, any of which could be nil indicating failure 2) computations yielding either a sequence of results or nil in the case of failure. The more interesting combination is 1), because 2) is of little practical use: failure can be represented more easily and with no additional effort by returning an empty result sequence.

So how can we create a monad that puts the maybe monad functionality inside sequence monad values? Is there a way we can reuse the existing implementations of the maybe monad and the sequence monad? It turns out that this is not possible, but we can keep one and rewrite the other one as a monad transformer, which we can then apply to the sequence monad (or in fact some other monad) to get the desired result. To get the combination we want, we need to turn the maybe monad into a transformer and apply it to the sequence monad.

First, as a reminder, the definitions of the maybe and the sequence monads:

(defmonad maybe-m
   [m-zero   nil
    m-result (fn [v] v)
    m-bind   (fn [mv f]
               (if (nil? mv) nil (f mv)))
    m-plus   (fn [& mvs]
               (first (drop-while nil? mvs)))
    ])

(defmonad sequence-m
   [m-result (fn [v]
               (list v))
    m-bind   (fn [mv f]
               (apply concat (map f mv)))
    m-zero   (list)
    m-plus   (fn [& mvs]
               (apply concat mvs))
    ])

And now the definition of the maybe monad transformer:

(defn maybe-t
  [m]
  (monad [m-result (with-monad m m-result)
          m-bind   (with-monad m
                     (fn [mv f]
                       (m-bind mv
                               (fn [x]
                                 (if (nil? x)
                                   (m-result nil)
                                   (f x))))))
          m-zero   (with-monad m m-zero)
          m-plus   (with-monad m m-plus)
          ]))

The real definition in clojure.algo.monads is a bit more complicated, and I will explain the differences later, but for now this basic version is good enough. The combined monad is constructed by

(def maybe-in-sequence-m (maybe-t sequence-m))

which is a straightforward function call, the result of which is a monad. Let’s first look at what m-result does. The m-result of maybe-m is the identity function, so we’d expect that our combined monad m-result is just the one from sequence-m. This is indeed the case, as (with-monad m m-result) returns the m-result function from monad m. We see the same construct for m-zero and m-plus, meaning that all we need to understand is m-bind.

The combined m-bind calls the m-bind of the base monad (sequence-m in our case), but it modifies the function argument, i.e. the function that represents the rest of the computation. Before calling it, it first checks if its argument would be nil. If it isn't, the original function is called, meaning that the combined monad behaves just like the base monad as long as no computation ever returns nil. If there is a nil value, the maybe monad says that no further computation should take place and that the final result should immediately be nil. However, we can’t just return nil, as we must return a valid monadic value in the combined monad (in our example, a sequence of possibly-nil values). So we feed nil into the base monad's m-result, which takes care of wrapping up nil in the required data structure.

Let's see it in action:

(domonad maybe-in-sequence-m
  [x [1 2 nil 4]
   y [10 nil 30 40]]
  (+ x y))

The output is:

(11 nil 31 41 12 nil 32 42 nil 14 nil 34 44)

As expected, there are all the combinations of non-nil values in both input sequences. However, it is surprising at first sight that there are four nil entries. Shouldn’t there be eight, resulting from the combinations of a nil in one sequence with the four values in the other sequence?

To understand why there are four nils, let’s look again at how the m-bind definition in maybe-t handles them. At the top level, it will be called with the vector [1 2 nil 4] as the monadic value. It hands this to the m-bind of sequence-m, which calls the anonymous function in maybe-t's m-bind four times, once for each element of the vector. For the three non-nil values, no special treatment is added. For the one nil value, the net result of the computation is nil and the rest of the computation is never called. The nil in the first input vector thus accounts for one nil in the result, and the rest of the computation is called three times. Each of these three rounds produces then three valid results and one nil. We thus have 3×3 valid results, 3×1 nil from the second vector, plus the one nil from the first vector. That makes nine valid results and four nils.

Is there a way to get all sixteen combinations, with all the possible nil results in the result? Yes, but not using the maybe-t transformer. You have to use the maybe and the sequence monads separately, for example like this:

(with-monad maybe-m
  (def maybe-+ (m-lift 2 +)))

(domonad sequence-m
  [x [1 2 nil 4]
   y [10 nil 30 40]]
  (maybe-+ x y))

When you use maybe-t, you always get the shortcutting behaviour seen above: as soon as there is a nil, the total result is nil and the rest of the computation is never executed. In most situations, that’s what you want.

The combination of maybe-t and sequence-m is not so useful in practice because a much easier (and more efficient) way to handle invalid results is to remove them from the sequences before any further processing happens. But the example is simple and thus fine for explaining the basics. You are now ready for a more realistic example: the use of maybe-t with the probability distribution monad.

The probability distribution monad is made for working with finite probability distributions, i.e. probability distributions in which a finite set of values has a non-zero probability. Such a distribution is represented by a map from the values to their probabilities. The monad and various useful functions for working with finite distributions is defined in the library clojure.contrib.probabilities.finite-distributions (NOTE: this module has not yet been migrated to the new Clojure contrib library set.).

A simple example of a finite distribution:

(use 'clojure.contrib.probabilities.finite-distributions)
(def die (uniform #{1 2 3 4 5 6}))
(prob odd? die)

This prints 1/2, the probability that throwing a single die yields an odd number. The value of die is the probability distribution of the outcome of throwing a die:

{6 1/6, 5 1/6, 4 1/6, 3 1/6, 2 1/6, 1 1/6}

Suppose we throw the die twice and look at the sum of the two values. What is its probability distribution? That’s where the monad comes in:

(domonad dist-m
  [d1 die
   d2 die]
  (+ d1 d2))

The result is:

{2 1/36, 3 1/18, 4 1/12, 5 1/9, 6 5/36, 7 1/6, 8 5/36, 9 1/9, 10 1/12, 11 1/18, 12 1/36}

You can read the above domonad block as "draw a value from the distribution die and call it d1, draw a value from the distribution die and call it d2, then give me the distribution of (+ d1 d2)". This is a very simple example; in general, each distribution can depend on the values drawn from the preceding ones, thus creating the joint distribution of several variables. This approach is known as 'ancestral sampling'.

The monad dist-m applies the basic rule of combining probabilities: if event A has probability p and event B has probability q, and if the events are independent (or at least uncorrelated), then the probability of the combined event (A and B) is p×q. Here is the definition of dist-m:

(defmonad dist-m
   [m-result (fn [v] {v 1})
    m-bind   (fn [mv f]
               (letfn [(add-prob [dist [x p]]
                         (assoc dist x (+ (get dist x 0) p)))]
                 (reduce add-prob {}
                   (for [[x p] mv  [y q] (f x)]
                     [y (* q p)]))))
    ])

As usual, the interesting stuff happens in m-bind. Its first argument, mv, is a map representing a probability distribution. Its second argument, f, is a function representing the rest of the calculation. It is called for each possible value in the probability distribution in the for form. This for form iterates over both the possible values of the input distribution and the possible values of the distribution returned by (f x), combining the probabilities by multiplication and putting them into the output distribution. This is done by reducing over the helper function add-prob, which checks if the value is already present in the map, and if so, adds the probability to the previously obtained one. This is necessary because the samples from the (f x) distribution can contain the same value more than once if they were obtained for different x.

For a more interesting example, let’s consider the famous Monty Hall problem. In a game show, the player faces three doors. A prize is waiting for him behind one of them, but there is nothing behind the two other ones. If he picks the right door, he gets the prize. Up to there, the problem is simple: the probability of winning is 1/3.

But there is a twist. After the player makes his choice, the game host open one of the two other doors, which shows an empty space. He then asks the player if he wants to change his mind and choose the last remaining door instead of his initial choice. Is this a good strategy?

To make this a well-defined problem, we have to assume that the game host knows where the prize is and that he would not open the corresponding door. Then we can start coding:

(def doors #{:A :B :C})

(domonad dist-m
  [prize  (uniform doors)
   choice (uniform doors)]
  (if (= choice prize) :win :loose))

Let's go through this step by step. First, we choose the prize door by drawing from a uniform distribution over the three doors :A, :B, and :C. That represents what happens before the player comes in. Then the player's initial choice is made, drawing from the same distribution. Finally, we ask for the distribution of the outcome of the game, code>:win or :loose. The answer is, unsurprisingly, {:win 1/3, :loose 2/3}.

This covers the case in which the player does not accept the host's proposition to change his mind. If he does, the game becomes more complicated:

(domonad dist-m
  [prize  (uniform doors)
   choice (uniform doors)
   opened (uniform (disj doors prize choice))
   choice (uniform (disj doors opened choice))]
  (if (= choice prize) :win :loose))

The third step is the most interesting one: the game host opens a door which is neither the prize door nor the initial choice of the player. We model this by removing both prize and choice from the set of doors, and draw uniformly from the resulting set, which can have one or two elements depending on prize and choice. The player then changes his mind and chooses from the set of doors other than the open one and his initial choice. With the standard three-door game, that set has exactly one element, but the code above also works for a larger number of doors - try it out yourself!

Evaluating this piece of code yields {:loose 1/3, :win 2/3}, indicating that the change-your-mind strategy is indeed the better one.

Back to the maybe-t transformer. The finite-distribution library defines a second monad by

(def cond-dist-m (maybe-t dist-m))

This makes nil a special value in distributions, which is used to represent events that we don't want to consider as possible ones. With the definitions of maybe-t and dist-m, you can guess how nil values are propagated when distributions are combined: for any nil value, the distributions that potentially depend on it are never evaluated, and the nil value's probability is transferred entirely to the probability of nil in the output distribution. But how does nil ever get into a distribution? And, most of all, what is that good for?

Let's start with the last question. The goal of this nil-containing distributions is to eliminate certain values. Once the final distribution is obtained, the nil value is removed, and the remaining distribution is normalized to make the sum of the probabilities of the remaining values equal to one. This nil-removal and normalization is performed by the utility function normalize-cond. The cond-dist-m monad is thus a sophisticated way to compute conditional probabilities, and in particular to facilitate Bayesian inference, which is an important technique in all kinds of data analysis.

As a first exercice, let's calculate a simple conditional probability from an input distribution and a predicate. The output distribution should contain only the values satisfying the predicate, but be normalized to one:

(defn cond-prob [pred dist]
  (normalize-cond (domonad cond-dist-m
                    [v dist
                     :when (pred v)]
                    v))))

The important line is the one with the :when condition. As I have explained in parts 1 and 2, the domonad form becomes

(m-bind dist
        (fn [v]
          (if (pred v)
            (m-result v)
             m-zero)))

If you have been following carefully, you should complain now: with the definitions of dist-m and maybe-t I have given above, cond-dist-m should not have any m-zero! But as I said earlier, the maybe-t shown here is a simplified version. The real one checks if the base monad has m-zero, and if it hasn't, it substitutes its own, which is (with-monad m (m-result nil)). Therefore the m-zero of cond-dist-m is {nil 1}, the distribution whose only value is nil.

The net effect of the domonad form in this example is thus to keep all values that satisfy the predicate with their initial probabilities, but to transfer the probability of all values to nil. The call to normalize-cond then takes out the nil and re-distributes its probability to the other values. Example:

(cond-prob odd? die)
-> {5 1/3, 3 1/3, 1 1/3}

The cond-dist-m monad really becomes interesting for Bayesian inference problems. Bayesian inference is technique for drawing conclusions from incomplete observations. It has a wide range of applications, from spam filters to weather forecasts. For an introduction to the technique and its mathematical basis, you can start with the Wikipedia article.

Here I will discuss a very simple inference problem and its solution in Clojure. Suppose someone has three dice, one with six faces, one with eight, and one with twelve. This person picks one die, throws it a few times, and gives us the numbers, but doesn't tell us which die it was. Given these observations, we would like to infer the probabilities for each of the three dice to have been picked. We start by defining a function that returns the distribution of a die with n faces:

(defn die-n [n] (uniform (range 1 (inc n))))

Next, we come to the core of Bayesian inference. One central ingredient is the probability for throwing a given number under the assumption that die X was used. We thus need the probability distributions for each of our three dice:

(def dice {:six     (die-n 6)
           :eight   (die-n 8 )
           :twelve  (die-n 12)})

The other central ingredient is a distribution representing our 'prior knowledge' about the chosen die. We actually know nothing at all, so each die has the same weight in this distribution:

(def prior (uniform (keys dice)))

Now we can write the inference function. It takes as input the prior-knowledge distribution and a number that was obtained from the die. It returns the a posteriori distribution that combines the prior information with the information from the observation.

(defn add-observation [prior observation]
  (normalize-cond
    (domonad cond-dist-m
      [die    prior
       number (get dice die)
       :when  (= number observation)]
      die)))

Let's look at the domonad form. The first step picks one die according to the prior knowledge. The second line "throws" that die, obtaining a number. The third line eliminates the numbers that don't match the observation. And then we ask for the distribution of the die.

It is instructive to compare this function with the mathematical formula for Bayes' theorem, which is the basis of Bayesian inference. Bayes' theorem is P(H|E) = P(E|H) P(H) / P(E), where H stands for the hypothesis ("the die chosen was X") and E stands for the evidence ("the number thrown was N"). P(H) is the prior knowledge. The formula must be evaluated for a fixed value of E, which is the observation.

The first line of our domonad form implements P(H), the second line implements P(E|H). These two lines together thus sample P(E, H) using ancestral sampling, as we have seen before. The :when line represents the observation; we wish to apply Bayes' theorem for a fixed value of E. Once E has been fixed, P(E) is just a number, required for normalization. This is handled by normalize-cond in our code.

Let's see what happens when we add a single observation:

(add-observation prior 1)
-> {:twelve 2/9, :eight 1/3, :six 4/9}

We see that the highest probability is given to :six, then :eight, and finally :twelve. This happens because 1 is a possible value for all dice, but it is more probable as a result of throwing a six-faced die (1/6) than as a result of throwing an eight-faced die (1/8) or a twelve-faced die (1/12). The observation thus favours a die with a small number of faces.

If we have three observations, we can call add-observation repeatedly:

(-> prior (add-observation 1)
          (add-observation 3)
          (add-observation 7))
-> {:twelve 8/35, :eight 27/35}

Now we see that the candidate :six has disappeared. In fact, the observed value of 7 rules it out completely. Moreover, the observed numbers strongly favour :eight over :twelve, which is again due to the preference for the smallest possible die in the game.

This inference problem is very similar to how a spam filter works. In that case, the three dice are replaced by the choices :spam or :no-spam. For each of them, we have a distribution of words, obtained by analyzing large quantities of e-mail messages. The function add-observation is strictly the same, we'd just pick different variable names. And then we'd call it for each word in the message we wish to evaluate, starting from a prior distribution defined by the total number of :spam and :no-spam messages in our database.

To end this introduction to monad transformers, I will explain the m-zero problem in maybe-t. As you know, the maybe monad has an m-zero definition (nil) and an m-plus definition, and those two can be carried over into a monad created by applying maybe-t to some base monad. This is what we have seen in the case of cond-dist-m. However, the base monad might have its own m-zero and m-plus, as we have seen in the case of sequence-m. Which set of definitions should the combined monad have? Only the user of maybe-t can make that decision, so maybe-t has an optional parameter for this (see its documentation for the details). The only clear case is a base monad without m-zero and m-plus; in that case, nothing is lost if maybe-t imposes its own.

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