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Wednesday, March 27, 2013

partition-by-equivalence


;; I keep finding that I need a function which will partition a sequence into runs of things.

;; For instance, you might want

'(1 1 1 2 3 3 4 4 4 3 3 3 5 5)

;; to go to:

'((1 1 1) (2) (3 3) (4 4 4) (3 3 3) (5 5))

;; Which is what partition-by does:

(partition-by identity '(1 1 1 2 3 3 4 4 4 3 3 3 5 5)) ;-> ((1 1 1) (2) (3 3) (4 4 4) (3 3 3) (5 5))

;; But partition-by isn't quite what I want.

;; I'd like to be able to turn

'( 1 2 3 4 1 2 3 2 3 4 6 9 10)

;; Into

'((1 2 3 4) (1 2 3) (2 3 4) (6) (9 10))

;; By defining a comparator like:

(defn ^:dynamic sameish [a b] ( = (inc a) b))

;; And then saying:
;; (partition-by-equivalence sameish '( 1 2 3 4 1 2 3 2 3 4 6 9 10))

;; Doing this by hand:

;; () () '( 1 2 3 4 1 2 3 2 3 4 6 9 10)
;; ->
;; () (1) (2 3 4 1 2 3 2 3 4 6 9 10)
;; ->
;; () (1 2) (3 4 1 2 3 2 3 4 6 9 10)
;; ->
;; () (1 2 3) (4 1 2 3 2 3 4 6 9 10)
;; ->
;; () (1 2 3 4) (1 2 3 2 3 4 6 9 10)
;; -> test is false, start a new list
;; ((1 2 3 4)) (1) (2 3 2 3 4 6 9 10)
;; ->
;; ((1 2 3 4)) (1 2) (3 2 3 4 6 9 10)
;; ->
;; ((1 2 3 4)) (1 2 3) (2 3 4 6 9 10)
;; -> test is false, start a new list
;; ((1 2 3 4) (1 2 3))  (2 3 4 6 9 10)
;; ->
;; ((1 2 3 4) (1 2 3) (2))  (3 4 6 9 10)
;; -> etc

;; makes me think that this looks like a tail recursion with two accumulators

(defn ^:dynamic recaccacc [ f acc1 acc2 coll]
  (if (empty? coll) (cons acc2 acc1)
      (if (empty? acc2) (recaccacc f acc1 (cons (first coll) acc2) (rest coll))
          (if (f (first acc2) (first coll))
            (recaccacc f acc1 (cons (first coll) acc2) (rest coll))
            (recaccacc f (cons acc2 acc1) '() coll)))))



;; Unfortunately, this comes out backwards
(use 'clojure.tools.trace)

(dotrace [recaccacc] (recaccacc  sameish '() '() '(1 2 3 4 1 2 3 2 3 4 6 9 10)))

;; TRACE t1169: (recaccacc #<user$sameish user$sameish@11b99c4> () () (1 2 3 4 1 2 3 2 3 4 6 9 10))
;; TRACE t1170: | (recaccacc #<user$sameish user$sameish@11b99c4> () (1) (2 3 4 1 2 3 2 3 4 6 9 10))
;; TRACE t1171: | | (recaccacc #<user$sameish user$sameish@11b99c4> () (2 1) (3 4 1 2 3 2 3 4 6 9 10))
;; TRACE t1172: | | | (recaccacc #<user$sameish user$sameish@11b99c4> () (3 2 1) (4 1 2 3 2 3 4 6 9 10))
;; TRACE t1173: | | | | (recaccacc #<user$sameish user$sameish@11b99c4> () (4 3 2 1) (1 2 3 2 3 4 6 9 10))
;; TRACE t1174: | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((4 3 2 1)) () (1 2 3 2 3 4 6 9 10))
;; TRACE t1175: | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((4 3 2 1)) (1) (2 3 2 3 4 6 9 10))
;; TRACE t1176: | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((4 3 2 1)) (2 1) (3 2 3 4 6 9 10))
;; TRACE t1177: | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((4 3 2 1)) (3 2 1) (2 3 4 6 9 10))
;; TRACE t1178: | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((3 2 1) (4 3 2 1)) () (2 3 4 6 9 10))
;; TRACE t1179: | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((3 2 1) (4 3 2 1)) (2) (3 4 6 9 10))
;; TRACE t1180: | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((3 2 1) (4 3 2 1)) (3 2) (4 6 9 10))
;; TRACE t1181: | | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((3 2 1) (4 3 2 1)) (4 3 2) (6 9 10))
;; TRACE t1182: | | | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((4 3 2) (3 2 1) (4 3 2 1)) () (6 9 10))
;; TRACE t1183: | | | | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((4 3 2) (3 2 1) (4 3 2 1)) (6) (9 10))
;; TRACE t1184: | | | | | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((6) (4 3 2) (3 2 1) (4 3 2 1)) () (9 10))
;; TRACE t1185: | | | | | | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((6) (4 3 2) (3 2 1) (4 3 2 1)) (9) (10))
;; TRACE t1186: | | | | | | | | | | | | | | | | | (recaccacc #<user$sameish user$sameish@11b99c4> ((6) (4 3 2) (3 2 1) (4 3 2 1)) (10 9) ())
;; TRACE t1186: | | | | | | | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1185: | | | | | | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1184: | | | | | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1183: | | | | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1182: | | | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1181: | | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1180: | | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1179: | | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1178: | | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1177: | | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1176: | | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1175: | | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1174: | | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1173: | | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1172: | | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1171: | | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1170: | => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))
;; TRACE t1169: => ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))



;-> ((10 9) (6) (4 3 2) (3 2 1) (4 3 2 1))

;; Which we can fix:

(reverse (map reverse (recaccacc sameish '() '() '(1 2 3 4 1 2 3 2 3 4 6 9 10))))
;-> ((1 2 3 4) (1 2 3) (2 3 4) (6) (9 10))

;; Hooray!

;; So our first definition is

(defn partition-by-equivalence [f coll]
  (let [recaccacc (fn [f acc1 acc2 coll]
                    (if (empty? coll) (reverse (cons (reverse acc2) acc1))
                        (if (empty? acc2) (recur f acc1 (cons (first coll) acc2) (rest coll))
                            (if (f (first acc2) (first coll))
                              (recur f acc1 (cons (first coll) acc2) (rest coll))
                              (recur f (cons (reverse acc2) acc1) '() coll)))))]
    (recaccacc f '() '() coll)))



(partition-by-equivalence sameish '(1 2 3 4 1 2 3 2 3 4 6 9 10)) ;-> ((1 2 3 4) (1 2 3) (2 3 4) (6) (9 10))

(partition-by-equivalence sameish '()) ;-> (())
(partition-by-equivalence sameish '(1)) ;-> ((1))
(partition-by-equivalence sameish '(1 1)) ;-> ((1) (1))
(partition-by-equivalence sameish '(1 2)) ;-> ((1 2))
(partition-by-equivalence sameish '(1 2 1)) ;-> ((1 2) (1))
(partition-by-equivalence sameish '(1 2 1 1)) ;-> ((1 2) (1) (1))
(partition-by-equivalence sameish '(1 2 1 1 2 2)) ;-> ((1 2) (1) (1 2) (2))

;; Here's some incomprehensible maths-stuff about numbers of digits and logarithms and so on.
(map count (partition-by (fn[a] (int (Math/log a))) (range 1 10000))) ; (2 5 13 34 94 255 693 1884 5123 1896)
(partition-by-equivalence (fn [a b] (= (int (Math/log a)) (int (Math/log b)))) (range 1 100)) ;-> ((1 2) (3 4 5 6 7) (8 9 10 11 12 13 14 15 16 17 18 19 20) (21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54) (55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99))
(partition-by-equivalence (fn [a b] (= (int (Math/log10 a)) (int (Math/log10 b)))) (range 1 100)) ;-> ((1 2 3 4 5 6 7 8 9) (10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99))

;; ascending subsequences
(partition-by-equivalence <= '(1 2 3 3 4 5 7 8 9 1 2 5 6 1 7 8))
;-> ((1 2 3 3 4 5 7 8 9) (1 2 5 6) (1 7 8))

;; strictly ascending subsequences
(partition-by-equivalence < '(1 2 3 3 4 5 7 8 9 1 2 5 6 1 7 8))
;-> ((1 2 3) (3 4 5 7 8 9) (1 2 5 6) (1 7 8))


;; lengths of increasing runs
(map count (partition-by-equivalence <= '(1 2 3 3 4 5 7 8 9 1 2 5 6 1 7 8)) ) ;-> (9 4 3)
;; lengths of decreasing ones
(map count (partition-by-equivalence >= '(1 2 3 3 4 5 7 8 9 1 2 5 6 1 7 8)) ) ;-> (1 1 2 1 1 1 1 2 1 1 2 1 1)

;; and finally, a simplified version of the latest problem I actually needed this for, pulling a sequence of lists of scores out of a log file
;; so that each full score list only appears once, and all its ancestors are discarded.
(map last (partition-by-equivalence (fn[a b] (= a (drop 1 b))) '( () (1) (2 1) (3 2 1) () (9) (7 9)))) ;-> ((3 2 1) (7 9))




;; It's a strict generalization of partition-by
(defn my-partition-by [f coll]
  (partition-by-equivalence (fn[a b] (= (f a) (f b))) coll))

(map #(/ (Math/log %) (Math/log 2)) (range 1 100))
(my-partition-by #(int(/ (Math/log %) (Math/log 2))) (range 1 100))


;; And I think it's a really nice function, which is helpful in all sort of situations.

;; It should be possible to make it completely lazy, so that it can take infinite inputs without wolfing the lot.

Rerum Cognoscere Causas IV : How to Deal with Larger Samples


;; Rerum Cognoscere Causas IV : How to Deal with Larger Samples.

;; So far we have:

;; Our data, generated at random from a secret algorithm:

(def ^:dynamic *randomizer* (java.util.Random. 0))

(defn rand-int [n] (.nextInt *randomizer* n))

(defn D6 [] (inc (rand-int 6)))

(defn three-D6 [] (reduce + (repeatedly 3 D6)))

(defn three-from-four-D6 [] (reduce + (drop 1 (sort (repeatedly 4 D6)))))

(defn mixed []
  (if (zero? (rand-int 10))
    (three-from-four-D6)
    (three-D6)))

(defn first-edition [] {:str (three-D6) :int (three-D6)})

(defn second-edition [] {:str (mixed) :int (mixed)})

(defn third-edition []
  (if (zero? (rand-int 10))
    {:str (three-from-four-D6) :int (three-from-four-D6)}
    {:str (three-D6) :int (three-D6)}))

(def village
  (binding [*randomizer* (java.util.Random. 0)]
    (doall (repeatedly 100 (case (rand-int 3)
                    0 first-edition
                    1 second-edition
                    2 third-edition)))))


;; And the calculations of our sages, who have determined with prodigious effort:
;; the sides on a six sided die
(def r16 (range 1 7))

;; the probabilities of each result for each suggested method of generating a characteristic
(def threed6f (frequencies (for [i r16 j r16 k r16] (reduce + [i j k]))))

(def fourd6drop1f (frequencies (for [i r16 j r16 k r16 l r16]
                     (reduce + (drop 1 (sort [i j k l]))))))

(defn p3d6 [char] (/ (threed6f char) (reduce + (vals threed6f))))

(defn p4d6drop1 [char] (/ (fourd6drop1f char) (reduce + (vals fourd6drop1f))))

(defn pmixed [char] (+ (* 9/10 (p3d6 char)) (* 1/10 (p4d6drop1 char))))

;; And thus the probabilities of a villager with particular characteristics coming into being under their scheme

(def ptrad (memoize (fn [{:keys [str int]}]  (* (p3d6 int) (p3d6 str)))))

(def pindep(memoize (fn [{:keys [str int]}] (* (pmixed int) (pmixed str)))))

(def pcommon (memoize (fn [{:keys [str int]}]
  (+
   (* 9/10 (p3d6 int) (p3d6 str))
   (* 1/10 (p4d6drop1 int) (p4d6drop1 str))))))


;; using these calculations, we can take beliefs
(def prior [1 1 1])

;; and update them when we find new evidence.
(defn update [beliefs villager] 
  (map * ((juxt ptrad pindep pcommon) villager) beliefs))

;; since ratios are a bit unreadable, this function will allow us to render them as (approximate) percentages.
(defn approx-odds [[a b c]]
  (let [m (/ (+ a b c) 100)]
    (mapv int [(/ a m) (/ b m) (/ c m)])))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;; Earlier we found that there's not really enough information in our village to convince anyone one way or the other.
(approx-odds (reduce update prior village)) ;-> [15 31 52]

;; It points half-heartedly towards a common cause model, but we wouldn't be surprised if that was the wrong answer.

;; As it happens, our village is just one of ten in the district
(def district
  (binding [*randomizer* (java.util.Random. 0)]
    (doall (repeatedly 1000 (case (rand-int 3)
                    0 first-edition
                    1 second-edition
                    2 third-edition)))))

(def villages (partition 100 district))

;; paranoid check
(= village (first villages)) ;-> true

;; So let's see what conclusions we can draw from each individual village

(for [v villages] 
  (approx-odds (reduce update prior v)))
;-> ([15 31 52] [53 29 17] [29 38 32] [63 21 14] [34 37 27] [36 27 36] [39 38 22] [49 30 19] [48 24 27] [20 37 42])

;; Some villages are pointing one way, and some the other. 

;; We might want to consider the district as a whole:
;; (approx-odds (reduce update prior district))

;; But unfortunately this expression takes a while to evaluate. Can you see why?

;; Here's a clue:
(reduce update [1.0 1.0 1.0] (take 100 district)) ;-> (1.0019121633199549E-224 2.0261656590064221E-224 3.398128024413593E-224)
;; And rather more worryingly
(reduce update [1.0 1.0 1.0] district) ;-> (0.0 0.0 0.0)

;; It looks as though mathematics itself is failing us!

;; We need to find a new way to multiply numbers.

;; Fortunately we can add their logarithms instead:

(+ (Math/log 6) (Math/log 6)) ;-> 3.58351893845611
(Math/log (* 6 6)) ;-> 3.58351893845611

;; So let's make a logarithmic version of our update function
(defn log-update [beliefs villager] 
  (map + (map #(Math/log %) ((juxt ptrad pindep pcommon) villager)) beliefs))

;; And of our prior:
(def log-prior (map #(Math/log %) prior))

;; And try that on the village
(reduce log-update log-prior village)
;; (-515.7771504932035 -515.0729156616442 -514.5558361317242)

;; If we know the logs, we can get the numbers we actually want:
(map #(Math/exp %) (reduce log-update log-prior village)) 
;-> (1.0019121633198053E-224 2.0261656590065323E-224 3.398128024414225E-224)
;; Notice that these numbers are not exactly the same as the numbers calculated above. 

;; Exercise for the reader: which of the two answers we can calculate is closer to the answer we would like to calculate but can't?

;; We can then turn those numbers into percentages
(approx-odds (map #(Math/exp %) (reduce log-update log-prior village))) 
;-> [15 31 52]

;; So we can do our calculation on the district as a whole:

(reduce log-update log-prior district)
;;-> (-4967.368738149676 -4968.862029975447 -4970.195021140233)

;; Hooray, now we have the logarithms of the numbers we really want!

;; That's done our probability calculation for us, but unfortunately we can't actually recover
;; the number that -4967.36... is the logarithm of:
(Math/exp -4967)
;; Because it's too small to fit into floating point arithmetic.

;; So we need to pull another rabbit out of our hat:

;; All we're interested in is the ratios of the three likelihoods.

;; That's not affected by multiplying through by a constant
(approx-odds [2 1 1]) ; [50 25 25] 
(approx-odds [6 3 3]) ; [50 25 25] 
(approx-odds [1332 666 666]) ; [50 25 25]

;; Similarly, if the likelihoods are expressed as logarithms, the odds ratio
;; isn't affected by adding a constant.
(approx-odds (map #(Math/exp %) [2 1 1])) ; [57 21 21]
(approx-odds (map #(Math/exp %) [3 2 2])) ; [57 21 21]
(approx-odds (map #(Math/exp %) [4 3 3])) ; [57 21 21]

;; What we'd like to be able to calculate is what this would be:
;; (approx-odds (map #(Math/exp %) '(-4967.368738149676 -4968.862029975447 -4970.195021140233)))

;; But we can't, since the Math/exp function is broken on numbers this low.


;; So we'll calculate instead (adding 4967 to each log-likelihood):
(approx-odds (map #(Math/exp %)
                  '(-0.368738149676 -1.862029975447 -3.195021140233))) 

;-> [77 17 4]


;; So after looking at a full thousand people, with perfect models and
;; a perfect prior, looking for what you would have thought was a
;; pretty obvious effect, the existence of gifted superbeings, we're
;; still in a bit of a dubious position.

;; If we decide 'that's good enough', and declare that we live in a
;; first e'dition world, then we've still got a fair chance of being
;; wrong.

;; We can generalize our procedure thus:

(defn log-update [beliefs villager] 
  (doall (map + (map #(Math/log %) ((juxt ptrad pindep pcommon) villager)) beliefs))) ; the doall is papering over a bug in clojure's lazy sequences

(defn log-posterior [log-prior data]
  (reduce log-update log-prior data))

(defn log-prior [& s]
  (map #(Math/log %) s))

(defn percentages-from-log-beliefs [beliefs]
  (let [bmax (apply max beliefs)
        odds (for [b beliefs] (Math/exp (- b bmax)))]
    (approx-odds odds)))

;; paranoid checking again:
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) village)) ;-> [15 31 52]
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) district)) ;-> [77 17 4]


;; And look at an even larger sample:
(def country
  (binding [*randomizer* (java.util.Random. 0)]
    (doall (repeatedly 10000 (case (rand-int 3)
                               0 first-edition
                               1 second-edition
                               2 third-edition)))))

(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) (take 100 country))) ; [15 31 52]
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) (take 300 country))) ; [27 39 32]
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) (take 1000 country))) ; [77 17 4]
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) (take 1500 country))) ; [96 3 0]

(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) (take 3000 country))) ; [99 0 0]

;; Finally, this looks decisive!
;; And everyone agrees. (Their prior beliefs are *overwhelmed* by the evidence)
(percentages-from-log-beliefs (log-posterior (log-prior 10 1 1) (take 3000 country))) ; [99 0 0]
(percentages-from-log-beliefs (log-posterior (log-prior 1 10 1) (take 3000 country))) ; [99 0 0]
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 10) (take 3000 country))) ; [99 0 0]

;; If we're completely paranoid:
(percentages-from-log-beliefs (log-posterior (log-prior 1 1 1) (take 10000 country))) ; [100 0 0]

;; That's close enough for government work.





;; So, in summary, we're looking at a situation where we understand perfectly what's going on, but can't directly see which of the three alternatives

;; (a) everyone's on a bell curve
;; (b) it's not quite a bell curve, it's a bit biased towards good scores
;; (c) there is a sub-race of gifted superbeings

;; was chosen by our Dungeon Master. We just have to look at characteristic scores.

;; And all our measurements are perfect, there is no noise in the data at all, and no systematic bias in our sampling.

;; And we are using, as far as I know, platonically ideal statistical
;; methods perfectly suited to the problem in hand, and guaranteed to
;; extract absolutely all the significance from our data that there
;; is.

;; And still, we need one thousand data points to get a feel for which way the wind is blowing (one hundred was actively misleading!)
;; And rather more than that to make people change their minds.

;; I would be most interested to know if anyone thinks I've done this analysis wrongly. 
;; Things I am not quite certain of are the randomness of the random number generator and the behaviour of the floating point numbers.
;; I'm reasonably confident of the basic method of comparing the three models, and so that means I'll be even more interested and grateful
;; if someone can show me that I'm talking rubbish.

;; If it's true, it makes me wonder how it is possible to know anything at all about anything interesting.

;; Next time I see a study claiming to show something, I might see if I can make this sort of analysis work on it.

Rerum Cognoscere Causas III : What can we tell from our small sample?


;; Rerum Cognoscere Causas III : What can we tell from our small sample?

;; So far we have:

;; Our data, generated at random from a secret algorithm:

(def ^:dynamic *randomizer* (java.util.Random. 0))

(defn rand-int [n] (.nextInt *randomizer* n))

(defn D6 [] (inc (rand-int 6)))

(defn three-D6 [] (reduce + (repeatedly 3 D6)))

(defn three-from-four-D6 [] (reduce + (drop 1 (sort (repeatedly 4 D6)))))

(defn mixed []
  (if (zero? (rand-int 10))
    (three-from-four-D6)
    (three-D6)))

(defn first-edition [] {:str (three-D6) :int (three-D6)})

(defn second-edition [] {:str (mixed) :int (mixed)})

(defn third-edition []
  (if (zero? (rand-int 10))
    {:str (three-from-four-D6) :int (three-from-four-D6)}
    {:str (three-D6) :int (three-D6)}))

(def village
  (binding [*randomizer* (java.util.Random. 0)]
    (doall (repeatedly 100 (case (rand-int 3)
                    0 first-edition
                    1 second-edition
                    2 third-edition)))))


village ;-> ({:str 13, :int 18} {:str 11, :int 18} {:str 14, :int 15} {:str 6, :int 12} {:str 14, :int 13} {:str 18, :int 10} {:str 15, :int 11} {:str 12, :int 15} {:str 7, :int 8} {:str 16, :int 12} {:str 8, :int 7} {:str 9, :int 14} {:str 10, :int 9} {:str 11, :int 10} {:str 5, :int 10} {:str 7, :int 9} {:str 9, :int 13} {:str 12, :int 9} {:str 13, :int 9} {:str 5, :int 9} {:str 8, :int 13} {:str 9, :int 11} {:str 13, :int 14} {:str 12, :int 14} {:str 12, :int 17} {:str 14, :int 9} {:str 10, :int 11} {:str 18, :int 17} {:str 11, :int 9} {:str 8, :int 9} {:str 15, :int 13} {:str 8, :int 5} {:str 11, :int 9} {:str 10, :int 8} {:str 9, :int 12} {:str 5, :int 11} {:str 10, :int 7} {:str 9, :int 14} {:str 11, :int 9} {:str 11, :int 12} {:str 12, :int 13} {:str 15, :int 9} {:str 12, :int 12} {:str 6, :int 13} {:str 5, :int 4} {:str 12, :int 13} {:str 15, :int 10} {:str 14, :int 14} {:str 11, :int 4} {:str 12, :int 9} {:str 10, :int 12} {:str 7, :int 12} {:str 8, :int 11} {:str 10, :int 10} {:str 9, :int 8} {:str 8, :int 12} {:str 7, :int 9} {:str 13, :int 3} {:str 14, :int 9} {:str 8, :int 9} {:str 10, :int 11} {:str 15, :int 4} {:str 10, :int 11} {:str 8, :int 10} {:str 15, :int 10} {:str 8, :int 13} {:str 12, :int 5} {:str 8, :int 16} {:str 4, :int 8} {:str 10, :int 18} {:str 12, :int 12} {:str 11, :int 10} {:str 12, :int 8} {:str 12, :int 13} {:str 8, :int 12} {:str 9, :int 12} {:str 12, :int 10} {:str 15, :int 10} {:str 8, :int 11} {:str 7, :int 11} {:str 4, :int 8} {:str 12, :int 11} {:str 13, :int 9} {:str 14, :int 13} {:str 5, :int 9} {:str 17, :int 10} {:str 8, :int 13} {:str 9, :int 10} {:str 5, :int 14} {:str 15, :int 12} {:str 13, :int 13} {:str 11, :int 8} {:str 8, :int 6} {:str 12, :int 8} {:str 10, :int 3} {:str 14, :int 9} {:str 15, :int 12} {:str 15, :int 14} {:str 6, :int 10} {:str 16, :int 13})

;; And the calculations of our sages, who have determined with prodigious effort:
;; the sides on a six sided die
(def r16 (range 1 7))

;; the probabilities of each result for each suggested method of generating a characteristic
(def threed6f (frequencies (for [i r16 j r16 k r16] (reduce + [i j k]))))

(def fourd6drop1f (frequencies (for [i r16 j r16 k r16 l r16]
                     (reduce + (drop 1 (sort [i j k l]))))))

(defn p3d6 [char] (/ (threed6f char) (reduce + (vals threed6f))))

(defn p4d6drop1 [char] (/ (fourd6drop1f char) (reduce + (vals fourd6drop1f))))

(defn pmixed [char] (+ (* 9/10 (p3d6 char)) (* 1/10 (p4d6drop1 char))))

;; And thus the probabilities of a villager with particular characteristics coming into being under their scheme

(def ptrad (memoize (fn [{:keys [str int]}]  (* (p3d6 int) (p3d6 str)))))

(def pindep(memoize (fn [{:keys [str int]}] (* (pmixed int) (pmixed str)))))

(def pcommon (memoize (fn [{:keys [str int]}]
  (+
   (* 9/10 (p3d6 int) (p3d6 str))
   (* 1/10 (p4d6drop1 int) (p4d6drop1 str))))))


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;; Very strong, very clever villagers are more likely under the common
;; cause model than under the independent mixed model and even less
;; likely under the traditional model:

(apply < ((juxt ptrad pindep pcommon) {:str 18 :int 18})) ;-> true

;; Let us imagine a neutral observer, employed to fairly determine the correctness of the three schools.

;; He starts off looking at each school's argument, and can see no way
;; to decide between the two, so he assigns them odds of 1:1:1 ,
;; meaning that he thinks each one is equally likely, or equivalently,
;; that he will take or place bets on any of the three schools at odds
;; of 2:1 against ( 1 gold piece gets you 2 if you picked right ),
;; (plus a small commission for bookkeeping and accepting risk.)

(def prior [1 1 1])

;; He now considers the first villager:

(def Olaf (first village))

;; Who is strong, and extremely clever:

Olaf ;-> {:str 13, :int 18}

;; And he considers the probability that Olaf would exist under each of the models.

(ptrad Olaf)   ;-> 7/15552
(pindep Olaf)  ;-> 653/1119744
(pcommon Olaf) ;-> 217/349920

;; Having observed some data, he now considers that he should re-weight his beliefs in the three models accordingly:

(map * ((juxt ptrad pindep pcommon) Olaf) prior) ;-> (7/15552 653/1119744 217/349920)

;; A ratio of odds can always be rescaled. 10:5 is the same as 2:1.

(map #(float (* 15552/7 %)) (map * ((juxt ptrad pindep pcommon) Olaf) prior))
;-> (1.0 1.2956349 1.3777778)

;; So here's a function to take any odds ratio and turn it into (approximate) percentages
(defn approx-odds [[a b c]]
  (let [m (/ (+ a b c) 100)]
    (mapv int [(/ a m) (/ b m) (/ c m)])))

(approx-odds (map * ((juxt ptrad pindep pcommon) Olaf) prior)) ;-> [27 35 37]

;; And we can now see that the arbitrator's judgement has shifted a little away from the traditionalists, and towards the second and third e'ditions.

;; This is reasonable, given that he has just observed a man who is both cleverer and stronger than would be expected by the traditionalists

;; Suppose he had seen Magnus instead:
(def Magnus (second village))

Magnus ;-> {:str 11, :int 18}

;; Magnus is average in strength, but very clever. This is probably more likely under the independent rules than it is under the common cause rules.

(approx-odds (map * ((juxt ptrad pindep pcommon) Magnus) prior)) ;-> [28 35 35]

;; Seeing Magnus should change a neutral person's beliefs towards the second e'dition, mostly at the expense of the first.

;; But of course, our observer has seen both:

(approx-odds
 (map * ((juxt ptrad pindep pcommon) Magnus)
      (map * ((juxt ptrad pindep pcommon) Olaf)
           prior))) ;-> [23 37 39]

;; Implying that the combined effect of both men is to discredit the traditional school while slightly favouring the common-cause hypothesis.

;; We could generate the series representing how the assessors beliefs should change as he 
;; considers each villager like this:

(reductions 
 (fn [beliefs villager] (map * ((juxt ptrad pindep pcommon) villager) beliefs))
 prior
 (list Magnus Olaf))
;-> ([1 1 1] (1/1728 803/1119744 247/349920) (7/26873856 524359/1253826625536 53599/122444006400))

;; More readably, we can separate the function which updates our beliefs given a datum.
(defn update [beliefs villager] 
  (map * ((juxt ptrad pindep pcommon) villager) beliefs))

(map approx-odds
 (reductions update prior (list Magnus Olaf)))
;-> ([33 33 33] [28 35 35] [23 37 39])

;; What if we look at the first ten villagers?
(map approx-odds
 (reductions update prior (take 10 village)))
;-> ([33 33 33] [27 35 37] [23 37 39] [19 37 42] [20 37 41] [18 38 43] [16 40 43] [14 41 44] [13 41 45] [14 40 45] [12 40 46])

;; The first twenty:
(approx-odds
 (reduce update prior (take 20 village)))
;-> [19 36 44]

;; And at the whole village?
(approx-odds
 (reduce update prior (take 100 village)))
;;-> [15 31 52]


;; So if we look at our whole village, it looks as though we'd be
;; slightly more confident that we lived in a third e'dition world.

;; But we're really not terribly confident about that.
;; We know that the models and priors are spot on, since
;; we've seen the source code for the world.

;; But even then, if we declared on the basis of this one village that
;; we lived in a third edition world, we'd literally expect to be
;; wrong half the time.  

;; That's only a slight improvement on being wrong two thirds of the
;; time, which we'd expect if we hadn't bothered to look at any data at
;; all. Most of our opinion is coming from our prior beliefs and there
;; is really very little evidence in the data that we have.

;; Another way to look at this is to ask what happens to the beliefs of the three competing philosophical schools.

;; Suppose each of the three is initially so convinced of its own
;; correctness that it will place or lay bets at 5:1 on its pet
;; hypothesis. But that each school will adjust its beliefs as it sees evidence.

;; So the first e'dition guys will suffer a serious loss of confidence
(approx-odds
 (reduce update [10 1 1] (take 100 village))) ; [64 13 22]
;; But not actually change their minds, just the strength of their convictions.

;; The second e'dition guys barely notice.
(approx-odds
 (reduce update [1 10 1] (take 100 village))) ; [4 82 13]

;; While the third e'dition find their beliefs slightly strengthened
;; after looking at the evidence.
(approx-odds
 (reduce update [1 1 10] (take 100 village))) ; [2 5 91]

;; Clearly more research is needed!

Tuesday, March 5, 2013

Rerum Cognoscere Causas II : In Which Divers Probabilities are Deduced Simply by Counting


;; Rerum Cognoscere Causas II : In Which Divers Probabilities are Deduced Simply by Counting

;; The problem: Which e'dition's rules were used to generate a village?

(def ^:dynamic *randomizer* (java.util.Random. 0))

(defn rand-int [n]
  (.nextInt *randomizer* n))

(defn D6 [] (inc (rand-int 6)))

(defn three-D6 []
  (reduce + (for [i (range 3)] (D6))))

(defn three-from-four-D6 []
  (reduce + (drop 1 (sort (for [i (range 4)] (D6))))))

(defn mixed []
  (if (zero? (rand-int 10))
    (three-from-four-D6)
    (three-D6)))

(defn first-edition [] {:str (three-D6) :int (three-D6)})

(defn second-edition [] {:str (mixed) :int (mixed)})

(defn third-edition []
  (if (zero? (rand-int 10))
    {:str (three-from-four-D6) :int (three-from-four-D6)}
    {:str (three-D6) :int (three-D6)}))

(def village
  (binding [*randomizer* (java.util.Random. 0)]
    (doall (repeatedly 100 (case (rand-int 3)
                    0 first-edition 
                    1 second-edition
                    2 third-edition)))))


village ;-> ({:str 13, :int 18} {:str 11, :int 18} {:str 14, :int 15} {:str 6, :int 12} {:str 14, :int 13} {:str 18, :int 10} {:str 15, :int 11} {:str 12, :int 15} {:str 7, :int 8} {:str 16, :int 12} {:str 8, :int 7} {:str 9, :int 14} {:str 10, :int 9} {:str 11, :int 10} {:str 5, :int 10} {:str 7, :int 9} {:str 9, :int 13} {:str 12, :int 9} {:str 13, :int 9} {:str 5, :int 9} {:str 8, :int 13} {:str 9, :int 11} {:str 13, :int 14} {:str 12, :int 14} {:str 12, :int 17} {:str 14, :int 9} {:str 10, :int 11} {:str 18, :int 17} {:str 11, :int 9} {:str 8, :int 9} {:str 15, :int 13} {:str 8, :int 5} {:str 11, :int 9} {:str 10, :int 8} {:str 9, :int 12} {:str 5, :int 11} {:str 10, :int 7} {:str 9, :int 14} {:str 11, :int 9} {:str 11, :int 12} {:str 12, :int 13} {:str 15, :int 9} {:str 12, :int 12} {:str 6, :int 13} {:str 5, :int 4} {:str 12, :int 13} {:str 15, :int 10} {:str 14, :int 14} {:str 11, :int 4} {:str 12, :int 9} {:str 10, :int 12} {:str 7, :int 12} {:str 8, :int 11} {:str 10, :int 10} {:str 9, :int 8} {:str 8, :int 12} {:str 7, :int 9} {:str 13, :int 3} {:str 14, :int 9} {:str 8, :int 9} {:str 10, :int 11} {:str 15, :int 4} {:str 10, :int 11} {:str 8, :int 10} {:str 15, :int 10} {:str 8, :int 13} {:str 12, :int 5} {:str 8, :int 16} {:str 4, :int 8} {:str 10, :int 18} {:str 12, :int 12} {:str 11, :int 10} {:str 12, :int 8} {:str 12, :int 13} {:str 8, :int 12} {:str 9, :int 12} {:str 12, :int 10} {:str 15, :int 10} {:str 8, :int 11} {:str 7, :int 11} {:str 4, :int 8} {:str 12, :int 11} {:str 13, :int 9} {:str 14, :int 13} {:str 5, :int 9} {:str 17, :int 10} {:str 8, :int 13} {:str 9, :int 10} {:str 5, :int 14} {:str 15, :int 12} {:str 13, :int 13} {:str 11, :int 8} {:str 8, :int 6} {:str 12, :int 8} {:str 10, :int 3} {:str 14, :int 9} {:str 15, :int 12} {:str 15, :int 14} {:str 6, :int 10} {:str 16, :int 13})

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;; The first thing that our philosophers should look at is the distribution of scores according to their models.

;; What do the results of rolling 3D6 and adding look like? We'll just
;; enumerate the 216 possible ways the dice can fall and count how
;; many ways there are to score the various possible numbers:

(def threed6f (frequencies 
           (for [i (range 1 7) j (range 1 7) k (range 1 7)] 
             (reduce + [i j k]))))

;; {3 1, 4 3, 5 6, 6 10, 7 15, 8 21, 9 25, 10 27, 11 27, 12 25, 13 21, 14 15, 15 10, 16 6, 17 3, 18 1}

;; What about 4D6 and discarding the lowest?
(def fourd6drop1f (frequencies 
                   (for [i (range 1 7) j (range 1 7) k (range 1 7) l (range 1 7)] 
                     (reduce + (drop 1 (sort [i j k l])))))) 

;; {3 1, 4 4, 5 10, 6 21, 7 38, 8 62, 9 91, 10 122, 11 148, 12 167, 13 172, 14 160, 15 131, 16 94, 17 54, 18 21}

;; So the probability distributions for the dice rolling methods are: 

(defn p3d6 [char] (/ (threed6f char) (reduce + (vals threed6f))))

(defn p4d6drop1 [char] (/ (fourd6drop1f char) (reduce + (vals fourd6drop1f))))

(defn pmixed [char] (+ (* 9/10 (p3d6 char)) (* 1/10 (p4d6drop1 char))))

;; Sanity checks!

;; As every schoolboy once knew:

;; You get 3 sixes only one time in ~200, so 18s are very rare
(p3d6 18) ;-> 1/216
(p3d6 17) ;-> 1/72
;; In fact only one roll in ten is in the 15-18 range:
(reduce + (map p3d6 '(18 17 16 15)))

;; But you see 10 or 11 very often, in fact about a quarter of the time.
(p3d6 10) ;-> 1/8
(p3d6 11) ;-> 1/8

;; However if you get to roll four dice and ignore one of them, 
;; then the high scores are more likely.

(p4d6drop1 18) ;-> 7/432
(p4d6drop1 17) ;-> 1/24
(reduce + (map p4d6drop1 '(18 17 16 15))) ;-> 25/108
;; And in fact you have a quarter chance of an exceptional score 15-18.

;; 10 and 11 are slightly less likely than they were, and 11 is more likely than 10
(p4d6drop1 11) ;-> 37/324
(p4d6drop1 10) ;-> 61/648

;; Truly bad scores are very hard to get
(p4d6drop1 3) ;-> 1/1296
;; In fact only one in six scores are 'below average'.
(reduce + (map p4d6drop1 '(3 4 5 6 7 8 9))) ;-> 227/1296


;; However in the mixed distribution where you might be doing it one
;; way and you might be doing it the other, these differences are not
;; nearly as large:
(pmixed 18) ;-> 5/864
(pmixed 3) ;-> 11/2592

; A subtle bias indeed. 
(/ (pmixed 18) (pmixed 3)) ;-> 15/11

;; Just as a sanity check, check that our three distributions add up to 1.
(reduce + (map p4d6drop1 (range 3 19))) ;-> 1N
(reduce + (map p3d6 (range 3 19))) ;-> 1N
(reduce + (map pmixed (range 3 19))) ;-> 1N


;; Our three schools, with their different beliefs about how characteristics are generated, can now write down the chances
;; of combinations of characteristics:

;; The traditionalists of the first e'dition say that your chance of
;; getting strength 18, intelligence 18 is simply the chance of
;; getting both scores independently.

(defn ptrad [{:keys [str int]}]
  (* (p3d6 int) (p3d6 str)))

;; Being truly gifted is terribly unlikely:
(ptrad {:str 18 :int 18}) ;-> 1/46656 
;; As is being truly disadvantaged:
(ptrad {:str 3  :int  3}) ;-> 1/46656

;; But then, only 1 person in 16 is truly average:
(reduce + (map ptrad (for [s '(10 11) i '(10 11)] {:str s :int i}))) ;-> 1/16

;; Being a weedy genius is just as likely as being a genius who can bend swords.
(ptrad {:str 3 :int 18}) ;-> 1/46656

;; The second e'dition guys still model the two scores as independent things, 
(defn pindep [{:keys [str int]}]
  (* (pmixed int) (pmixed str)))

;; But they predict more beefy einsteins
(pindep {:str 18 :int 18}) ;-> 25/746496

;; And fewer weedy vegetables
(pindep {:str 3 :int 3}) ;-> 121/6718464

;; One good, one bad 
(pindep {:str 3 :int 18}) ;-> 55/2239488
;; Has about the same frequency as in the traditionalists' model
(apply / ((juxt pindep ptrad) {:str 3 :int 18})) ;-> 55/48

;; The third e'dition guys think that you use either the traditional system for both characteristics,
;; or you use the enhanced system for both. They say that there is a common cause, being a 'player character'.
;; And they say that one in ten people are like that.

(defn pcommon [{:keys [str int]}]
  (+ 
   (* 9/10 (p3d6 int) (p3d6 str))
   (* 1/10 (p4d6drop1 int) (p4d6drop1 str))))

;; If we just look at the frequencies of a characteristic in
;; isolation, then we can't tell the difference between the second and
;; third schools. 
;; They make the same guesses about the number of very strong people:

;; The chance of having STR 18 is the chance of having
;; STR 18 and any intelligence
(reduce + (map pcommon (for [i (range 3 19)] {:str 18 :int i}))) ;-> 5/864
(reduce + (map pindep  (for [i (range 3 19)] {:str 18 :int i})))  ;-> 5/864
(reduce + (map ptrad   (for [i (range 3 19)] {:str 18 :int i})))   ;-> 1/216

;; It's only by looking at both scores together that we can see
;; differences between what the world will look like if there's a
;; common cause and what it will look like if there isn't:

;; Both later e'ditions predict more supermen, but the common causers predict even more than the independents:
(map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 18 :int 18})) ;-> (45 21 33)

;; And they both predict slightly fewer basket cases, but the common causers expect less of a drop.
(map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 3 :int 3}))   ;-> (19 21 18)

;; The traditionalists predict slightly more cases of people who are good at only one thing
(map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 18 :int 3}))  ;-> (20 21 24)
(map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 3 :int 18}))  ;-> (20 21 24)

;; Again, we ought to sanity check our distributions:
(reduce + (map pcommon (for [i (range 3 19) j (range 3 19)] {:int i :str j}))) ;-> 1N
(reduce + (map ptrad   (for [i (range 3 19) j (range 3 19)] {:int i :str j}))) ;-> 1N
(reduce + (map pindep  (for [i (range 3 19) j (range 3 19)] {:int i :str j}))) ;-> 1N

;; Premature optimization is the root of all evil, and there is no
;; place to optimize more premature than an introductory tutorial
;; essay, but I am worried about my transistors wearing out if I have
;; to use these functions a lot.

(def pcommon (memoize pcommon))
(def pmixed  (memoize pmixed))
(def ptrad   (memoize ptrad))

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