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Wednesday, March 27, 2013

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!

3 comments:

  1. Hey John,

    No offense, I love your blog, but the clicksor ads make it impossible to read without a popup blocker. Is that intentional? Just wanted to let you know in case you want to check it out and do something about it.

    Otherwise, this is my favorite Clojure blog, keep it up!

    Cheers,
    Craig

    ReplyDelete
    Replies
    1. Craig, I am appalled to find that there is advertising on this blog at all, let alone something that makes it hard to read. I haven't put any on myself, and I can't see them in any of my own browsers. Are google sneaking something in or has something been hacked?

      There should be no ads on this blog at all. Any that appear are being introduced without my consent, and I'd appreciate knowing about them so I can either get rid of them or move this blog to a platform that doesn't do this evil thing.

      Thanks for letting me know, and any details you can give me will be helpful.

      Delete
    2. It appears that some stupid social media gadget 'Sociable' was introducing (fairly discreet) advertising. I've removed it. Has that killed them all?

      Delete

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