;; 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.

## Wednesday, March 27, 2013

### Rerum Cognoscere Causas IV : How to Deal with Larger Samples

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