Three Concepts: Probability Henry Tirri, Petri Myllymäki

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1 6..6 robability as a masur o bli Thr Conpts: robability Hnry Tirri, tri Myllymäki

2 robabilitis ar to b intrprtd Ditionary dinition: probability han liklihood probability? Thr Conpts: robability Hnry Tirri, tri Myllymäki

3 Numrial masurs o bli Bli in a proposition,, an b masurd in trms o a numbr btwn dinitly als and dinitly tru this is th probability o has a probability btwn and, dosn t man is tru to som dgr, but mans that you ar ignorant o its truth valu. robability is a masur o your ignoran Thr Conpts: robability Hnry Tirri, tri Myllymäki

4 Random Variabls A random variabl is a trm in a languag that an tak on o a numbr o dirnt valus domx, th domain o a variabl, is th st o valus x an tak a tupl o random variabls x,, x variabl with domain domx... domx a proposition is a Boolan ormula mad rom assignmnts o valus to variabls > is a omplx random Thr Conpts: robability Hnry Tirri, tri Myllymäki

5 ossibl world smantis A possibl world spiis an assignmnt o on valu to ah random variabl w x mans variabl x is assignd valu v in world w v logial onntivs hav thir standard maning w α β i w α β i w α or w β w α i w α w α and w β Thr Conpts: robability Hnry Tirri, tri Myllymäki

6 Smantis o probability For a init numbr o variabls with init domains: µ!! " µ ω w Thr Conpts: robability Hnry Tirri, tri Myllymäki

7 Axioms o probability Axiom. g i g is a tautology. That is, logially quivalnt ormula hav th sam probability. Axiom. or any ormula. Axiom 3. τ i τ is a tautology. Axiom 4. g + g i g is a tautology. Ths axioms ar sound and omplt with rspt to th smantis Thr Conpts: robability Hnry Tirri, tri Myllymäki

8 Conditioning Spiis how to rvis blis basd on nw inormation Building a probabilisti modl starts by taking all bakground inormation into aount. This givs th prior probability. All othr inormation must b onditiond on. I vidn is all th ino obtaind subsquntly, th onditional probability h o h givn is th postrior probability o h Thr Conpts: robability Hnry Tirri, tri Myllymäki

9 Thr Conpts: robability Hnry Tirri, tri Myllymäki Smantis o onditional probability Evidn ruls out possibl worlds inompatibl with. Evidn indus a nw masur, µ #, ovr possibl worlds i i ω ω ω µ µ Th onditional probability o ormula h givn vidn is h h w h ω µ

10 Thr Conpts: robability Hnry Tirri, tri Myllymäki roprtis o onditional probabilitis Chain rul n i i i n n n

11 Law o total probability us a wightd avrag o onditional probabilitis to alulat a probability ropositions W W, W W ~ disjointnss W W + W ~ W W W ~ W ~ ~ W W + W ~ ~ Thr Conpts: robability Hnry Tirri, tri Myllymäki

12 & 5 A , >.3 & 4 83 G 4 83 & : 4 - D & > 83 - ; L 3 8: & 5 C & : - ; L 3 V U $.3.3 Bating lassiirs ; 5 : 4 5 * ' $ /. EI - H F B ; 5 : F 5 A & 5 E - D BC 8 H 5 ; 5 : A NO, C & M 5 83 L 4 L 5 3 & 8 83 K > 8 4 $ 3 NO Q ; & - H F 5 L. E - H F B S M. A & B 5 NR O L 8 - D 4 4 L 5 M $ E NR T L 8 X 8 > : K 8& & M L 8 M 5 ; 3 W 8 ; Thr Conpts: robability Hnry Tirri, tri Myllymäki

13 Bating lassiirs ropositions B B B bating 4 4, ~ By th law o, B total against DTC, ~ probability.35 ~ against NBC, Thr Conpts: robability Hnry Tirri, tri Myllymäki

14 Bays Rul lt us rvrs th prvious situation: assuming sam probability assignmnts, you tll m that YFC outprormd th omptitor. Whih lassiir DTC or NBC did you ompt with? Thr Conpts: robability Hnry Tirri, tri Myllymäki

15 Thr Conpts: robability Hnry Tirri, tri Myllymäki Bays Rul probability w hav total using th law o w hav rom hain rul W W W W W W ~ ~ W W W W +

16 In th xampl... ropositions B B bating B By Bays 4, ~, B Rul against DTC, ~ ~ 4 against NBC, 3 Thr Conpts: robability Hnry Tirri, tri Myllymäki 998-6

17 Z h } y p { kl z ` d k ko n [ gd Y p d k h ` j o [k k k^ k ` ko o [k k o k ^ o gd Y Z h { ^ o Z h p ^ o ` o Z h ^ o m ` k l Sid not Not that B + B ~ is not nssarily, but B + ~B is! Why? [ g ^ d `ba ` _^ Z\[ Y m` o y wx tv tu s ko rq d n j ^ nn o ` l k^ j m d [ k^ ij m s v~ `\[ o d n j ^ nno ` l k^ j m d r ~ _ƒ [ g ^ `\ d [ dg [ i i }` l ` [ m`\[ [ i i }` l [k ` }` [k k^ ij o[k ` p ^ }` o i i }` l } n l np ` m [ o Mnmoni rul o thumb: y ƒ v w t u [ ` a ` o n m y t } ` } o [ m d Š ˆ m ka g ^ o k^ ij m` [ [ j d [ d n m y tv } ` } o [ m d \Ž Œ m ka g ^ o k^ ij y t u n m y tv u p j } iv o n ^ da m ka g ^ o k^ ij Thr Conpts: robability Hnry Tirri, tri Myllymäki 998-6

18 Marginalization A B C A,B,C A? A,B? B A?, X X, X,5 X X,5 X,3,5,,5,,5,5,5/,5,3 Thr Conpts: robability Hnry Tirri, tri Myllymäki

19 robabilisti inrn marginalization H: somthing you do not know and want to know U: vrything you do not know and do not nd to know nuisan I: bakground knowldg D: obsrvd data H D, I U H D, I, U U D, I Thr Conpts: robability Hnry Tirri, tri Myllymäki

20 Anothr vrsion o BR W W W + W ~ ~ divid by th numrator W W ~ + W ~ I w xprss ~ W with this xprssion, w gt ~ W W ~ ~ W W Thr Conpts: robability Hnry Tirri, tri Myllymäki

21 Bays Rul in Trms o Odds Bays ator in avor o liklihood W ~ W W W ~ ~ ostrior odds in avor o rior odds in avor o Thr Conpts: robability Hnry Tirri, tri Myllymäki

22 Exampl: Whih di? two di: 4-sidr and -sidr ah sid qually likly or ah di F pik 4-sidr, T~F pik -sidr or you F~F/ I roll th di pikd. Th rsult is 3. Whih di did I pik? Thr Conpts: robability Hnry Tirri, tri Myllymäki 998-6

23 Exampl: Whih di? liklihoods Bays' s F 3 rul Th Bays ar 3 says F 3 T 3 F 3 F F F F + 3 T T 4 Thr Conpts: robability Hnry Tirri, tri Myllymäki ator in avor o F and T

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th