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Probability theory

Probability theory. LING 570 Fei Xia Week 2: 10/01/07. Misc. Patas account and dropbox Course website, “Collect it”, and GoPost. Mailing list Received message on Thursday? Questions about hw1?. Outline. Quiz #1 Unix commands Linguistics Elementary Probability theory: M&S 2.1.

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Probability theory

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  1. Probability theory LING 570 Fei Xia Week 2: 10/01/07

  2. Misc. • Patas account and dropbox • Course website, “Collect it”, and GoPost. • Mailing list • Received message on Thursday? • Questions about hw1?

  3. Outline • Quiz #1 • Unix commands • Linguistics • Elementary Probability theory: M&S 2.1

  4. Quiz #1 Five areas: weight ave • Programming: 4.0 (3.74) • Try Perl or Python • Unix commands: 1.2 (0.99) • Probability: 2.0 (1.09) • Regular expression: 2.0 (1.62) • Linguistics knowledge: 0.8 (0.71)

  5. Results • 9.0-10: 4 • 8.0-8.9: 8 • < 8.0: 8

  6. Unix commands • ls (list), cp (copy), rm (remove) • more, less, cat • cd, mkdir, rmdir, pwd • chmod: to change file permission • tar, gzip: to tar/zip files • ssh, sftp: to log on or ftp files • man: to learn a command

  7. Unix commands (cont) • compilers: javac, gcc, g++, perl, … • ps, top, • which • Pipe: cat input_file | eng_tokenizer.sh | make_voc.sh > output_file • sort, unique, awk, grep grep “the” voc | awk ‘{print $2}’ | sort | uniq –c | sort -nr

  8. Examples • Set the permission of foo.pl so it is readable and executable by the user and the group. rwx rwx rwx => 101 101 000 chmod 550 foo.pl • Move a file, foo.pl, from your home dir to /tmp mv ~/foo.pl /tmp

  9. Linguistics: POS tags • Open class: Noun, verb, adjective, adverb • Auxiliary verb/modal: can, will, might, .. • Temporal noun: tomorrow • Adverb: adj+ly, always, still, not, … • Closed class: Preposition, conjunction, determiner, pron, • Conjunction: CC (and), SC (if, although) • Complementizer: that,

  10. Linguistics: syntactic structure • Two kinds: • Phrase structure (a.k.a. parse tree): • Dependency structure • Examples: • John said that he would call Mary tomorrow

  11. Outline • Quiz #1 • Unix commands • Linguistics • Elementary Probability theory

  12. Probability Theory

  13. Basic concepts • Sample space, event, event space • Random variable and random vector • Conditional probability, joint probability, marginal probability (prior)

  14. Sample space, event, event space • Sample space (Ω): the set of all possible outcomes. • Ex: toss a coin three times: {HHH, HHT, HTH, HTT, …} • Event: an event is a subset of Ω. • Ex: an event is {HHT, HTH, THH} • Event space (2Ω): the set of all possible events.

  15. Probability function • A probability function (a.k.a. a probability distribution) distributes a probability mass of 1 throughout the sample space . • It is a function from 2! [0,1] such that: P() = 1 For any disjoint sets Aj2 2, P( Aj) =  P(Aj) - Ex: P({HHT, HTH, HTT}) = P({HHT}) + P({HTH}) + P({HTT})

  16. The coin example • The prob of getting a head is 0.1 for one toss. What is the prob of getting two heads out of three tosses? • P(“Getting two heads”) = P({HHT, HTH, THH}) = P(HHT) + P(HTH) + P(THH) = 0.1*0.1*0.9 + 0.1*0.9*0.1+0.9*0.1*0.1 = 3*0.1*0.1*0.9

  17. Random variable • The outcome of an experiment need not be a number. • We often want to represent outcomes as numbers. • A random variable X is a function: ΩR. • Ex: the number of heads with three tosses: X(HHT)=2, X(HTH)=2, X(HTT)=1, …

  18. The coin example (cont) • X = the number of heads with three tosses • P(X=2) = P({HHT, HTH, THH}) = P({HHT}) + P({HTH}) + P({THH})

  19. Two types of random variables • Discrete: X takes on only a countable number of possible values. • Ex: Toss a coin three times. X is the number of heads that are noted. • Continuous: X takes on an uncountable number of possible values. • Ex: X is the speed of a car

  20. Common trick #1: Maximum likelihood estimation • An example: toss a coin 3 times, and got two heads. What is the probability of getting a head with one toss? • Maximum likelihood: (ML) * = arg max P(data | ) • In the example, • P(X=2) = 3 * p * p * (1-p) e.g., the prob is 3/8 when p=1/2, and is 12/27 when p=2/3 3/8 < 12/27

  21. Random vector • Random vector is a finite-dimensional vector of random variables: X=[X1,…,Xk]. • P(x) = P(x1,x2,…,xn)=P(X1=x1,…., Xn=xn) • Ex: P(w1, …, wn, t1, …, tn)

  22. Notation • X, Y, Xi, Yi are random variables. • x, y, xi are values. • P(X=x) is written as P(x) • P(X=x | Y=y) is written as P(x | y).

  23. Three types of probability • Joint prob: P(x,y)= prob of X=x and Y=y happening together • Conditional prob: P(x | y) = prob of X=x given a specific value of Y=y • Marginal prob: P(x) = prob of X=x for all possible values of Y.

  24. An example • There are two coins. Choose a coin and then toss it. Do that 10 times. • Coin 1 is chosen 4 times: one head and three tails. • Coin 2 is chosen six times: four heads and two tails. • Let’s calculate the probabilities.

  25. Probabilities • P(C=1) = 4/10, P(C=2) = 6/10 • P(X=h) = 5/10, P(X=t) = 5/10 • P(X=h | C=1) = ¼, P(X=h |C=2) =4/6 • P(X=t | C=1) = ¾, P(X=t |C=2) = 2/6 • P(X=h, C=1) =1/10, P(X=h, C=2)= 4/10 • P(X=t, C=1) = 3/10, P(X=t | C=2) = 2/10

  26. Relation between different types of probabilities P(X=h, C=1) = P(C=1) * P(X=h | C=1) = 4/10 * ¼ = 1/10 P(X=h) = P(X=h, C=1) + P(X=h, C=2) = 1/10 + 4/10 = 5/10

  27. Common trick #2:Chain rule

  28. Common trick #3:joint probMarginal prob

  29. Common trick #4:Bayes’ rule

  30. Independent random variables • Two random variables X and Y are independent iff the value of X has no influence on the value of Y and vice versa. • P(X,Y) = P(X) P(Y) • P(Y|X) = P(Y) • P(X|Y) = P(X) • Our previous examples: P(X, C) != P(X) P(C)

  31. Conditional independence Once we know C, the value of A does not affect the value of B and vice versa. • P(A,B | C) = P(A|C) P(B|C) • P(A|B,C) = P(A | C) • P(B|A, C) = P(B |C)

  32. Independence and conditional independence • If A and B are independent, are they conditional independent? • Example: • Burglar, Earthquake • Alarm

  33. Common trick #5:Independence assumption

  34. An example • P(w1 w2 … wn) = P(w1) P(w2 | w1) P(w3 | w1 w2) * … * P(wn | w1 …, wn-1) ¼ P(w1) P(w2 | w1) …. P(wn | wn-1) • Why do we make independence assumption which we know are not true?

  35. Summary of elementaryprobability theory • Basic concepts: sample space, event space, random variable, random vector • Joint / conditional /marginal probability • Independence and conditional independence • Five common tricks: • Max likelihood estimation • Chain rule • Calculating marginal probability from joint probability • Bayes’ rule • Independence assumption

  36. Outline • Quiz #1 • Unix commands • Linguistics • Elementary Probability theory

  37. Next time • J&M Chapt 2 • Formal language and formal grammar • Regular expression • Hw1 is due at 3pm on Wed.

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