1. September 2003 1 SOME BASIC NOTIONS OF PROBABILITY THEORY Universita’ di Venezia 29 Settembre 2003

2. September 2003 2 What probability theory is for Suppose that we have a fair dice, with six faces, and that we keep throwing (‘casting’) it. For what proportion of the throws will we get – a particular value (say, 4)? – an even value? – a value greater than or equal to 3? PROBABILITY THEORY was developed to give us a vocabulary to talk about the LIKELIHOOD of certain EVENTS Where an EVENT is any result of a TRIAL (‘experiment’) – Getting the value 4 when casting a die, – Getting a value greater than 3, – But also: winning a race, getting a ‘tail’ result when flipping a coin, encountering a certain word

3. September 2003 3 EVENTS and OUTCOMES More precisely, we call every possible result of a trial an OUTCOME – E.g., any of the numbers on the die, such as 4, constitutes an outcome An EVENT is defined as a set of possible OUTCOMES (possibly just one): – E1 = {4} – E2 = {2,4,6} – E3 = {3,4,5,6}

4. September 2003 4 SAMPLE SPACES The SAMPLE SPACE is the set of all possible outcomes: – For the case of a dice, sample space S = {1,2,3,4,5,6} Another example: – Writing down a word is a TRIAL, – The word written down is an OUTCOME, – EVENTS which result from this trial are: writing that particular word, writing that word in uppercase letters, etc – The set of all possible spellings is the SAMPLE SPACE (NB: sometime the sample space is NOT finite)

5. September 2003 5 Probability Functions The likelihood of an event is indicated using a PROBABILITY FUNCTION The probability of an event E is specified by a function P(E), with values between 0 and 1 – P(E) = 1: the event is CERTAIN to occur – P(E) = 0: the event is certain NOT to occur Example: in the case of die casting, – P(E’ = ‘getting as a result a number between 1 and 6’) = P({1,2,3,4,5,6}) = 1 – P(E’’ = ‘getting as a result 7’) = 0

6. September 2003 6 Probabilities and relative frequencies In the case of a die, we know all of the possible outcomes ahead of time, and we also know a priori what the likelihood of a certain outcome is. But in many other situations in which we would like to estimate the likelihood of an event, this is not the case. For example, suppose that we would like to bet on horses rather than on dice. Harry is a race horse: we do not know ahead of time how likely it is for Harry to win. The best we can do is to ESTIMATE P(WIN) using the RELATIVE FREQUENCY of the outcome `Harry wins’ Suppose Harry raced 100 times, and won 20 races overall. Then – P(WIN) = WIN/TOTAL NUMBER OF RACES =.2 – P(LOSE) =.8 The use of probabilities we are interested in (estimate the probability of certain sequences of words) is of this type

7. September 2003 7 Conjunctions of events We are often interested in the probability of TWO events happening: – When throwing a die TWICE, the probability of getting a 6 both times – The probability of finding a sequence of two words: `the’ and `car’ We use the notation A&B to indicate the conjunction of two events, and P(A&B) to indicate the probability of such conjunction – Because events are SETS, the probability is often also written as We use the same notation with WORDS: P(‘the’ & ‘car’)

8. September 2003 8 Prior probability vs. conditional probability The prior probability P(WIN) is the likelihood of an event occurring irrespective of anything else we know about the world Often however we DO have additional information, that can help us making a more informed guess about the likelihood of a certain event E.g, take again the case of Harry the horse. Suppose we know that it was raining during 30 of the races that Harry raced, and that Harry won 15 of these races. Intuitively, the probability of Harry winning when it’s raining is.5 - HIGHER than the probability of Harry winning overall – We can make a more informed guess We indicate the probability of an event A happening given that we know that event B happened as well – the CONDITIONAL PROBABILITY of A given B – as P(A|B)

9. September 2003 9 Conditional probability Conditional probability is DEFINED as follows: Intuitively, you RESTRICT the range of trials in consideration to those in which event B took place, as well (most easily seen when thinking in terms of relative frequency)

10. September 2003 10 Example Consider the case of Harry the horse again: Where: – P(WIN&RAIN) = 15/100 =.15 – P(RAIN) = 30/100 =.30 This gives: (in agreement with our intuitions)

11. September 2003 11 The chain rule The definition of conditional probability can we rewritten as: – P(A&B) = P(A|B) P(B) – P(A&B) = P(B|A) P(A) These equation generalize to the so-called CHAIN RULE: – P(w 1,w 2,w 3,….w n ) = P(w 1 ) P(w 2 |w 1 ) P(w 3 |w 1,w 2 ) …. P(w n |w 1 …. w n-1 ) The chain rule plays an important role in statistical NLE: – P(the big dog) = P(the) P(big|the) P(dog|the big)

12. September 2003 12 Independence Additional information does not always help. For example, knowing the color of a dice usually doesn’t help us predicting the result of a throw; knowing the name of the jockey’s girlfriend doesn’t help predicting how well the horse he rides will do in a race; etc. When this is the case, we say that two events are INDEPENDENT The notion of independence is defined in probability theory using the definition of conditional probability Consider again the basic form of the chain rule: – P(A&B) = P(A|B) P(B) We say that two events are INDEPENDENT if: – P(A&B) = P(A) P(B) – P(A|B) = P(A)

13. September 2003 13 Bayes’ Theorem Bayes’ Theorem is a pretty trivial consequence of the definition of conditional probability, but it is very useful in that it allows us to use one conditional probability to compute another We already saw that the definition of conditional probability can be rewritten equivalently as: – P(A&B) = P(A|B) P(B) – P(A&B) = P(B|A) P(A) If we equate the two left sides, we get Bayes’ theorem

14. September 2003 14 Statistical NLE What’s the connection between this and natural language? A number of NL interpretation (and generation) tasks can be formulated in terms of CHOICE BETWEEN ALTERNATIVES: choosing the most likely – continuation of a certain sentence – POS tag or meaning for a word – Parse for a sentence In all of these cases, we can formalize `likelihood’ using probabilities, and choose the alternative with THE HIGHEST PROBABILITY Tomorrow we will see the first (and simplest) example of this: choosing the most likely next word This task can be viewed as the task of choosing the w that maximizes: P(w | W 1 …. W N-1 )

15. September 2003 15 Using corpora to estimate probabilities But where do we get these probabilities? Idea: estimate them by RELATIVE FREQUENCY. The simplest method: Maximum Likelihood Estimate (MLE). Count the number of words in a corpus, then count how many times a given sequence is encountered.

16. September 2003 16 Readings Krenn and Samuelsson, The Linguist’s Guide to Statistics (on the Web site) The Statistics GlossaryStatistics Glossary Further reading: Manning and Schuetze, chapter 2