Using Probabilistic Information - PowerPoint PPT Presentation

Using Probabilistic Information

Read J & M Chapter 5, pages 141 - 156

• Noisy channel:

I haf to go. geneology

• Sentences are flat. Knowledge is structured.

Joe hit the ball with the bat.

• It would be too inefficient to have to say everything.

He bought it.

• Our programs still don’t know as much as people do.

Definition: P(A | B) = P(A  B)

P(B)

Intuition:

B A A B

A B

A

B

A

A

B B

Our task: find the object (word, structure, or whatever) that is most likely given our observation.

w= argmax P(w|O) = argmax P(w  O)

w  V w  V P(O)

Example:

P(word=“have” | sound=“haf”) = P(word=“have”  sound=“haf”)

P(sound = “haf”)

• But what do we actually know:
• P(sound = “haf” | word = “have”)
• P(word = “have”)

P(A | B) = P(A  B)

P(B)

= P(A  B)  P(A)

P(B) P(A)

= P(A  B)  P(A)

P(A) P(B)

= P(B | A)  P(A)

P(B)

P(A | B) = P(B | A)  P(A)

P(B)

Example:

P(word=“have” | sound=“haf”)

= P(sound=“haf” | word=“have”)  P(word=“have”)

P(sound = “haf”)

But, if we are comparing candidate interpretations for “haf”, we can ignore the denominator since they are all the same.

Common assumption: just one mistake (covers about 80% of nonword errors). Four kinds of mistakes: insertion, deletion, transposition, substitution.

Example:

c= argmax P(c|t) = argmax P(t|c) P(c)

c  C c C

Example: observed word: acress

Note: P(c)’s include adding .5 for smoothing.

c= argmax P(c|t) = argmax P(t|c) P(c)

w  V w  V

Example: What is P(deleting t following c)?

Answer: We need to collect data from a training set and encode them in some useful way: confusion matrices contain counts:

del[x,y], ins[x,y], sub[x,y], trans[x,y]

From these counts, we can compute probabilities:

Deletion: P(t|c) (which involves deleting the i’th character, which happens to be x, where the i-1st character is y

= del[ci-1,ci] / count[ci-1ci]

Typed word = acress

Intended word = ?

P(word = “britney” | O = “britne”)

= P(O=“britne” | word=“Britney”)  P(word=“britney”)

P(O = “britne”)

The data below shows some of the misspellings detected by our spelling correction system for the query [ britney spears ], and the count of how many different users spelled her name that way. Each of these variations was entered by at least two different unique users within a three month period, and was corrected to [britney spears] by our spelling correction system (data for the correctly spelled query is shown for comparison).

From http://www.google.com/jobs/britney.html

488941 britney spears 40134 brittany spears 36315 brittney spears 24342 britany spears  7331 britny spears  6633 briteny spears  2696 britteny spears  1807 briney spears  1635 brittny spears  1479 brintey spears  1479 britanny spears  1338 britiny spears  1211 britnet spears  1096 britiney spears   991 britaney spears   991 britnay spears

811 brithney spears   811 brtiney spears   664 birtney spears   664 brintney spears   664 briteney spears   601 bitney spears   601 brinty spears   544 brittaney spears   544 brittnay spears   364 britey spears   364 brittiny spears   329 brtney spears   269 bretney spears   269 britneys spears   244 britne spears   244 brytney spears

  220 breatney spears   220 britiany spears   199 britnney spears   163 britnry spears   147 breatny spears   147 brittiney spears   147 britty spears   147 brotney spears   147 brutney spears   133 britteney spears   133 briyney spears   121 bittany spears   121 bridney spears   121 britainy spears   121 britmey spears

  109 brietney spears   109 brithny spears   109 britni spears   109 brittant spears    98 bittney spears    98 brithey spears    98 brittiany spears    98 btitney spears    89 brietny spears    89 brinety spears    89 brintny spears    89 britnie spears    89 brittey spears    89 brittnet spears    89 brity spears    89 ritney spears

80 bretny spears    80 britnany spears    73 brinteny spears    73 brittainy spears    73 pritney spears    66 brintany spears    66 britnery spears    59 briitney spears    59 britinay spears    54 britneay spears    54 britner spears    54 britney's spears    54 britnye spears    54 britt spears    54 brttany spears

48 bitany spears    48 briny spears    48 brirney spears    48 britant spears    48 britnety spears    48 brittanny spears    48 brttney spears    44 birttany spears    44 brittani spears    44 brityney spears    44 brtitney spears    39 brienty spears    39 brritney spears    36 bbritney spears    36 briitany spears

  36 britanney spears    36 briterny spears    36 britneey spears    36 britnei spears    36 britniy spears    32 britbey spears    32 britneu spears

   2 brtittny spears     2 brttiny spears     2 brtttany spears     2 brydney spears     2 brynty spears     2 brythey spears     2 bryttney spears     2 btiany spears     2 btirtney spears     2 btitiney spears     2 btittny spears     2 btritany spears     2 buttney spears     2 grittney spears     2 prietny spears     2 pritany spears     2 prittany spears

• We observe Joe wearing glasses. We want to decide whether it is more likely that Joe is a salesman or a librarian. Here are the facts (L means librarian, S means salesman, G means glasses):
• P(G) = .1
• P(L) = .0001
• P(S) = .01
• P(G|L) = 1
• P(G|S) = .05
• P(L|G) = P(G|L) × P(L) / P(G)
• = 1 × 0.0001/.1
• = 0.001
• P(S|G) = P(G|S) × P(S) / P(G)
• = .05 × 0.01/.1
• = 0.005

• We want to compute the probability that Joe uses heroin given that he tests positive for it. Here are the facts (H means heroin use, E means a positive test for heroin):
• Sensitivity = P(E|H) = 0.95
• Specificity = 1 − P(E|~H) = 0.90
• Baseline "prior" probability = P(H) = 0.03.
• P(H|E) = P(E|H) × P(H)
• P(E)
• = 0.95 × 0.03/[0.03×0.95 + 0.97×0.1]
• = 0.1255.