Probabilistic Retrieval

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1 CS 630 Lectue 4: 02/07/2006 Lectue: Lillin Lee Scibes: Pete Bbinski, Dvid Lin Pobbilistic Retievl I. Nïve Beginnings. Motivtions b. Flse Stt : A Pobbilistic Model without Vition? II. Fomultion. Tems nd Definitions b. Mking Stuff Computble III. Relted Questions I. Nïve Beginnings. Motivtions Afte few dys of tlking bout the vecto spce model nd convincing ouselves tht such n d-hoc ppoch woks, it seems stnge to tke step bck nd decide to stt ove. Howeve, it is cle tht in defining the vecto spce model, we mde few potentilly contovesil judgments. Fo exmple, why does bg of wods fetue vecto mke sense nymoe thn some othe fetue vecto? Why not use fetue vectos tht tcks counts of the ten most common wods, the lengths of those wods, the numbe of pgphs, nd the numbe of idioms used in document? Why do we use this thing clled tem fequency, TF, nd invese document fequency, IDF, in the simility function we use to nk the documents by thei eltion to quey? These e the types of questions tht led eseches to conside pobbilistic model. Cedits: The eseches tht developed the pobbilistic model fo infomtion etievl include Robetson, Späk Jones, nd Wlke, mong mny othes Gols: Gin bette theoeticl undestnding of infomtion etievl. Obtin stte-of-the-t pefomnce. 1

2 b. Flse Stt : A Pobbilistic Model without Vition? Note: Fo the following discussion, unless othewise stted, we e ssuming tht q, the quey, is fixed. We stt by mentioning tht the fundmentl pinciple of ou model is to nk documents by the pobbility tht they e elevnt, noting tht this ide of elevnce is vey fuzzy. In othe wods, o the without using English wods, we wnt to compute the following: (1) P(elevnt document) Only few seconds into ou discussion, we see gling poblem. Whee e the ndom vibles in this eqution? Fo those who do not hve much expeience with ndom vibles, we mention the following ules: Use cpitl lettes fo ndom vibles Use lowe cse lettes fo vlues of ndom vible (.v.) Ex. We cn decle.v. X to be the vlue of fi coin nd cn compute the pobbility Heds,Tils. P(X = x), whee x { } In esponse to ou lck of ndom vibles, we cn ssign ndom vible to the elevnce, R, nd the document, D, nd ewite eqution (1) s the following: (2) P(R = D = Now tht we hve ndom vibles, we hve to decide wht possible vlues they cn tke. Fo exmple, we cn let R tke on the vlues { yes, no} o { 0,.1,.2, K,.9,1}. Whethe we use the fome o the ltte, thee is still no vition, becuse in the fome cse, we ledy know whethe document is eithe elevnt o not. In the ltte cse, thee is still no diffeence, s we ledy know if document hs elevnce ting of.5 o 0. We lso need to define wht possible vlues D cn tke. We cn eithe use d to epesent document in given copus, C, o s document fom the spce of ll possible documents. We lso need to tke ce tht P(D = > 0 fo ll documents d, egdless of which option we choose. Regdless of ou choices hee, we still hve not mnged to intoduce vince into ou pobbility fomul, (2). 2

3 Let s conside few of the possible souces of vince tht we could dd: Vition in specific use s judgment of whethe document is elevnt o not. The use my chnge his mind bout whethe document is elevnt bsed on when you sk him. Fo exmple, the use my use the sme tem to men diffeent things t diffeent times. Vition ove diffeent uses [Mon nd Kuhns (1960)]. Ech use my hve diffeent opinion on whethe document is elevnt o not to given quey. The min concen is how we would chcteize this vition without some model of humn pefeences nd beliefs. Vition becuse of the document epesenttion [Robetson nd Späck Jones (1976)]. Some documents with cetin epesenttion e elevnt, while some e not. This ide stems fom the fct tht if we use document epesenttion tht incopotes tems without the vious lnguge constucts o context they e found in, then some documents with cetin fetue vecto my be moe elevnt to quey thn othes with the sme fetue vecto. Fo exmple, document contining the phse the White House is hs diffeent topic thn document tht only contins the phse the house is white. Fo lck of bette choice, nd since using educed document epesenttions is fily stndd pctice in IR, we will ty the oute of intoducing vition in elevnce due to the document epesenttion. Still, is thee nothe souce of vition we could intoduce tht we en t consideing? II. Fomultion Note: Fo the following discussion, unless othewise stted, we e ssuming tht q, the quey, is fixed.. Tems nd Definitions Peviously, we decided tht we would intoduce vition into ou pobbilistic etievl model by llowing document with cetin epesenttion to be elevnt whee nothe document with the sme epesenttion is not elevnt. This fetue of ou model motivtes ou ensuing discussion on binning the documents into clsses. We will epesent ech document with set of ttibutes: A =,..., ( A ) T 1 A m whee A j is n.v. fo ttibute j s vlue Theefoe, fo ech document d, thee is n ssocited vecto:,..., ( = ( ( ( )) T m d 1 3

4 ( j) 1 if v in d j = 0 else m+1 ( = length(] We cn ledy see tht thee my be some poblems with mbiguity in this epesenttion. Ex. j ( = the numbe of times v (j) occus in d o ( [o even Ex. Tke the documents white house = d (1) nd house white = d (2). If the quey is sking fo documents deling with U.S. politics, then only document (1) is elevnt, even d( 1) = d(2). though ( ) ( ) Using ou ttibute vectos, we cn ewite ou pobbility fomul fom eqution (2) s the following: (3) P(R = y A (d = ) ) Some immedite concens ise howeve. Wsn t ou choice of fetues fo this epesenttion of the document just s bity s the bg of wods epesenttion we decided upon fo the VSM (vecto spce model)? b. Mking Stuff Computble We next exmine if we cn deive the VSM fom ou pobbilistic etievl model s some specil cse. In ode fo this esult, we will need to eliminte o compute the tems tht we hve left fom eqution (3). We obseve tht the event tht A (d = ) is likely vey e unless we educe ou ttibute spce such tht moe documents e in ech bin. Thee is lso lck of elevnce sttistics (ecll tht we hve no elevnce-lbeled documents). Theefoe, we will ty to use Byes Rule to see if we cn obtin pobbility fomul tht we ctully cn compute. Recll tht Byes Rule is the following: P(B A) P(A) (4) P(A B) = P(B) Applying this fomul to (3), we obtin: (5) P(R = y A (d ( R= y) P(R= y) = ) ) = ( ) And unde nking equivlence [nking equivlence mens tht we cn eliminte tems tht do not depend on the documents since they will be ignoed when we nk documents by this function] fo documents, (5) becomes: 4

5 ( R= y) P(R= y) nk ( R= y) (6) ( ( The next step equies us to mke some bod ssumptions tht don t seem possible to justify. We need to bek down this tem some moe, so we ty to ssume conditionl independence between the elements of A nd R nd independence between elements of A in genel. As Coope suggests, this modeling ssumption is mjo inconsistency in the pobbilistic etievl model. He in tun suppots his sttement with n exmple whee ssuming both independence nd conditionl independence gives logicl inconsistency. In esponse, we cn insted ely on sot of wek link dependence ssumption to obtin the following: P( A = ( R= y) α j j ( R= y) w.l.d. j (7) α> 0, α 1 ( P( Aj= j In tun, theα disppes when nking documents in ny cse. Note some effects of this choice. Thee e some ttctive fetues to this decomposition. Ex. Sy we hve d (1) = White nd d (2) = House. Let A 1 = does white ppe? A 2 = does house ppe? (1) (2) ( = (1, 0) ( = (0, 1) We would find tht the pobbility of White House ppeing is computed by this #(" White") #(" House") eqution,. Although no document contins White House, # docs # docs this pobbility we computed will be gete thn zeo, which is esonble. We still hve poblems with these tems conditioned on the elevnce of the document, which we still hve no wy to compute. Wht othe infomtion cn we tke dvntge of? We still hve ou quey, q, which my be ou only clue to the vlue of R. Let q j = 1 0 if v else ( j) in q 5

6 Intuitively, we cn tke eqution (7) nd split the tems in the poduct sum by whethe they occu in the quey o not (whethe q j is equl to 1 o 0). Thus we hve the following eqution: P( Aj= R= y) P( Aj= R= y) (8) j = 1 P( Aj= j= 0 P( Aj= j : q j: q ) Once gin, we mke simplifying ssumption. Wht if the tems not in the quey (q j = 0) hve the sme distibution ove elevnt documents s ove ll othe documents? If we cn ffod to mke this ssumption, then the second tem in (8) will disppe entiely unde equivlence nking becuse P( Aj= R= y) = P( Aj = j) if qj = 0. Theefoe, we find ouselves t the following simple, but still difficult to compute function: P( Aj= R= y) (9) j= 1 P( Aj= j : q ) In conclusion, we see tht ou ppoch is indeed helping us to simplify the tem. Howeve, we still e not t point whee we cn estimte the pt of the eqution elting to R. We will need to continue simplifying (9) to ech point whee we cn use some (ny?) estimtion to del with ou lck of elevnce infomtion. III. Relted Questions Let us exmine how we might use ou pobbilistic model if we hd some tining dt with elevnce infomtion lbeled fo fixed quey. Sy we hve m = 5 fetues, listed below: Element of j Fetue compute science esech pogm fishing Ech element of j is equl to the numbe of occuences of fetue if tht fetue is pesent in the document. In ode to clculte pobbilities, we will need to use counts ove ou copus, which is only the fou documents below. We cn estimte the pobbility of ny fetue in ou fetue vecto by summing the occuences of tht fetue vlue mong the documents (esticted to elevnt ones in the cse of R = y) nd dividing by the numbe of documents. 6

7 Given Tining Documents: d (0) = some new esech in compute chitectue, R = y d (1) = compute hve yet to chnge ge-old methods of fishing, R = n d (2) = compute ided esech in lue dynmics, R = n d (3) = compute science hs ceted pletho of new pogm, R = y A) Compute the ttibute vectos fo the documents bove, nd then compute P(R = y), P( A j, nd P( A j= 1 R= y) j. Answe: (d (0) ) = (1, 0, 1, 0, 0) T (d (1) ) = (1, 0, 0, 0, 1) T (d (2) ) = (1, 0, 1, 0, 0) T (d (3) ) = (1, 1, 0, 1, 0) T P(R = y) = # elevnt docs 2 = =.5 P(R = n) = 1 P(R = y) =.5 # docs 4 Recll tht the pobbility P( Aj = ) fo tem v (j) is computed by the following: # docs s. t. ) = P( Aj = = # docs And fo the pobbility P( Aj = j ( R= y) fo tem v (j) is computed by the following: # docs s. t. is elevnt nd ) = P( Aj = R= y) = # elevnt docs P( A 1 = 1 P( A 2 =.25 P( A 3 =.5 P( A 4 =.25 P( A 5 =.25 P( A 1 y) = 1 P( A 2 y) =.5 P( A 3 y) =.5 P( A 4 y) =.5 P( A 5 y) = 0 Convesely, we lso know the pobbilities of ech ttibute vecto tem being 0. 7

8 B) Immeditely we noticed one of the poblems tht we consideed duing ou discussion. Since the wod fishing ws neve found mong ou elevnt documents, the pobbility P( A 5 y) = 0 becme zeo. Howeve, we would like to be ble to ccount fo cses whee we hven t seen wod in ou copus. This sitution my occu if we e obtining new documents tht e not in ou tining copus nd would like to immeditely clssify them s elevnt o not elevnt to fixed quey. If we wnt to bette estimte counts fo ou model, how could we modify the function we use to estimte pobbilities? Recll we used fomul like this: # docs s. t. ) = P( Aj = = # docs Answe: Thee e sevel wys in which we could modify how we estimte pobbilities fom ou counts. The simplest method would be to dd one to ll ou counts s.t. the new eqution is: [# docs s. t. ) = ] + 1 P( Aj = = # docs m is the numbe of fetues in ou fetue vecto. Anothe slightly moe complicted method is to estimte the counts of fetues tht we hven t seen by using the counts of fetues tht we ve only seen in one document. We could edistibute some of the pobbility mss fom ll wods seen once to those wods not yet seen in ou fetue set. A still smte ide would be to conside ny low numbe of counts (less thn 5 fo exmple) to be unelible nd edistibute these counts to those counts tht e zeo. Regdless, these smoothing methods e hedy topic on thei own. E.g. Chen nd Goodmn 98. C) Wht if we decided to incopote gete detil of esolution in ou elevnce lbels? Fo exmple, we could let R { 0,.1,.2, K,.9,1} insted of just y o n. Wht is one poblem tht such chnge would cuse when we tied to ctully estimte the pobbilities fo such model? 8

9 Answe: By now you likely know tht dding dditionl clsses would only decese the mount of documents of ech elevnce level tht we could use to estimte the pobbilities, which would likely hut the pefomnce of this estimted model. As n open ended question, conside wht would hppen if we mgiclly hd enough dt to popely tin ou model? Knowing tht we intend to nk the documents though some scoing scheme nywy, would bette elevnce esolution scheme necessily help us? 9

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