Rather than reweighting in a vector space…
If user has told us some relevant and some irrelevant documents, then we can proceed to build a probabilistic classifier, such as a Naive Bayes model:
P(t_{k}R) = D_{rk} / D_{r}
P(t_{k}NR) = D_{nrk} / D_{nr}
t_{k} is a term; D_{r} is the set of known relevant documents; D_{rk} is the subset that contain t_{k}; D_{nr} is the set of known irrelevant documents; D_{nrk} is the subset that contain tk.
In traditional IR systems, matching between each document and query is attempted in a semantically imprecise space of index terms.
Probabilities provide a principled foundation for uncertain reasoning. Can we use probabilities to quantify our uncertainties?
Classical probabilistic retrieval model
Probability ranking principle, etc.
(Naïve) Bayesian Text Categorization
Bayesian networks for text retrieval
Language model approach to IR
An important emphasis in recent work
Probabilistic methods are one of the oldest but also one of the currently hottest topics in IR.
Traditionally: neat ideas, but they’ve never won on performance. It may be different now.
We have a collection of documents
User issues a query
A list of documents needs to be returned
Ranking method is core of an IR system:
In what order do we present documents to the user?
We want the “best” document to be first, second best second, etc….
Idea: Rank by probability of relevance of the document w.r.t. information need
P(relevantdocumenti, query)
Let x be a document in the collection.
Let R represent relevance of a document w.r.t. given (fixed) query and let NR represent nonrelevance. ⇐ R={0,1} vs. NR/R
p(xR), p(xNR)  probability that if a relevant (nonrelevant) document is retrieved, it is x.
Simple case: no selection costs or other utility concerns that would differentially weight errors
Bayes’ Optimal Decision Rule
x is relevant iff p(Rx) > p(NRx)
PRP in action: Rank all documents by p(Rx)
Theorem:
Using the PRP is optimal, in that it minimizes the loss (Bayes risk) under 1/0 loss
Provable if all probabilities correct, etc. [e.g., Ripley 1996]
More complex case: retrieval costs.
Let d be a document
C  cost of retrieval of relevant document
C’  cost of retrieval of nonrelevant document
Probability Ranking Principle: if
for all d’ not yet retrieved, then d is the next document to be retrieved
We won’t further consider loss/utility from now on
How do we compute all those probabilities?
Do not know exact probabilities, have to use estimates
Binary Independence Retrieval (BIR) – which we discuss later today – is the simplest model
Questionable assumptions
“Relevance” of each document is independent of relevance of other documents.
Really, it’s bad to keep on returning duplicates
Boolean model of relevance
That one has a single step information need
Seeing a range of results might let user refine query
Estimate how terms contribute to relevance
How do things like tf, df, and length influence your judgments about document relevance?
One answer is the Okapi formulae (S. Robertson)
Combine to find document relevance probability
Order documents by decreasing probability
“If a reference retrieval system's response to each request is a ranking of the documents in the collection in order of decreasing probability of relevance to the user who submitted the request, where the probabilities are estimated as accurately as possible on the basis of whatever data have been made available to the system for this purpose, the overall effectiveness of the system to its user will be the best that is obtainable on the basis of those data.”
[1960s/1970s] S. Robertson, W.S. Cooper, M.E. Maron; van Rijsbergen (1979:113); Manning & Schütze (1999:538)
Basic concept:
"For a given query, if we know some documents that are relevant, terms that occur in those documents should be given greater weighting in searching for other relevant documents.
By making assumptions about the distribution of terms and applying Bayes Theorem, it is possible to derive weights theoretically."
Van Rijsbergen
Traditionally used in conjunction with PRP
“Binary” = Boolean: documents are represented as binary incidence vectors of terms (cf. lecture 1):
“Independence”: terms occur in documents independently
Different documents can be modeled as same vector
Queries: binary term incidence vectors
Given query q,
for each document d need to compute p(Rq,d).
replace with computing p(Rq,x) where x is binary term incidence vector representing d Interested only in ranking
Will use odds and Bayes’ Rule:
Using Independence Assumption:
So :
Since xi is either 0 or 1:
Let Pi = P(xi = 1  R,q); rt = p (x_{i} =1  NR,q)
Assume, for all terms not occurring in the query (q_{i}=0) pi = r_{i}
Then... (This can be changed (e.g.,
in relevance feedback)
Retrieval Status Value:
All boils down to computing RSV.
So, how do we compute ci’s from our data ?
Estimating RSV coefficients.
For each term i look at this table of document counts:
log (1– ri_{)}/r_{i} = log (N– n)/n ≈ log N/n = IDF!
pi (probability of occurrence in relevant documents) can be estimated in various ways:
from relevant documents if know some
Relevance weighting can be used in feedback loop
constant (Croft and Harper combination match) – then just get idf weighting of terms
proportional to prob. of occurrence in collection
more accurately, to log of this (Greiff, SIGIR 1998)
Assume that pi constant over all xi in query
pi = 0.5 (even odds) for any given doc
Determine guess of relevant document set:
V is fixed size set of highest ranked documents on this model (note: now a bit like tf.idf!)
We need to improve our guesses for pi and ri, so
Use distribution of xi in docs in V. Let Vi be set of documents containing xi
pi = Vi / V
Assume if not retrieved then not relevant
ri = (ni – Vi) / (N – V)
Go to 2. until converges then return ranking
Interact with the user to refine the description: learn some definite members of R and NR
Repeat, thus generating a succession of approximations to R.
Getting reasonable approximations of probabilities is possible.
Requires restrictive assumptions:
term independence
terms not in query don’t affect the outcome
boolean representation of documents/queries/relevance
document relevance values are independent
Some of these assumptions can be removed
Problem: either require partial relevance information or only can derive somewhat inferior term weights

Think through the differences between standard tf.idf and the probabilistic retrieval model in the first iteration
Think through the differences between vector space (pseudo) relevance feedback and probabilistic (pseudo) relevance feedback
Standard Vector Space Model
Empirical for the most part; success measured by results
Few properties provable
Probabilistic Model Advantages
Based on a firm theoretical foundation
Theoretically justified optimal ranking scheme
Disadvantages
Making the initial guess to get V
Binary wordindoc weights (not using term frequencies)
Independence of terms (can be alleviated)
Amount of computation
Has never worked convincingly better in practice
Standard probabilistic model assumes you can’t estimate P(RD,Q)
Instead assume independence and use P(DR)
But maybe you can with a Bayesian network*
What is a Bayesian network?
A directed acyclic graph
Nodes
Events or Variables
Assume values.
For our purposes, all Boolean
Links
model direct dependencies between nodes
a,b,c  propositions (events). Bayesian networks model causal relations between events

For more information see:
R.G. Cowell, A.P. Dawid, S.L. Lauritzen, and D.J. Spiegelhalter. 1999. Probabilistic Networks and Expert Systems. Springer Verlag.
J. Pearl. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. MorganKaufman.

Evidence  a node takes on some value
Inference
Compute belief (probabilities) of other nodes
conditioned on the known evidence
Two kinds of inference: Diagnostic and Predictive
Computational complexity
General network: NPhard
Treelike networks are easily tractable
Much other work on efficient exact and approximate Bayesian network inference
Clever dynamic programming
Approximate inference (“loopy belief propagation”)
Goal
Given a user’s information need (evidence), find probability a doc satisfies need
Retrieval model
Model docs in a document network
Model information need in a query network
Construct Document Network (once !)
For each query
Construct best Query Network
Attach it to Document Network
Find subset of di’s which maximizes the probability value of node I (best subset).
Retrieve these di’s as the answer to query.


*conditional probability table
Prior probs don’t have to be 1/n.
“User information need” doesn’t have to be a query  can be words typed, in docs read, any combination …
Phrases, interdocument links
Link matrices can be modified over time.
User feedback.
The promise of “personalization”
Document network built at indexing time
Query network built/scored at query time
Representation:
Link matrices from docs to any single term are like the postings entry for that term
Canonical link matrices are efficient to store and compute
Attach evidence only at roots of network
Can do single pass from roots to leaves
Flexible ways of combining term weights, which can generalize previous approaches
Boolean model
Binary independence model
Probabilistic models with weaker assumptions
Efficient largescale implementation
InQuery text retrieval system from U Mass
Turtle and Croft (1990) [Commercial version defunct?]
Need approximations to avoid intractable inference
Need to estimate all the probabilities by some means (whether more or less ad hoc)
Much new Bayes net technology yet to be applied?
S. E. Robertson and K. Spärck Jones. 1976. Relevance Weighting of Search Terms. Journal of the American Society for Information Sciences 27(3): 129–146.
C. J. van Rijsbergen. 1979. Information Retrieval. 2nd ed. London: Butterworths, chapter 6. [Most details of math] http://www.dcs.gla.ac.uk/Keith/Preface.html
N. Fuhr. 1992. Probabilistic Models in Information Retrieval. The Computer Journal, 35(3),243–255. [Easiest read, with BNs]
F. Crestani, M. Lalmas, C. J. van Rijsbergen, and I. Campbell. 1998. Is This Document Relevant? ... Probably: A Survey of Probabilistic Models in Information Retrieval. ACM Computing Surveys 30(4): 528–552.
http://www.acm.org/pubs/citations/journals/surveys/1998304/p528crestani/
[Adds very little material that isn’t in van Rijsbergen or Fuhr ]
H.R. Turtle and W.B. Croft. 1990. Inference Networks for Document Retrieval. Proc. ACM SIGIR: 124.
E. Charniak. Bayesian nets without tears. AI Magazine 12(4): 5063 (1991). http://www.aaai.org/Library/Magazine/Vol12/1204/vol1204.html
D. Heckerman. 1995. A Tutorial on Learning with Bayesian Networks. Microsoft Technical Report MSRTR9506
http://www.research.microsoft.com/~heckerman/
N. Fuhr. 2000. Probabilistic Datalog: Implementing Logical Information Retrieval for Advanced Applications. Journal of the American Society for Information Science 51(2): 95–110.
R. K. Belew. 2001. Finding Out About: A Cognitive Perspective on Search Engine Technology and the WWW. Cambridge UP 2001.
MIR 2.5.4, 2.8