26:010:680 Current Topics in Accounting Research

Transcription

1 26:010:680 Current Topics in Accounting Research Dr. Peter R. Gillett Associate Professor Department of Accounting and Information Systems Rutgers Business School Newark and New Brunswick

2 OVERVIEW ADAPT Homework 4 Valuation Networks * Lauritzen & Spiegelhalter * Aalborg * Shenoy-Shafer * Gillett Homework 5 March 2, 2011 Dr. Peter R. Gillett 2

3 ADAPT Describes a project to create an expert system for audit program(me) tailoring Goals * To ensure that sufficient audit evidence is planned * To ensure that excessive audit evidence is not planned * To produce an optimal audit approach March 2, 2011 Dr. Peter R. Gillett 3

4 ADAPT The principal design philosophies incorporated in ADAPT are: * the user retains ultimate control of the process * the software is capable both of generating programmes, and of assessing the effectiveness of programmes determined by the auditor * two techniques are used in audit programme generation heuristic rules may include appropriate audit procedures to the extent that both the user and heuristics leave the programme indeterminate, a mathematical evaluation based on the assurance that is derivable from various procedures is used to plan the audit programme * this combination of techniques is designed to provide a means by which relevant auditor expertise in the designing of audit programmes may be combined with detailed knowledge of the client to ensure that the audit programme produced is relevant, effective and efficient to ensure that ADAPT can nevertheless operate with a relatively small number of heuristic rules, requiring only limited effort in supplying responses to questions to generate an efficient and cost-effective audit programme by taking into account the structure of audit evidence, and the relationships between related assertions * ADAPT was implemented initially with a small number of heuristic rules; more were added as necessary during pilot testing, particularly in response to issues arising during validation of the programmes produced by the system * built-in expertise is used in conjunction with the auditor's own knowledge March 2, 2011 Dr. Peter R. Gillett 4

5 ADAPT The Assertion Hierarchy OR F1 F2 F3 F4 etc. A1 A2 A3 A4 etc. IR CR DR IR CR DR IR CR DR P1 P2 P3 etc. March 2, 2011 Dr. Peter R. Gillett 5

6 ADAPT Derivable Assurance, Evidential Power and Scope * Evidential Power = Relevance * Reliability * Maximum Derivable Assurance (MDA)= Evidential Power * Scope * Maximum Derivable Assurance (MDA)= Relevance * Reliability * Scope * Evidential Value = MDA for all relevant assertions * Diversity and Diversity Factors (b = 1 - (30 + 2*min(p, 5) - f*(13-f)) 2 /1568 ) March 2, 2011 Dr. Peter R. Gillett 6

7 ADAPT Aggregation Rules * Direct Assurance DA = AsC b (X 1, X 2,... X n ) = 1 - (1-X 1 )*[(1-X 2 )...(1-X n )] b where the procedures are sorted so that X 1 > X 2 >... > X n * Indirect Assurance IA = Asxx(Y 1, Y 2,... Y q ) = Y 1 *Y 2 *...*Y q * Direct and Indirect Assurance AsC(DA, IA) = 1 - (1 - DA)*(1 - IA) Model Rules * Audit Risk = Inherent Risk * Control Risk * Detection Risk * Detection Risk = Audit Risk / (Inherent Risk * Control Risk) * Target Assurance = 1 - Detection Risk * Bayesian and Belief Function alternatives Hierarchical Rules March 2, 2011 Dr. Peter R. Gillett 7

8 ADAPT Control Strategy: for each financial statement area in turn, as selected: 1 Ignore Unimportant assertions (ie those not designated as Important by the user) throughout. 2 Identify Balance Sheet assertions and Transaction Stream assertions from the Assertions Database. 3 Identify Strategic assertions: 3.1 assertions that are normally Strategic are identified on the Assertions Database 3.2 heuristics may modify this determination 3.3 users may override these determinations of Strategic assertions. 4 Set a Target Assurance for each assertion: 4.1 using model rules based on the Audit Risk Model (or other specified alternative) 4.2 applying any appropriate heuristics. 5 Include any procedures designated as Mandatory by heuristics. 6 Ignore any procedures designated as Ignored by heuristics. 7 Include any procedures designated as Necessary by heuristics, except if they have been designated as Excluded by the user. 8 Exclude any procedures designated as Unnecessary by heuristics, except if they have been designated as Included by the user. 9 Include any procedures designated as Included by the user. 10 Include any Additional procedures supplied by the user that are relevant to Important assertions, except if they have subsequently been designated as Excluded by the user. 11 Exclude any procedures designated as Excluded by the user. 12 Include any procedures required by heuristics as a consequence of any procedures already included or excluded. 13 Evaluate the Planned Assurance achieved for each assertion. March 2, 2011 Dr. Peter R. Gillett 8

9 ADAPT 14 Select audit procedures for Strategic Balance Sheet assertions where the Target Assurance has not been achieved: 14.1 exclude available procedures as required by heuristics 14.2 for each assertion in turn select one of the available procedures: consider any procedures that, if added, would achieve the Target Assurance, and select the one with the lowest Cost,; if several procedures have the same lowest Cost, select the one with the greatest Evidential Value; if there are several of these, select the first one encountered; however, if this is the first procedure selected by the Control Strategy, do not select a sample unless there is no qualifying alternative available if there is no procedure that, if added, would achieve the Target Assurance and this is the first procedure selected by this algorithm, select the procedure with the greatest Maximum Derivable Assurance for the assertion; if several procedures have the same greatest Maximum Derivable Assurance, select the one with the lowest Cost; if several procedures have the same lowest Cost, select the one with the greatest Evidential Value; if there are several of these, select the first one encountered if there is no procedure that, if added, would achieve the Target Assurance but this is not the first procedure selected by this algorithm, select the procedure with the highest ratio of Maximum Derivable Assurance/Cost for the assertion; if several procedures have the same highest ratio of Maximum Derivable Assurance/Cost, select the one with the greatest Maximum Derivable Assurance for the assertion; if several procedures have the same Maximum Derivable Assurance, select the one with the greatest Evidential Value; if there are several of these, select the first one encountered but obey any heuristics overriding these choices 14.3 select any procedures now required by heuristics 14.4 revise the Planned Assurance for all appropriate assertions 14.5 iterate CS14.1 to CS14.4 until: there are no more procedures directly relevant or the Target Assurance for the assertion is achieved. March 2, 2011 Dr. Peter R. Gillett 9

10 ADAPT 15 Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 16 Select audit procedures for Strategic Transaction Stream assertions where the Target Assurance has not been achieved, as described in CS CS Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 18 Select audit procedures for non-strategic Balance Sheet assertions where the Target Assurance has not been achieved, as described in CS CS Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 20 Select audit procedures for non-strategic Transaction Stream assertions where the Target Assurance has not been achieved, as described in CS CS Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 22 For each assertion for which the Target Assurance has not been achieved, consider whether additional assurance can be achieved by extending the Scope of any selected non-sampling procedures: 22.1 consider Analytical Procedures only if the user has indicated that the quality of the accounting systems and the level of detailed data available may permit this 22.2 for each permitted Analytical Procedure or Test of Details that has been selected, ask the user to confirm whether it is possible to extend the Scope of the procedure 22.3 for the procedures where the user has confirmed the possibility of extending the Scope, select procedures as described in CS14.2 and increase the Scope as indicated in the ADAPT procedures database (AUDPROCS), until the Target Assurance for the assertion is achieved. 23 Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. March 2, 2011 Dr. Peter R. Gillett 10

11 ADAPT 24 For each assertion in turn for which the Target Assurance has not been achieved, select remaining procedures relevant to related assertions that provide appropriate indirect evidence as follows: 24.1 identify the related assertion with the lowest Planned Assurance for which there are relevant procedures available 24.2 select one of the available procedures relevant to this assertion, in accordance with CS revise the Planned Assurance for all appropriate assertions 24.4 iterate CS24.1 to CS24.3 until: there are no more relevant procedures available or the Target Assurance is achieved for the problem assertion or the Planned Assurances for all the assertions related to the problem assertion have reached the level set by the Governor. 25 Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 26 For each assertion for which the Target Assurance has not been achieved, extend the Scope of any of the procedures added during CS24, in accordance with CS CS Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 28 Minimise the sample sizes for the selected sampling procedures as follows: 28.1 build a network of related samples by selecting any sample not already included in a previous network, identifying all the assertions to which it is directly relevant for which the Target Assurance has been achieved, and all the assertions to which they provide indirect assurance for which the Target Assurance has been reached, then including all the samples relevant to any of these assertions, and iterating until no further samples are added to the network March 2, 2011 Dr. Peter R. Gillett 11

12 ADAPT 28.2 for each sample in the network, in descending order of Cost and, where Costs are equal, in ascending order of Evidential Value, consider whether it can be given a scope of zero (ie removed from the programme) without there being: an assertion for which the Target Assurance would no longer be achieved, although it would be achieved if the procedure were retained or an assertion for which the Target Assurance was not previously achieved, where the Planned Assurance would be reduced if the procedure were eliminated and if so, remove it, and iterate from CS for each of the remaining samples in the network, determine a minimum sample size assuming that the Scope of this procedure alone is reduced, as follows: determine an upper limit for the Sampling Confidence, which will be the lower of the Maximum Sampling Confidence permitted and the Sampling Confidence that would be needed for the procedure in question to achieve the Target Assurance alone determine a lower limit for the Sampling Confidence, which will be the Sampling Confidence needed to ensure that the minimum sample size is not less than 10 items or, for statistical samples, , the Sampling Confidence at which the sampling interval is equal to the acceptable level of imprecision (Touchstone), as this is greater determine the minimum Sampling Confidence for the sample by using a Binary Chop, rejecting the lower limit whenever there is : an assertion for which the Target Assurance would no longer be achieved, although it would be achieved if the sample size were retained at the present upper limit or an assertion for which the Target Assurance was not previously achieved, where the Planned Assurance would be reduced if the sample size were reduced to this lower limit generate the sample size for the minimum Sampling Confidence (using the Grant Thornton Sampling Plan) March 2, 2011 Dr. Peter R. Gillett 12

13 ADAPT 28.4 ascertain which of the possible reductions in sample sizes corresponding to the minima determined in CS28.3 achieves the greatest reduction in the total Cost of all the sampling, and reduce the Sampling Confidence for the appropriate procedure to the calculated minimum 28.5 iterate CS28.3 to CS28.4, calculating new minima following the previous Sampling Confidence reduction, until no further reductions are possible in the network 28.6 iterate CS28.1 to CS28.5 until all samples included in the audit programme have been included in a network, their Sampling Confidences have been reduced so far as possible, and appropriate sample sizes have been calculated. 29 Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. 30 For each procedure in turn that has been selected by the system using the Aggregation rules, in descending order of Cost and, where Costs are equal, in ascending order of Evidential Value: 30.1 eliminate the procedure unless: there is an assertion for which the Target Assurance would no longer be achieved, although it would be achieved if the procedure were retained or there is an assertion for which the Target Assurance was not previously achieved, where the Planned Assurance would be reduced if the procedure were eliminated 30.2 if there is any procedure selected by heuristic as a result of the procedure under consideration in CS30.1 being selected, then heuristics should enable both to be considered for elimination or retention together. 31 Revise the Planned Assurance for all assertions to take account of the direct and indirect evidence available from the procedures selected. March 2, 2011 Dr. Peter R. Gillett 13

14 ADAPT 32 If there is any assertion for which the Target Assurance has not been reached, there is insufficient evidence planned. 33 If there is any procedure selected by the user, or as a result of a heuristic rule, that could be eliminated without there being: an assertion for which the Target Assurance would no longer be achieved, although it would be achieved if the procedure were retained or an assertion for which the Target Assurance was not previously achieved, where the Planned Assurance would be reduced if the procedure were eliminated there is excessive evidence planned. Heuristics and Eval function Control Panel External representation of knowledge and heuristics * five system databases Extensive knowledge engineering project involving many people System verification and validation Outstanding research issues March 2, 2011 Dr. Peter R. Gillett 14

15 Homework 4 March 2, 2011 Dr. Peter R. Gillett 15

16 Assignments for Week 7 Lauritzen, Steffan. L., and David. J. Spiegelhalter "Local Computations with Probabilities on Graphical Structures and their Application to Expert Systems". Journal of the Royal Statistical Society. Shenoy, Prakash., and Glenn. Shafer "Axioms for Probability and Belief-Function Propagation". Uncertainty in Artificial Intelligence. Gillett, Peter. R A Comparative Study of Audit Evidence and Audit Planning Models Using Uncertain Reasoning. UMI. March 2, 2011 Dr. Peter R. Gillett 16

17 Valuation Networks Last week we studied approximate methods for propagation probabilities A brute force approach to computing probabilities is not realistic, and so we look for ways to carry out local propagation (i.e., to combine probabilities with neighboring probabilities in a network, but not combine all probabilities at once) March 2, 2011 Dr. Peter R. Gillett 17

18 Valuation Networks Brute force * 35 binary variables * 2 35 = 34,359,738,368 rows * 46 evidence nodes * 16 relation nodes * 2 calculation columns * 64 x x 2 = cells * = bytes * = 1,024 PCs with 2Gb memory each March 2, 2011 Dr. Peter R. Gillett 18

19 Valuation Networks Brute force * 2 35 x 61 = multiplications * 2 35 divisions * x (2 34-1) additions * 2 35 subtractions * ==> 4 hours at 2 GHz * ==> days(s) when overhead added * ==> many months/years to build an audit plan! March 2, 2011 Dr. Peter R. Gillett 19

20 Valuation Networks A great deal of work on formal propagation of probabilities in a network has been carried out by a number of researchers Perhaps most notable among them is Judea Pearl The biggest obstacle was the problem created when the network did not have a tree structure and the key breakthrough came in Lauritzen & Spiegelhalter s paper March 2, 2011 Dr. Peter R. Gillett 20

21 Lauritzen & Spiegelhalter Emphasis on efficient computational schemes Assumes a fixed model currently being entertained and the numerical assessments are precisely specified Based on development of MUNIN Acknowledges much prior work March 2, 2011 Dr. Peter R. Gillett 21

22 Lauritzen & Spiegelhalter Main Issues * Initialization * Absorption of evidence * Global propagation * Hypothesizing * Planning * Influential findings March 2, 2011 Dr. Peter R. Gillett 22

23 Lauritzen & Spiegelhalter As in Pearl s scheme, there is a fundamental assumption that the joint distribution is given as a product of priors and conditionals; e.g., ( ) α τ ξ ε δ λ β σ P,,,,,,, = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) α τα ξε ετδ δ ε β λσ βσ σ P P P P, P, P P P Or, working with proportionality, as potentials: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) α τ, α ξ, ε ε, τ, λ δ, ε, β λ, σ β, α σ ψ ψ ψ ψ ψ ψ ψ ψ March 2, 2011 Dr. Peter R. Gillett 23

24 Lauritzen & Spiegelhalter Evidence potentials are not uniquely determined They may not, in general, be easy to interpret other than through the joint probability being proportional to their product Initially, however, we may set the potentials equal to the probability factors March 2, 2011 Dr. Peter R. Gillett 24

25 Lauritzen & Spiegelhalter Building the tree of cliques for propagation * Form the moral graph by: dropping directions on the links; and marrying parents * Add fill-ins to create a triangulated graph * Identify the cliques of the triangulated graph * Label nodes using maximal cardinality search, rank cliques according to the highest labeled node, to form a set chain with the running intersection property * This is, of course, a join tree; arbitrarily select a root March 2, 2011 Dr. Peter R. Gillett 25

26 Lauritzen & Spiegelhalter A subset of nodes of a graph is complete if every member is connected to every other member by an edge A clique is a maximal complete subset March 2, 2011 Dr. Peter R. Gillett 26

27 Lauritzen & Spiegelhalter Maximum cardinality search * Assign 1 to an arbitrary node * Number the nodes consecutively, choosing as the next a node with a maximum number of previously numbered neighbors * Break ties arbitrarily March 2, 2011 Dr. Peter R. Gillett 27

28 Lauritzen & Spiegelhalter Maximum cardinality fill-in * Assume the nodes are numbered as for maximum cardinality search * Working backwards from the last node to the first, consider the set bd ( i ) { 1,..., i 1} where bd ( i ) is the set of numbers of the neighbors of the i th node * If this set is not complete, add whatever fill-in edges are necessary to make it so March 2, 2011 Dr. Peter R. Gillett 28

29 Lauritzen & Spiegelhalter Running intersection property * Suppose we have an ordered set of cliques * Then the set of cliques has the running intersection property if, for each i, all the nodes of C i that are in any earlier cliques are all members of one particular earlier clique; i.e., i 2, j < i : C C... C C ( ) i 1 i-1 j C i March 2, 2011 Dr. Peter R. Gillett 29

30 Lauritzen & Spiegelhalter Rules for propagation * Inward pass Each node waits to send its message to its inward neighbor until it has received messages from all its outward neighbors When a node is ready to send its message to its inward neighbor, it computes the message by marginalizing its current potential to its intersection with its inward neighbor; it sends this marginal to the inward neighbor, and then divides its own current potential by it When a node receives a message from its outward neighbor, it replaces its own current potential with the product of that potential and the message March 2, 2011 Dr. Peter R. Gillett 30

31 Lauritzen & Spiegelhalter Rules for propagation * Outward pass Each node waits to send its message to its outward neighbors until it has received the message from its inward neighbor When a node is ready to send its message to its outward neighbor, it computes the message by marginalizing its current potential to its intersection with its outward neighbor; it sends this marginal to the outward neighbor When a node receives a message from its inward neighbor, it replaces its own current potential with the product of that potential and the message March 2, 2011 Dr. Peter R. Gillett 31

32 Example Current Topics in Accounting Research Lauritzen & Spiegelhalter * Bayesian Network for student behavioral problems Overworked Insomnia Tired Cross ( o) = ( i) = ( toi) = ( to i) = ( t oi) = ( t o i) = ( coi) ( co i) ( c oi) ( c o i) P 0.2; P 0.01; P, 0.98; P, 0.8; P, 0.9; P, 0.05; P, = 0.96; P, = 0.9; P, = 0.6; P, = 0.01 March 2, 2011 Dr. Peter R. Gillett 32

33 Lauritzen & Spiegelhalter Moral graph Overworked Insomnia Tired Cross March 2, 2011 Dr. Peter R. Gillett 33

34 Lauritzen & Spiegelhalter This is clearly triangulated... Overworked Insomnia Tired Cross March 2, 2011 Dr. Peter R. Gillett 34

35 Lauritzen & Spiegelhalter Maximum cardinality 1 Overworked 2 Insomnia 3 Tired 4 Cross March 2, 2011 Dr. Peter R. Gillett 35

36 Lauritzen & Spiegelhalter No fill-ins needed because { } { } bd( 4) 1, 2, 3 = 1, 2 { } { } bd(3) 1, 2 = 1, 2 { } { } bd(2) 1 = 1 { } { } bd(1) = March 2, 2011 Dr. Peter R. Gillett 36

37 Lauritzen & Spiegelhalter Cliques ordered by highest numbered node: 1, 2, 3, 1, 2, 4 = OIT,,, OIC,, { } { } { } { } Since there are only two cliques, they vacuously have the running intersection property! i Clique C i Residual R i Separator S i Parent(s) 1 {O,I,T} {O,I,T} 2 {O,I,C} {C} {O,I} 1 March 2, 2011 Dr. Peter R. Gillett 37

38 Assignment 4 - Question 2 {O,I,C} {O,I,T} It would be natural to take {O,I,T} as root, but since this is a tree I can in fact choose any node, and I have chosen {O,I,C} as root in the next slide! March 2, 2011 Dr. Peter R. Gillett 38

39 Lauritzen & Spiegelhalter March 2, 2011 Dr. Peter R. Gillett 39

40 Aalborg Architecture Building the Junction Tree * Construct a tree of cliques with the running intersection property as for the Lauritzen & Spiegelhalter architecture Note that more efficient techniques for fill-in such as a onestep look-ahead may be used instead of maximum cardinality Alternatively, a tree of cliques may be constructed by building a Join Tree (Markov Tree) using the Shenoy-Shafer architecture, and then dropping all nodes that are subsets of adjacent nodes * Place a separator between each two adjacent cliques containing their intersection * This is, of course, a join tree; arbitrarily select a root March 2, 2011 Dr. Peter R. Gillett 40

41 Aalborg Architecture Rules for propagation * If separators initially contain tables of 1s, then for both the Inward pass and Outward pass: Each non-root node waits to send a message to a given neighbor until it has received messages from all its other neighbors The root waits to send messages to its neighbors until it has received messages from them all When a node is ready to send its message to a particular neighbor, it computes the message by marginalizing its current potential to its intersection with this neighbor, and then sends the message to the separator between it and the neighbor March 2, 2011 Dr. Peter R. Gillett 41

42 Aalborg Architecture Rules for propagation * If separators initially contain tables of 1s, then for both the Inward pass and Outward pass: When a separator receives a message from one of its two nodes, it divides the message by its current potential, sends the quotient on to the other node, and replaces its current potential with the new message When a node receives a message, it replaces its current potential with the product of that potential and the message March 2, 2011 Dr. Peter R. Gillett 42

43 Aalborg Architecture Junction Tree {O,I,C} oi, o, i oi, o, i {O,I} oi, 1 o, i 1 oi, 1 o, i 1 {O,I,T} oi, 0.98 o, i 0.80 oi, 0.90 o, i 0.05 March 2, 2011 Dr. Peter R. Gillett 43

44 Aalborg Architecture Inward Pass oi, 0.98 o, i 0.80 oi, 0.90 o, i 0.05 oi, 0.98 o, i 0.80 oi, 0.90 o, i 0.05 {O,I,C} {O,I} {O,I,T} oi, = o, i = oi, = o, i = oi, 0.98 o, i 0.80 oi, 0.90 o, i 0.05 oi, 0.98 o, i 0.80 oi, 0.90 o, i 0.05 March 2, 2011 Dr. Peter R. Gillett 44

45 Aalborg Architecture Outward Pass oi, o, i oi, o, i oi, o, i oi, o, i {O,I,C} {O,I} {O,I,T} oi, o, i oi, o, i oi, o, i oi, o, i oi, = o, i = oi, = o, i = March 2, 2011 Dr. Peter R. Gillett 45

46 Aalborg Architecture Hence, after normalization: P ( o ) = = and: P ( ) ( ) ( ) ( ) ( i ) = = March 2, 2011 Dr. Peter R. Gillett 46

47 Aalborg Architecture The Aalborg architecture has been implemented in Bayesian Network software called HUGIN Download a demonstration version from hugin-lite/200-hugin-lite-windows Recompute this example using HUGIN Add to the HUGIN model the observation that the student suffers from Insomnia, and examine what happens to the posterior probability of the student being Overworked as a result March 2, 2011 Dr. Peter R. Gillett 47

48 Textbooks Current Topics in Accounting Research Aalborg Architecture * Introduction to Bayesian Networks Finn V. Jensen * Probabilistic Networks and Expert Systems Robert G. Cowell, Steffen L. Lauritzen, David J. Spiegelhater * Expert Systems and Probabilistic Network Models Enrique Castillo, Jose M. Gutierrez, Ali S. Hadi March 2, 2011 Dr. Peter R. Gillett 48

49 Shenoy-Shafer Architecture Building the Join Tree * Detailed algorithms were provided in the readings for Shenoy-Shafer Join Trees * Later work has proposed a refinement that is more efficient Binary Join Trees * The Shenoy-Shafer algorithm was designed to work with both probabilities and belief functions * The axioms are fairly general, and the algorithm has been found to apply to many other valuations, such as Possibilites and Spohn s Epistemic (Dis)beliefs March 2, 2011 Dr. Peter R. Gillett 49

50 Shenoy-Shafer Architecture Rules for propagation * There is no distinction between inward and outward passes Each node waits to send its message to a given neighbor until it has received messages from all its other neighbors When a node is ready to send its message to a particular neighbor, it computes the message by collecting all its messages from other neighbors, multiplying its own potential by these messages, and marginalizing the product to its intersection with the neighbor to which it is sending March 2, 2011 Dr. Peter R. Gillett 50

51 Shenoy-Shafer Architecture Let s return to our example of cross, tired doctoral students but, now adding the knowledge that the student suffers from Insomnia March 2, 2011 Dr. Peter R. Gillett 51

52 Shenoy-Shafer Architecture Join Tree {O} {C} {O,I,C} {O,I} {O,I,T} {T} {I} March 2, 2011 Dr. Peter R. Gillett 52

53 Shenoy-Shafer Architecture Initially o.2 o.8 {O} oit,,.98 oi,, t.02 o, it,.80 o, i, t.20 oit,,.90 oi,, t.10 o, it,.05 o, i, t.95 c 1 c 0 {C} {O,I,C} {O,I} {O,I,T} {T} t t 1 0 oic,,.96 oi,, c.04 o, ic,.90 o, i, c.10 oic,,.60 oi,, c.40 o, ic,.01 o, i, c.99 i.01 i.99 {I} i i 1 0 March 2, 2011 Dr. Peter R. Gillett 53

54 Shenoy-Shafer Architecture Propagation o.2 c 1 c 0 o.8 oi,.96 o, i.90 oi,.60 o, i.01 {O} o.2 o.8 oit,,.98 oi,, t.02 o, it,.80 o, i, t.20 oit,,.90 oi,, t.10 o, it,.05 o, i, t.95 c 1 c 0 {C} oic,,.96 oi,, c.04 o, ic,.90 o, i, c.10 oic,,.60 oi,, c.40 o, ic,.01 o, i, c.99 {O,I,C} i i i.01 i {O,I} {I} oi,.98 o, i.80 oi,.90 o, i.05 i 1 i 0 {O,I,T} t t 1 0 {T} t t 1 0 March 2, 2011 Dr. Peter R. Gillett 54

55 Shenoy-Shafer Architecture Propagation o.2 c 1 c 0 o.8 oi,.96 o, i.90 oi,.60 o, i.01 {O} o.2 o.8 oit,,.98 oi,, t.02 o, it,.80 o, i, t.20 oit,,.90 oi,, t.10 o, it,.05 o, i, t.95 c 1 c 0 oic,,.96 oi,, c.04 o, ic,.90 o, i, c.10 oic,,.60 oi,, c.40 o, ic,.01 o, i, c.99 {C} {O,I,C} i i i.01 i.99 {O,I} {I} oi,.98 o, i.80 oi,.90 o, i.05 i 1 i 0 {O,I,T} t t 1 0 {T} t t 1 0 March 2, 2011 Dr. Peter R. Gillett 55

56 Shenoy-Shafer Architecture Propagation o.2 c c o.8 oi, o, i oi, o, i {O} o.2 o.8 oit,,.98 oi,, t.02 o, it,.80 o, i, t.20 oit,,.90 oi,, t.10 o, it,.05 o, i, t.95 c 1 c 0 {C} oic,,.96 oi,, c.04 o, ic,.90 o, i, c.10 oic,,.60 oi,, c.40 o, ic,.01 o, i, c.99 {O,I,C} i i i.01 i {O,I} {I} oi,.98 o, i.80 oi,.90 o, i.05 i 1 i 0 {O,I,T} t t 1 0 {T} t t 1 0 March 2, 2011 Dr. Peter R. Gillett 56

57 Shenoy-Shafer Architecture Propagation o.2 c 1 c 0 o.8 oi,.96 o, i.90 oi,.60 o, i.01 {O} o.2 o.8 oit,,.98 oi,, t.02 o, it,.80 o, i, t.20 oit,,.90 oi,, t.10 o, it,.05 o, i, t.95 c 1 c 0 {C} {O,I,C} {O,I} {O,I,T} {T} t t 1 0 oic,,.96 oi,, c.04 o, ic,.90 o, i, c.10 oic,,.60 oi,, c.40 o, ic,.01 o, i, c.99 i i i.01 i {I} oi, o, i oi, o, i i 1 i 0 t t March 2, 2011 Dr. Peter R. Gillett 57

58 Shenoy-Shafer Architecture Propagation o.2 c 1 c 0 o.8 oi,.96 o, i.90 oi,.60 o, i.01 {O} o =.9408 o =.5400 oit,,.98 oi,, t.02 o, it,.80 o, i, t.20 oit,,.90 oi,, t.10 o, it,.05 o, i, t.95 c 1 c 0 {C} {O,I,C} {O,I} {O,I,T} {T} t t 1 0 oic,,.96 oi,, c.04 o, ic,.90 o, i, c.10 oic,,.60 oi,, c.40 o, ic,.01 o, i, c.99 i i i.01 i {I} oi,.98 o, i.80 oi,.90 o, i.05 i 1 i 0 t t 1 0 March 2, 2011 Dr. Peter R. Gillett 58

59 Shenoy-Shafer Architecture Hence the posterior for O is given by o = ~ o March 2, 2011 Dr. Peter R. Gillett 59

60 Chapter 3 Current Topics in Accounting Research A Comparative Study... * Introduces Shenoy s Valuation-Based Systems * Defines Variables Configurations Valuations Marginalization Combination * Introduces Valuation Networks March 2, 2011 Dr. Peter R. Gillett 60

61 Chapter 3 Current Topics in Accounting Research A Comparative Study... * Axioms for Local Computation Order of deletion does not matter Combination is commutative and associative Marginalization distributes over combination * Fusion and Computation Algorithms * Join Tree construction * Relationship with Auditing research * Illustrative models March 2, 2011 Dr. Peter R. Gillett 61

62 A Comparative Study... Chapter 4 Probability Theory * Introduction * Probability as VBS * AND nodes * Discounted AND nodes * Independent and non-independent evidence * Hierarchical aggregation * Example March 2, 2011 Dr. Peter R. Gillett 62

63 A Comparative Study... Chapter 5 Belief Functions * Introduction * Belief Functions as VBS * AND nodes * Discounted AND nodes * Independent and non-independent evidence * Statistical Audit Evidence * Hierarchical aggregation * Example March 2, 2011 Dr. Peter R. Gillett 63

64 A Comparative Study... Chapter 6 Spohn s Epistemic Beliefs * Introduction * Spohn s Epistemic Beliefs as VBS * AND nodes * No Discounted AND nodes! * Independent and non-independent evidence * Statistical Audit Evidence undesirable? * Hierarchical aggregation * Example March 2, 2011 Dr. Peter R. Gillett 64

65 A Comparative Study... Chapter 7 Possibility Theory * Introduction * Possibility Theory as VBS * AND nodes * No Discounted AND nodes * Independent and non-independent evidence * Statistical Audit Evidence undesirable? * Hierarchical aggregation * Example March 2, 2011 Dr. Peter R. Gillett 65

66 A Comparative Study... Chapter 8 The Decision problem * Compares VN Instantiations *... March 2, 2011 Dr. Peter R. Gillett 66

67 Chapter 9 A Comparative Study... * Not in readings * Discusses knowledge elicitation difficulties for a comparative study, and implements proposed solutions March 2, 2011 Dr. Peter R. Gillett 67

68 Possibility Theory Possibility Theory is a computational implementation (largely inspired by Dubois and Prade) of Zadeh s Fuzzy Logic Shenoy showed that it may be re-formulated as a variant of Valuation Networks, so that possibilities may be propagated in Join Trees using the Shenoy-Shafer algorithm March 2, 2011 Dr. Peter R. Gillett 68

69 Possibility Theory A possibility function is a function such that: and Ω p [ ] π: 2 0,1 x Ω : π x = 1 p ( ) Ω 2 : ( ) max{ ( ) } p a π a = π x x a March 2, 2011 Dr. Peter R. Gillett 69

70 Possibility Theory By virtue of this second condition, possibility functions are completely determined by their values for singleton elements Intuitively, the degree of possibility for a subset a is 1 π( ~a) and the degree of impossibility is 1 π( a) March 2, 2011 Dr. Peter R. Gillett 70

71 Marginalization Combination where { X} Current Topics in Accounting Research Possibility Theory a { X } ( ) ( ) { } y Ω : π y = max π yx, x Ω a π1 π 2 = K π x π x K 1 ( a) ( b ) ( ) 1 2 x 0 K = 0 max{ ( a ) ( ) } b K = π1 x π2 x x Ωa b 0 X March 2, 2011 Dr. Peter R. Gillett 71

72 Possibility Theory Example: Marginalization Objective 1 Objective 2 Possibility true true 1 true false.5 false true.3 false false.1 March 2, 2011 Dr. Peter R. Gillett 72

73 Possibility Theory Example: Marginalization Objective 1 Possibility true 1 false.3 Objective 2 Possibility true 1 false.5 March 2, 2011 Dr. Peter R. Gillett 73

74 Possibility Theory So for Objective 1, the degree of possibility for true is 0.7, and the degree of impossibility is 0 Similarly, for Objective 2, the degree of possibility for true is 0.5, and the degree of impossibility is 0 March 2, 2011 Dr. Peter R. Gillett 74

75 Possibility Theory Example: Combination Objective 1 Possibility true 1 false.4 Objective 1 Possibility true 1 false.6 March 2, 2011 Dr. Peter R. Gillett 75

76 Possibility Theory Example: Combination Objective 1 Possibility true 1 false.24 March 2, 2011 Dr. Peter R. Gillett 76

77 Possibility Theory Example: Combination Objective 1 Possibility true 1 false.4 Objective 1 Possibility true.6 false 1 March 2, 2011 Dr. Peter R. Gillett 77

78 Possibility Theory Example: Combination Objective 1 Possibility true.6 false.4 Objective 1 Possibility true 1 false.67 March 2, 2011 Dr. Peter R. Gillett 78

79 Possibility Theory We can represent complete ignorance using possibilities as follows: Objective 1 Objective 2 Possibility true true 1 true false 1 false true 1 false false 1 March 2, 2011 Dr. Peter R. Gillett 79

80 Possibility Theory We can define logical relationships such as AND nodes as follows: A= B&C: A B C Possibility a b c 1 a b ~c 0 a ~b c 0 a ~b ~c 0 ~a b c 0 ~a b ~c 1 ~a ~b c 1 ~a ~b ~c 1 March 2, 2011 Dr. Peter R. Gillett 80

81 Possibility Theory Discounted AND nodes as discussed for probabilities and belief functions CANNOT be defined Dubois and Prade argue that statistical sampling is contrary to the non-probabilistic nature of possibility theory: however, it seems that it could be incorporated using normalized maximum likelihood functions March 2, 2011 Dr. Peter R. Gillett 81

82 Possibility Theory Joint possibilities generally cannot be uniquely determined from marginals, as for probabilities and belief functions: X Y P1 P2 P3 x y x ~y ~x y ~x ~y have the same marginals for X and Y. However, when combined with an AND node, all three produce the same results!!! March 2, 2011 Dr. Peter R. Gillett 82

83 Possibility Theory When multiple nodes are combined in an AND node, the effect is that the degree of possibility for the conjunction is equal to the degree of possibility for the least possible conjunct This is significantly different from probabilities and belief functions, and is very relevant, for example, to auditing March 2, 2011 Dr. Peter R. Gillett 83

84 Spohn s Epistemic Calculus Spohn introduced his theory of epistemic states in order to represent plain human beliefs in a non-probabilistic way easily amenable to revision Initially they were based on functions mapping into the ordinals; later he changed this to the natural numbers March 2, 2011 Dr. Peter R. Gillett 84

85 Spohn s Epistemic Calculus For technical reasons, we will define disbeliefs as functions mapping to the natural numbers extended by adding a representation of infinity, March 2, 2011 Dr. Peter R. Gillett 85

86 Spohn s Epistemic Calculus A disbelief function is a function δ:2 d N such that: and Ω + x Ω : π x = 0 p ( ) a δ a = π x x a Ω 2 : ( ) min{ ( ) } d March 2, 2011 Dr. Peter R. Gillett 86

87 Spohn s Epistemic Calculus By virtue of this second condition, disbelief functions are completely determined by their values for singleton elements Intuitively, the degree of disbelief for a subset a is δ( a) and the degree of belief is δ( ~ a) March 2, 2011 Dr. Peter R. Gillett 87

88 Spohn s Epistemic Calculus Marginalization { X} a Combination { X } ( ) ( ) { } y Ω : δ y = min δ yx, x Ω a δ δ x =δ x +δ x K ( ) ( a) ( b) X where min{ ( a ) ( ) } b 1 2 a b K = δ x +δ x x Ω March 2, 2011 Dr. Peter R. Gillett 88

89 Spohn s Epistemic Calculus Example: Marginalization Objective 1 Objective 2 Disbelief true true 0 true false 5 false true 7 false false 9 March 2, 2011 Dr. Peter R. Gillett 89

90 Spohn s Epistemic Calculus Example: Marginalization Objective 1 Disbelief true 0 false 7 Objective 2 Disbelief true 0 false 5 March 2, 2011 Dr. Peter R. Gillett 90

91 Spohn s Epistemic Calculus Example: Combination Objective 1 Disbelief true 0 false 6 Objective 1 Disbelief true 0 false 4 March 2, 2011 Dr. Peter R. Gillett 91

92 Spohn s Epistemic Calculus Example: Combination Objective 1 Disbelief true 0 false 10 March 2, 2011 Dr. Peter R. Gillett 92

93 Spohn s Epistemic Calculus Example: Combination Objective 1 Disbelief true 0 false 6 Objective 1 Disbelief true 4 false 0 March 2, 2011 Dr. Peter R. Gillett 93

94 Spohn s Epistemic Calculus Example: Combination Objective 1 Disbelief true 4 false 6 Objective 1 Disbelief true 0 false 2 March 2, 2011 Dr. Peter R. Gillett 94

95 Spohn s Epistemic Calculus We can represent complete ignorance using disbeliefs as follows: Objective 1 Objective 2 Disbelief true true 0 true false 0 false true 0 false false 0 March 2, 2011 Dr. Peter R. Gillett 95

96 Spohn s Epistemic Calculus We can define logical relationships such as AND nodes as follows: A= B&C: A B C Disbelief a b c 0 a b ~c a ~b c a ~b ~c ~a b c ~a b ~c 0 ~a ~b c 0 ~a ~b ~c 0 March 2, 2011 Dr. Peter R. Gillett 96

97 Spohn s Epistemic Calculus Discounted AND nodes as discussed for probabilities and belief functions CANNOT be defined Statistical sampling is contrary to the intended ordinal nature of epistemic (dis)beliefs March 2, 2011 Dr. Peter R. Gillett 97

98 Spohn s Epistemic Calculus Joint disbeliefs generally cannot be uniquely determined from marginals, as for probabilities and belief functions: X Y P1 P2 P3 x y x ~y ~x y ~x ~y have the same marginals for X and Y. However, when combined with an AND node, all three produce the same results!!! March 2, 2011 Dr. Peter R. Gillett 98

99 Spohn s Epistemic Calculus When multiple nodes are combined in an AND node, the effect is that the degree of belief for the conjunction is equal to the degree of belief for the least believed conjunct This is significantly different from probabilities and belief functions, and is very relevant, for example, to auditing March 2, 2011 Dr. Peter R. Gillett 99

100 Spohn s Epistemic Calculus In addition, Spohn s system offers the prospect of elicitation of ordinal rankings rather than highly sensitive real values March 2, 2011 Dr. Peter R. Gillett 100

101 Homework 5 Homework 5 is a set of simple exercises using your knowledge of valuation networks The important thing is not the answers, but your workings and whether you understand the manipulations you are carrying out so by all means get help, but do the work yourself On this occasion, the actual answers are all in the readings, so it is the workings that count! Post your solutions as a WORD document on Blackboard as usual You will also need HUGIN March 2, 2011 Dr. Peter R. Gillett 101