We view a constructivist and model-revising epistemology
as a rapprochement between the empiricist, rationalist, and
pragmatist viewpoints. The constructivist hypothesizes that
all human understanding is the result of an interaction
between energy patterns in the world and mental categories
imposed on the world by an intelligent agent
Constructivists use the term schemata to describe the a priori structure used to mediate the experience of the external world. The term schemata is taken from the writing of the British psychologist Bartlett (1932) and its philosophical roots go back to Kant (1781/1964). On this viewpoint observation is not passive and neutral but active and interpretative. There are many current psychologists and philosophers that support and expand this pragmatic and teleological account of human developmental activity (Glymour 2001, Gopnik et al. 2010, Gopnik 2011a, 2011b, Kushnir et al. 2010).
Perceived information, Kant’s a posteriori knowledge, rarely fits precisely into our preconceived and a priori schemata. From this tension to comprehend and act as an agent, the schema-based biases a subject uses to organize experience are strengthened, modified, or replaced. This accommodation in the context of unsuccessful interactions with the environment drives a process of cognitive equilibration. The constructivist epistemology is one of cognitive evolution and continuous model refinement. An important consequence of constructivism is that the interpretation of any perception-based situation involves the imposition of the observers (biased) concepts and categories on what is perceived. This constitutes an inductive bias (Luger 2009, Ch 16).
When
There is a blending here of empiricist and rationalist traditions, mediated by the pragmatist requirement of agent survival. As embodied, agents can comprehend nothing except that which first passes through their senses. As accommodating, agents survive through learning the general patterns of an external world. What is perceived is mediated by what is expected; what is expected is influenced by what is perceived: these two functions can only be understood in terms of each other. A Bayesian model-refinement representation offers an appropriate model for critical components of this constructivist modelrevising epistemological stance (Luger et al. 2002, Luger 2012). Interestingly enough, David Hume acknowledged the epistemic foundation of all human activity (including, of course, the construction of AI artifacts) in A Treatise on Human Nature (1739/2000) when he stated “All the sciences have a relation, greater or less, to human nature; and ... however wide any of them may seem to run from it, they still return back by one passage or another. Even Mathematics, Natural Philosophy, and Natural Religion, are in some measure dependent on the science of MAN; since they lie under the cognizance of men, and are judged of by their powers and faculties.”
Thus, we can ask why a constructivist epistemology might be useful in addressing the problem of building programs that are “intelligent”. How can an agent within an environment understand its own understanding of that situation? We believe that constructivism also addresses this problem of epistemological access. For more than a century there has been a struggle in both philosophy and psychology between two factions: the positivist, who proposes to infer mental phenomena from observable € physical behavior, and a more phenomenological approach which allows the use of first person reporting to enable access to cognitive phenomena. This factionalism exists because both modes of access to cognitive phenomena require some form of model construction and inference.
In comparison to physical objects like chairs and doors, which often, naively, seem to be directly accessible, the mental states and dispositions of an agent seem to be particularly difficult to characterize. We contend that this dichotomy between direct access to physical phenomena and indirect access to mental phenomena is illusory. The constructivist analysis suggests that no experience of the external (or internal) world is possible without the use of some model or schema for organizing that experience. In scientific enquiry, as well as in our normal human cognitive experiences, this implies that all access to phenomena is through exploration, approximation, and continued model refinement.
Bayes theorem (1763) offers a plausible model of this constructivist rapprochement between the philosophical traditions we have just discussed. It is also an important modeling tool for much of modern AI, including AI programs for natural language understanding, robotics, and machine learning. With a high-level discussion of Bayes’ insights, we can describe the power of this approach.
Consider the general form of Bayes’ relationship used to determine the probability of a particular hypothesis, hi, given a set of evidence E: p(hi | E) = n
p(E | hi )p(hi ) ∑p(E | hk )p(hk ) k=1 p(hi|E) is the probability that a particular hypothesis, hi, is true given evidence E. p(hi) is the probability that hi is true overall. p(E|hi) is the probability of observing evidence E when hi is true. n is the number of possible hypotheses.
With the general form of Bayes’ theorem we have a functional (and computational!) description (model) for a particular situation happening given a set of perceptual evidence clues. Epistemologically, we have created on the right hand size of the equation a schema describing how prior accumulated knowledge of occurrences of phenomena can relate to the interpretation of a new situation, the left hand side of the equation. This relationship can be seen as an example of Piaget’s assimilation where encountered information fits (is interpreted by) the patterns created from prior experiences.
To describe further the pieces of Bayes formula: The probability of an hypothesis being true, given a set of evidence, is equal to the probability that the evidence is true given the hypothesis times the probability that the hypothesis occurs. This number is divided (normalized) by the probability of the evidence itself, p(E). This probability of evidence is represented as the sum over all hypotheses presenting the evidence times the probability of that hypothesis itself.
There are limitations to using Bayes’ theorem as just presented as an epistemological characterization of the phenomenon of interpreting new (a posteriori) data in the context of (prior) collected knowledge and experience. First, of course, is the fact that the epistemological subject is not a calculating machine. We simply don’t have all the prior (numerical) values for all the hypotheses and evidence that can fit a problem situation. In a complex situation such as medicine where there can be hundreds of hypothesized diseases and thousands of symptoms, this calculation is intractable (Luger 2009, Chapter 5).
A second objection is that in most situations the sets of evidence are NOT independent, given the set of hypotheses. This makes the calculation of p(E) in the denominator of Bayes rule as just presented unjustified. When this independence assumption is simply ignored, as we see shortly, the result is called naïve Bayes. More often, however, the rationalization of the probability of the occurrence of evidence across all hypotheses is seen as simply a normalizing factor, supporting the calculation of a realistic measure for the probability of the hypothesis given the evidence (the left side of Bayes’ equation). The same normalizing factor is utilized in determining the actual probability of any of the hi, given the evidence, and thus, as in most natural language processing applications, is usually ignored.
A final objection asserts that diagnostic reasoning is not about the calculation of probabilities; it is about determining the most likely explanation, given the accumulation of pieces of evidence. Humans are not doing real-time complex mathematical processing; rather we are looking for the most coherent explanation or possible hypothesis, given the amassed data. Thus, a much more intuitive form of Bayes rule ignores this p(E ) denominator entirely (as well as the associated assumption of evidence independence). The resulting formula determines the likelihood of any hypothesis given the evidence, as the product of the probability of the evidence given the hypothesis times the probability of the hypothesis itself p(E|hi) p(hi). In most diagnostic situations we are asked to determine which of a set of hypotheses hi is most likely to be supported. We refer to this as determining the argmax across all the set of hypotheses. Thus, if we wish to determine which of all the hi has the most support we look for the largest p(E|hi) p(hi): argmax(hi) p(E|hi) p(hi)
In a dynamic interpretation, as sets of evidence themselves change across time, we will call this argmax of hypotheses given a set of evidence at a particular time the greatest likelihood of that hypothesis at that time. We show this relationship, an extension of the Bayesian maximum a posteriori (or MAP) estimate, as a dynamic measure over time t: We&next&illustrate&the&Bayesian&approach&in&two&application&dom discrete&component&semiconductors&(Stern&et&al.&1997,&Chakraba expecrrte,atthineg&ptrhees&egnrceeateosft&laikberliehaokodsu&fpopro&hrytspoathneusmesb&aercroosfs&expanding& alterFniagtuivree&1h,&ypproetsheensetisn. gT&twheo&fsaeialurcrhesf&oofr&dtihsceremteo&scto mlikpeolnyent&semicon expl“aonpaetino”n,&ofro&rthae&bfraeilaukre&inb&aro&wadireen&scotnhneecetviindge&nccoempseoanrechn:ts&to&others&i Howthlea&rpgreesisenthcee&obfr&eaa&kb?reIaskt&hsuerpepaonryts&dai&sncuomlobraetrio&onf&areltleatrendative&hypoth to theexpbrlaenaka?tioWne&froert&hae&fraeilaunrye&(bpreoracdepetnusa<)hseo&euvniddsenorces&mseeallrsch:&How&larg wherneliattheadp&tpoe&ntheed&?brWeahka?t&Wweerree&tthheeres&aunltyi n&(gpecrocnedpittiuoanls)&soofunds&or&sme the croemsuplotinnegn&ctsoonfdtihtieonsyss&otefm&th?e&components&of&the&system?&
interpretation of new data, given evidence E at time t, is a function of whWiche&ninetxetr&pilrleutsattrioantei&tshme&oBsatyleiksiealyn&atoppprroodacuhce&in&two&application&domains.&In&the&diagnosis&of&failures&in& that evidence discrete&cotmponent&semiconductors&(Sttheartn&et&al.&1997,&Chakrabarti&et&al.&2005)&we&have&an&example&of& at time and the probability of interpretation itcsreelaftoinccgu&trhrien&gg.reatest&likelihood&for&hypotheses&across&expanding&data&sets.&Consider&the&situation&of&
By the eaFrliygur1e9&190,&ps,resmeunctihng&otfwoc&foamilupuretast&ioofn&d-bisacsreedte&component&semiconductors.&The&failure&type&is&called&an& language unde“rosptaennd”i,n&ogr&the&break&in&a&wire&connecting&components&to&others&in&the&system.&For&the&diagnostic&expert,& and generation was stochastic, including partshineg&p, respeanrtc-eo&fo-sf&pae&becrheak&tsaugpgipnogr,ts&are&nfeurmenbceer&of&alternative&hypotheses.&The&search&for&the&most&likely& resolution, andexdpislacnouartsioenp&froorc&aes&fsaiinlgu,reu&bsuraolalydeunssi&ntghet&oeovlisdence&search:&How&large&is&the&break?&Is&there&any&discoloration& related&to&the&break?&Were&there&any&(perceptual)&sounds&or&smells&when&it&happened?&What&were&the&& l2i0k0e9)g.reOattehsetr lraiekrseeulaislhtoinoogfd&caomrnteidfaiisctuiiaorelnssi&no(tJfeu&tlrlhaigefs&ecknoycme,apnoednspeMenctaisar<oliynf&the&system& ?& & & (a)a.&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&b.& wmaaycshintehelseearnuisnegs, boefcasmtoechmaosrtiec Btaeycehsnioanlo-gbyasefdo.rInpamttaenrny “Fcigounrnee&c1t.i&Tonw&bo&reoxkaemnp”&lfeasil&uofr&ed.i&screte&component&semiconductor recognition were another instantiation of the constructivist & tradition, as collected sets of patterns were used to Driven&by&the&data&search&supporting&multiple&possible&hypothes condition recognition of new patterns. notes&the&bambooing&effect&in&the&disconnected&wire,&Figure&1a.&T
Judea Pearl’s (1988) proposal for use of Bayesian belief likelihood&hypothesis&that&explains&the&open&as&a&break&created&b
nets (BBNs) and his assumption of their links reflecting caused&by&a&sequence&of&lowSfrequency&highScurrent&pulses.&The&
“causal” relationships (Pearl 2000) brought the use of the&example&of&Figure&1b,&where&the&break&is&seen&as&balled,&is&me
Bayesian technology to an entirely new importance. First, these&diagnostic&scenarios&have&been&implemented&by&an&expert&
the assumption of these networks being directed graphs – hypothesis&space&
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We next illilkusetlriahtoeodth&heypBotahyeessiiasn&thaatp&epxropalachinsi&tnhet&owpoen&as&a&brDeTraihkvi&ecsnr&ceubarytreetdhn&ebt&yed&xmaataemtsapella&ecr&rcdyhesmtsauolplnipzsoatrrttiaiontnegs&t&mhhaoutwl&twi&ptahlese&&lpmikooesslsyitb&&lliekely¤t& application domains. In the diagnosis of failures in discrete
caused&by&a&sequence&of&lowSfrequency&highScurrheynptob&ptehuseltss&eeesxsp.&tTlhahantea&cgtaironenae,t&gxeipsvtlea&linink&aet&lhpyea&hr“ytoipcpouetlnha”re,&stiitsmh&efeo&erax&ntphdee&ra&tonpn&heoyntep&osoft&hesis&spac component semt hiceo&enxdaumctporles&o(Sf&tFeirgnuerte&a1l.b,1& w99h7e,rCe&hthake&rbarbeaartki&is&seenth&aessb&beaatmsll&beoodfo&,h&iinys&gpmoeetfhlfteeincstegs&idn&auntehd&et&odd&aeitsxacc&oaencsnrseoivcsetse&&cdtiumwrreier,&enu,ts.F&iBnigogu&tthrhe&eo1&fm&a.ost&likely&hy et al. 2005) wtehheasev&ediaangneoxsatmicp&sleceonfarciroesa&thinagvet&hbeeegnr e&iamtepsltementeTdh&ibslyiks&uaegnlig&heeoxsoptsder&aht&yrsepyvositstehemdesSgilsrike.&aet&essetarlickhe<ihroooudghhy&apno&thesis that likelihood forhyhpyoptohtheessise&sspaaccreo&(sSsteerxnp&eatn&adli.n&1g99d7a)ta&as&well&as&refelxecptlaeidn&sin&at&hBeayeospieann&beaslief&anetb&(rCeahkakrcarbeartetdi&et&bayl.&20m0e5ta).l&& sets.
Consider the siFtuigautiroen&2o& pfrFeisgeunrtes&1a,&Bpareyseesnitainng&betwlieof&fnaeiltu&(rBesBN)&capctruysrgtianll(glhi&zti haEteitso)en=&athanardtg&owmtahasexlr(i&khrei)lyaptce(aEdu&td|sheiadi)gbpny(ohasitsi&ce&qsuiteunacteioonfsl.o&
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of discrete component semiconductors. The failure type is frequency high-current pulses. The greatest likely
called an “opTenh”e,&BoBrN,t&hweithboreuatk&neiwn&daatwa,i&rreepcroesnennectsti&nthge&a&priohryip&sottahtees&iosf&faonrðxepoerpte’ns&konf othweleexdagme&polfe&aonf&aFpigpulirceat1ibo,nw&here
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working&with&human&experts’&analysis&of&known&failures.&Thus,&the&BBN&can&be&said&to&capture&a&priori&
expert&knowledge&implicit&in&a&domain&of&interest.&When&new&(a&posteriori)&data&are&given&to&the&BBN,&
e.g.,&the&wire&is&“bambooed”,&the&color&of&the&copper&wire&is&normal,&etc,&the&belief&network&“infers”&the&
most&likely&explanation,&within&its&(a&priori)&model,&given&this&new&information.&There&are&many&inference&
rules&for&doing&this&(Luger&2009,&Chapter&9).&An&important&result&of&using&the&BBN&technology&is&that&as&
one&hypothesis&achieves&its&greatest&likelihood,&other&related&hypotheses&are&“explained&away”,&i.e.,&their&
likelihood&measures&decrease&within&the&BBN.&
current. Both of these diagnostic scenarios have been
implemented by an expert system-like search through an
hypothesis space
The BBN, without new data, represents the a priori state of an expert’s knowledge of an application domain. In fact, these networks of causal relationships are usually carefully crafted through many hours working with human experts’ analysis of known failures. Thus, the BBN can be said to capture a priori expert knowledge implicit in a domain of interest. When new (a posteriori) data are given to the BBN, e.g., the wire is “bambooed”, the color of the copper wire is normal, etc, the belief network “infers” the most likely explanation, within its (a priori) model, given this new information. There are many inference rules for doing this (Luger 2009, Chapter 9). An important result of using the BBN technology is that as one hypothesis achieves its greatest likelihood, other related hypotheses are “explained away”, i.e., their likelihood measures decrease within the BBN.
This current example demonstrates how the most likely current hypothesis can be used to determine the best explanation, given a particular time and an hypothesis space. We next demonstrate how considering sets of hypotheses and data across time, using the most likely hypothesis at time t, can produce a greatest likelihood hypothesis. gl(hi|Et) = argmax(hi) p(Et|hi) p(hi)
In this model the most likely interpretation of new data, given evidence E at time t, is a function of which interpretation is most likely to produce that evidence at time t and the probability of that interpretation itself occurring. If we want to expand this to the next time period, t + 1, we need to describe how models can evolve across time.
As an example of argmax processing, Chakrabarti et al. (2005, 2007) analyze a continuous data stream from a set of distributed sensors. The running “health” of the transmission of a Navy helicopter rotor system is represented by a steady stream of sensor data. This data consists of temperature, vibration, pressure, and other measurements reflecting the state of the various components of the running transmission system. An example of this data can be seen in the top portion of Figure 3, where the continuous data stream is broken into discrete and partial time slices.
A Fourier transform is then used to translate these signals into the frequency domain, as shown on the left side of the second row of Figure 3. These frequency readings were compared across time periods to diagnose the running health of the rotor system. The model used to diagnose rotor health the auto-regressive hidden Markov model (A-RHMM) of Figure 4. The observable states of the system are made up of the sequences of the segmented signals in the frequency domain while the hidden states are the imputed health states of the helicopter rotor system itself, as seen in the lower right of Figure 3.
The hidden Markov model (HMM) technology is an important stochastic technique that can be seen as a variant of a dynamic BBN. In the HMM, we attribute values to states of the network that are themselves not directly observable. For example, the HMM technique is widely used in the computer analysis of human speech, trying to determine the most likely word uttered, given a stream of acoustic signals (Jurasky and Martin 2009). In our helicopter example, training this system on streams of normal transmission data allowed the system to make the correct greatest likelihood measure of failure when these signals change to indicate a possible breakdown. The US Navy supplied data to train the normal running system as wIn&tehilsl&moddeal&tthae&mosset<iksely&interpretation&of&new&data,&ngisven&etvhidaetnce&E&at&time&t,&is&a&funcstieone&odf&ewhdich&
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Further, we contend that the greatest likelihood calculation trFanosiustorrise,&orr&ttinrkaenr&tsoyfso&S&rams&weoll&afs&ttoh&thee<animguaegess&ulsiecd&eto&mdaaktea&it&r(utnr.&Iat&nsimspfloy&rremqueiredd)&thie&nretsopoWnsehs&en&presented&with&novel&information&characterizing&a&new&diagnostic&situation,&th diagnostic reasoning. This includes the computation of a greatest likelihood measure of hypotheses, given new Modern&AI&research&has&proposed&probabilistic&representations&and&algorithms&for&the&realStime&integration& is cognitively plausible and offers an epistemological model,&until&it&finds&convergence&or&equilibrium,&in&the&form&of&a&particular&greatest&li
framework for understanding the phenomena of human 1968).&Among&these&algorithms&is&loopy,belief,propagation&(Pearl&1988,&2000)&that&captures&a&system&of&
The claim of this paper is that stochastic methods offer a sufficient account of hu plausible&beliefs&constantly&iterating&towards&equilibrium,&or&equilibration,&as&Piaget&might&describe&it.&A& When&presented&with&novel&information&characterizing&a&new&diagnostic&situation,&this&a&posteriori&data& cognitive&system&can&be&in&a,priori,equilibrium&with&its&continuing&states&of&learned&diagnostic&knowledge.&
areas such as diagnostic reasoning. This includes the computation of a greatest lik perturbs&the&equilibrium.&The&cognitive&system&then&infers,&using&prior&and&posterior&components&of&thhe&ypotheses, given new information and an expert’s a priori cognitive equilibrium. diagnostic and prognostic reasoning.
& that the greatest likelihood calculation is cognitively plausible and offers an episte The claim of this paper is that stochastic methods offer a sufficient account of human intelligence in areas such as diagnostic reasoning. This in&cludes the computation of a greatest likelihood measure foofr understanding the phenomena of human diagnostic and prognostic reasoning. O t
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