The Mathematical Sciences' Role in Homeland Security Proceedings of a Workshop (2004) / Chapter Skim
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Data Integration and Fusion
Data Integration and Fusion
Pages 457-566
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Prior to his current position, he was University Professor of Electrical, Computer and Systems Engineering at George Mason University in Fairfax, Virginia, and head of the System Architectures Laboratory of the C31 Center. For the last 20 years, Dr.
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My current position -- I have never worn a uniform in my life, but my current position is with the Air Force. I am chief scientist.
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From the discussion it is apparent that our knowledge is what we all hear from the newspapers, et cetera. I don't think even the White House has defined homeland security, let alone -- I don't know the difference between homeland defense and homeland security, by the way, but different people use them in different ways.
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You can characterize the uncertainty. We talked a lot about probability yesterday and a little bit today, but there are different kinds of uncertainty that are associated with this.
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The other one is the proactive; get them before they get a chance to get us, which is what we are doing right now, what the Air Force is actually doing. The term that the Air Force uses for this kind of a notion, that is, the last one, is predictive battle space awareness.
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A lot of our intelligence community is worrying about things like that. There are ways of trying to mark things that we eventually may want to find.
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463 don't want to solve the hard one. We need to keep this in mind.
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Since then, we have got people from the Air Force Research Laboratory, this is from Rome Lab up in upstate New York, the information directorate, and they were developing software over there which had some relation to mathematics, but it was not very explicit what the relationship between the software and the mathematics was. It was very heuristic in a way, but it did approximately the same things, but it is much more user friendly.
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This way they understood the influence nets and Bayesian networks, except for the Secretary, who does know Bayesian networks. Fortunately, I did my homework, and before I discussed those things I knew that he knew Bayesian networks, and I didn't say silly things to him.
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That is brute force, one can do it. It takes a few weeks, implement.
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Here is the alphabet soup of the intelligence community, providing the data and the updates. We are using it, responding to decision makers.
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468 is better math, because I am math limited in this area. So are the analysts.
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469 them. It is all good stuff, but it will take a long time to describe their accomplishments.
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What will happen next, so that it can be prevented In a proactive strategy, mathematics is used to create moclels and simulations of the problem. One idea, from the Air Force's Studies and Analysis group, is to use Bayesian networks to put ourselves in the minct of the adversary, and to clefine a set of effects that we then want to achieve.
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Dr. Levitt is a co-founder and member of the board of directors of the Association for Uncertainty in Artificial Intelligence.
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Certainly anything called homeland defense fulfills this in spaces. The title is reasoning about rare events.
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But first, to wave hands at what homeland defense might actually mean, any notion of it has to immediately acknowledge that it is a stupendous panorama, any way you look at it. This is one little cut Alex gave on it.
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That of course is spectacularly not the case. It is the source of what makes data fusion necessary.
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There are many actual organizations to consider. When you go to homeland defense, everything is much worse.
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These can be the same as they are at CDC, but they don't have to be. If you start asking to consciously integrate these, so that the information is being automatically sifted and compared, and regionally focused and then hierarchically fused up from regions, it is conceivable that a great deal more might be known.
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Then we have a standard one-stage Bayesian inference here. It is important to notice that in modeling this, typically the way we think of this in building actual applications, it need not be causal, but it is a powerful way to think about the model.
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Spiegelhalter developed the so-called joint triagramen, which solves any Bayesian network that is well structured automatically. Then in 1991, Bruce d'Ambrosia and Brendon Claverill developed the 478
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In this case, what you actually observe is a report at the Centers for Disease Control. That report may or may not accurately reflect the diagnosis that is supposed to be referred by a doctor because errors happen when reporting goes on, especially when it is done on a massive scale.
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This issue of what is relevant is place modeling, especially for something as broad as homeland defense might be defined to be. So in a deep hierarchy like this, odd things happen.
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481 what we might call a standard probability Pascalian type probability, and focus on conditional independence, whether or not it is true, between random variables, and the weight of evidence that a piece of evidence carries in its prepost area to an intermediate hypothesis or another piece of evidence, or the targetls hypotheses about what is going on that one is interested in. But as this example suggests, there is actually a lot of other issues that come into play when one goes to do large scale modeling of these scarce events, as opposed to, we are going to drive tanks into somebody, and we have a pretty good idea of how they are going to respond when we do that.
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That raises the issue of approximations. For instance, variational mean field approximation is an explicit attempt by Tommy Jeckle and Michael Jordan -- not the basketball player -- to come up with a robust computable approximation to very large scale Bayesian networks, et cetera.
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When you look at homeland security, this raises its head as a big issue. We simply don't know an awful lot.
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Something that is much less well known than the work of Professors Dempster and Schaeffer and Zotti is work by Jonathan Cohen, and more recently in causality with Glen Schaeffer and Yuta Proboth making very significant contributions. These approaches, Baconian they have been called, the Humeian approaches, address in the first case, instead of weight of evidence approach to reasoning, eliminative reasoning, attempting to come up with exhaustive explanations that essentially deny a hypothesis on logical grounds, and counting them.
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One of the things that happens when you look at homeland defense is a wide area of fusion. You are going to talk about evidence arising from all over at computers that are going to then do fusion.
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I am going to borrow heavily from his results in the following. He distinguishes weak versus rare evidence, and this is evidence of which I was speaking at the .
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What is interesting is that weak evidence in hierarchical based inference and solution algorithms and Bayesian nets and such, the order that you accrue that evidence, given a static set to accrue, the order of that accrual will not significantly affect the results except up to precision. In rare evidence, the inference order varies a lot.
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The weak evidence here is, if you had a doctor for instance who consistently misdiagnosed anthrax, then him saying things are normal is not particularly evidential to whether or not an attack is going on.
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In that case, this would be modeling that as rare evidence, not that it necessarily is; the one I pointed out as rare is in case three. But the point here is just to show how the numbers work, if you insert them in and just do hierarchical Bayesian accrual up.
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They are going to report some machine that is going to choose that with other information. The first thing to note is that when you distribute this sort of thing, this issue of rare evidence now when it arrives there is going to affect your conclusion.
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491 aerosols, you have disease incidents. Foot and mouth for instance can be trivially transmitted.
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In this case, you look at a computer network and look at what goes on in terms of a work flow model, which corresponds to the growth and death of things in a population. You model that.
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DR. LEVITT: No, it's not a is a work flow diagram that is differential equations for the varying being exchanged in the work Bayesian network.
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We need to solve the problem asynchronous arrival of rare evidence, the optimal placement combinatorial problem of sensors and assessors, especially from the point of view -- and this is the twist that hasn't been done, the rapid convergence of evidence in the fusion algorithms that are used, not just the placement to meet some objective function. Finally, the integration of these mathematical representations, or the interoperation of them, if you 494 completeness and of the optima]
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495 will, to transforation, from the continuous dynamical systems, boundary value type problems that you get when you look at diffuse phenomenon, to the inference webs and chains that have to occur if you are going to do fusion on machines. Thank you .
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Some areas of future work inciucte the following: Developing calculi for automated inference for fusion; Determining when information is relevant and how complete it must be in orcler to give a level of accuracy necessary for drawing conclusions; Handing the problem of asynchronous arrival of rare evidence; and Transitioning from the mathematical moclels that arise in studying diffuse phenomena to inference webs and chains that are required for data fusion.
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Her research interests include Bayesian inference and decision theory, multisource fusion, uncertainty in artificial intelligence, and situation assessment.
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The inventor of Bayesian networks was the father of Danny Pearl, so I would like to dedicate my talk to Danny Pearl and his father, and to say that I hope that Danny Pearl's memory will live on in those of us who apply his father's research to the problem of homeland security. I am going to talk about some basic requirements for inference and decision support for homeland security which you already heard some of in my comments at the microphone earlier.
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Now we are moving into information technology, where we are learning to apply technology to information processing. I believe that the impact on human society will be every bit as fundamental as in the agricultural and industrial revolutions.
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Level one is detecting individual entities, like, I see a tank there, or I saw an animal with anthrax there. Level two is to support situation assessment, which is militarily significant for the Joint Directors of Laboratories, but for our problem, a significant configuration of entities in the environment that has some meaning.
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In the homeland security problem, it is essential to be able to fuse both hard and soft types of knowledge. of disease transmission, but also, of, for example, the political conflicts evolve, the So not just the biology we need the knowledge scientist about how anthropologist, the social psychologist, the clinical psychologist on the mind of the terrorist.
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Let me talk a little bit -- this is where I am going to talk about Danny Pearl's father. I am going to talk a little bit about knowledge representation, but first let me talk about a paradigm shift that I see occurring in computing technology.
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We have got the explosion of micro chain Monte Carlo methods and statistics and variational methods. This is a recipe for getting a Ph.D thesis.
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Just like any paradigm shift, the old paradigm is a limited case of the new paradigm. Let me tell you very briefly what a knowledge based system is, because we are talking about representing knowledge.
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We need to have a unified theory for knowledge based systems. We see a synthesis occurring between traditional artificial intelligence, which looked at structured representations for knowledge, and complex search algorithms, probability and decision theory, which can deal with uncertainty and can deal with objectives and values.
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This is called an influence diagram or decision graph. The probability part of it is called a Bayesian network.
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I have had junior high kids playing with Bayesian networks, and they understand them and they like them. I can get students after one or two classes to build passable Bayesian networks.
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From a mathematician's perspective, there are a lot of fascinating problems there. In geometry, in inference, for example, if you take a Bayesian network and 508
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There are hundreds of papers on Bayesian networks now coming out every year. But that was a one size fits all model, in the sense that the types of applications people do.
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That is a little bit more complex. The final variation I am going to talk about is that they are both sneezing, and this time they are both allergy prone, and we don't know about their cold history, and they both saw scratches, but they are a continent apart.
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That is what people have been doing with Bayesian networks. You add variables that cover all the situational factors that you might want to model.
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So the question is, how much of the model do I construct and how much do I prune away. That is something that has got very interesting mathematical challenges.
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We need to think about the mathematical properties of these algorithms. This is an architecture for a system that retrieves Bayesian network or decision graph fragments and pastes them together.
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From that perspective, the modeling language provides a universal representation language that describes generic decision and inference problems. We need software tools that are theory based.
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We want to combine expert knowledge with designed experiments and observational data. Most of the learning methods cannot handle anything more complex than independent, identically distributed observations and a conjugate prior.
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There is lots of useful application experience. By the way, Clippy in your Microsoft office products is a Bayesian network, in case you didn't know.
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graphs associated with unclerlying probability distributions. She offered some examples to illustrate Bayesian networks' usefulness for representing common-sense knowiecige and assisting in clecision-making.
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He is coauthor of Ordinal Data Modeling with James Albert and author of Grade Inflation: A Crisis in College Education. His research interests include ordinal and rank data modeling, Bayesian image analysis, Bayesian reliability modeling, convergence diagnostics for Markov chain Monte Carlo algorithms, Bayesian goodness-of-fit diagnostics, and educational assessment.
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We are also trying to apply this type of methodology to the Ballistic Missile Defense -- or I guess, the Missile Defense Agency now. Here is a non-classified example of what we are doing.
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Then in the weapons program, we may have five or six different weapons systems that are all similar, and we want to model the similarity across these different systems. My talk is going to be a little bit different from some of the previous talks, in that I am going to become somewhat specific in how we are modeling this 520
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We model this in a beta type density function, but we haven't fixed the K So A is the maximum value of this beta density.
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We are going to see how consistent the expert is with the other data we have and with the other experts. For specific prior information, let me just emphasize that the A is fixed, but we are going to estimate the K, and we are going to use the beta density to do that.
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We can't for example compute a system-wide probability if we only have data on five or six of these terminal nodes. So we are going to assume another grouping type prior on the terminal nodes in our fault tree.
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As Kathryn mentioned, this is sort of like a Bayesian network. A lot of these conjugate priors - we use conjugate priors because they are convenient, but it is not necessary.
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That is sort of an important point, because if you have one system level test, all this information is transferring itself all the way up and down the fault tree now. So for example, if I have N trials at the system levels, the likelihood that comes from those N trials takes this form, where we have the product over the probabilities of all the terminal nodes minus that same probability.
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We had data from an anti-aircraft missile system, approximately 350 component level tests were performed on the system. I can't show you the actual data because it is proprietary.
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When we throw away all of the system level test and just use the component level tests, we get the red line. As you expect, the posterior distribution based on 527
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For one of the sub-systems that had a lot of test data, the posterior didn't change much when we got system level tests and when we incorporated that information, whereas we were looking for components where there was no data, the posteriors did change somewhat, because the system level tests did provide that additional information.
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Some extensions that we are currently modeling. Some missile test data that we had, we had a time associated with all the binomial data, so we are modeling now -- instead of just having a beta distribution at each node where we have data, we are looking at a logistic regression model.
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Based on just component level tests in general, we are going to have a broader distribution for the system, and when we get the system level information, it is not always going to fall right in the middle of that distribution. There are certain diagnostics we would like to do, for example, omitting some of the data in the nodes that we have data, and see if we predict well what that data would have done.
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JOHNSON: I think your question if the test data and the expert opinion was before 1992 when we stopped doing nuclear still speaks us confidence perform? is basically, all collected tests, do we really want to use it now.
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So in the last slide where I said we are trying to incorporate different sources of information into this model, this binomial system, we are really trying to look at some fairly sophisticated ways of putting information into these different nodes that isn't really in a conjugated form.
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This issue also arises in handling Air Force projects, such as the F-22 safety program, and working with the Army and the Missile Defense Agency. Consider a muiticomponent system in which every component must work in order for the system to work.
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534 Arthur Dempster "Remarks on Data integration and Fusion" Transcript of Presentation Summary of Presentation Video Presentation Arthur Dempster is a professor of theoretical statistics at Harvard University. His research interests include the methodology and logic of applied statistics; computational aspects of Bayesian and belief function inference; modeling and analysis of dynamic processes; statistical analysis of medical, social, and physical phenomena.
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From page 535... ...
I perhaps would have liked to hear a little bit more about gaming, the opposite to people are developing strategies, and we have to cope with their strategies and back and forth, multiple feedbacks and so on. Anyway, it goes much beyond passive description of the situation.
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From page 536... ...
A second comment that isn't perhaps very much on today's topic, but I'll back up a little bit -- I think discussants have been allowed to do this by the chairs -that is to say that data are dumb. What I mean by that is, in themselves, if they are just a string of bits, that means nothing at all and it is obvious.
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This session has been on fusion, and that is what I should try to address a little bit. We are talking about fusing different measurement sources that are measuring objective things with error.
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So the different information sources of evidential sources do have to have within them some representation that is independent from source to source before either Boolean logic or probabilistic combination or whatever you are doing, Bayesian combination, can operate.
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I'll say a little bit about graphical structures. It seems to me that this technology, certainly the one I am familiar with, the belief function technology, has been closely tied to join trees, tree structures, that kind of thing, since the mid-SOs when Schaeffer and Schnoy wrote a paper about it, and my student, Augustine Kahm, did a thesis in 1986 developing that subject.
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Maybe it is just the reductive aspect of the way mathematicians work, but to me, what I would like to see mathematics do is have new conceptions or frameworks for addressing scientific issues. I include all of these homeland security issues as scientific issues, in the sense that they need to be described, and what you know about them set up and formalized as much as possible and so on.
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One kind of thing that I am thinking about these days is a mathematical model of object recognition, very abstract, away from the real problem or what features, the complexities or dynamics or this kind of thing. But there was an example there of a medium characteristic.
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From page 542... ...
What I found when I thought about it was that the Bayesian aspect -- it has got a Bayesian label on it, but what the Bayesian would do, the standard Bayesian, would be to take the samples, the binomial samples and the 13 unknown probabilities and say, how am I going to assign a prior to those 13 probabilities that I can mix with the data. But those authors didn't do that.
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From page 543... ...
It is much more moving towards the belief function approach. I thought the way they used beta priors with those capital K and capital N values in them was quite ingenious, although it seemed to me to mix a little bit the notion of expert opinion, the U of I.J.
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From page 544... ...
544 So there is a kind of a philosophical attitude going on here, whereby there is objective science, something that I think Glen Schaeffer called nature, nature as the source of causation in the book, The Art of Causal Conjecture, that Tod mentioned. There is objective science of that nature.
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Dempster saint. "Maybe it's just the reductive aspect of the way mathematicians work, but what ~ wouict like to see mathematics do is have new conceptions or frameworks for acictressing scientific issuesinciucting homeianct security issues." Although one can set up a mocte]
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Dr. Grunbaum's current research is in medical imaging.
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The issue is, is there any way of bringing mathematical types in general, that includes all sorts of different things, to think about this new unfortunately national effort of homeland defense. I share fully what George mentioned yesterday and Dave McLaughlin and some other people, that it is very important to keep in mind the physics, the biology, the chemistry, the computational background that are part of these Problems.
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I will be pushing some of the community that I think I represent now, which is the community of inverse problems. There are a number of people that have worked in medical imaging, in geophysics.
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A number of pieces have been mentioned in these past two days that have appeared from the very beginning in the area of medical imaging, the question of high dimensionality being the main Back in the '70s, maybe one. even earlier than that, we finally realized something that by now everyone realizes is real, but it wasn't real at all back then.
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Nowadays this is part of the culture. An element that was mentioned yesterday in the very first session has to do with the training of analysts at NSA.
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From page 551... ...
I make propaganda for this journal published in I do this because I happen to be the editor right are finding that a collection, a whole community of people coming from many different areas, some of them even call themselves mathematicians, that have dealt with a number of issues that are very, very -- the way somebody was talking about at what level should we move in matters of analysis. This is a very classical problem in medical imaging; should we go for higher spatial resolution or quantum resolution.
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552 extremely complicated networks. What I have drawn here is supposed to be a Markov chain with finite states base.
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553 is describe the complete class of all solutions for the inverse problem. If we don't make any assumptions -- of course, everything depends on the number of things that you have, but even the simplest cases are complicated like this, actually a little bit more complicated than this one, is data source positions, vector ones.
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I think I already gave the answer. Many people cal this homeland security, as opposed to homeland defense.
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555 around and waiting to be hit next, and trying to prevent attacks. I'm not deprecating any of that.
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We have a huge intelligence establishment for which we pay a lot of taxpayers' dollars. Part of their job is to do the anticipatory and not even suggest, just bring the data up front for the appropriate decision makers to decide whether to be proactive or not.
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From page 557... ...
I think there is an enormous amount going on, but I don't know much of it, and therefore I wouldn't know, as far as germane to this workshop goes, where the mathematical challenges lay in offensive activity, which I think are highly focused. However, I completely agree with the scope of homeland defense.
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From page 558... ...
passively observing. In fact, I'm across that way, because it wasn't Inference can be used to passively observe, but you notice that level five of the five levels of fusion that I talked about was placing your sensors.
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From page 559... ...
However, things have changed a little bit, and you have to write more than one paragraph or one opening sentence. The old joke, everything looks like a worm.
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From page 560... ...
That office has kicked off a major homeland security initiative. It is not exactly new money yet, there will be new money in the future, but right now they are earmarking existing money everywhere that could be pointed in that direction.
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From page 561... ...
561 the inside, the paperwork, there is somebody that will have to write a paragraph that is that long on a form, explaining why your work is relevant to the problem. So you are passing the buck to somebody else.
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From page 562... ...
562 supportive to homeland security. I also think that there is a fair amount of what we already do that can almost be directly applied to some of the immediate problems, and it behooves us to seek out those opportunities and try to do that.
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From page 563... ...
It is all about uncertainty quantification and knowing when we have to go back and start testing again, or collect other types of information. That is true throughout all the homeland security problems.
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From page 564... ...
564 In the last session, I think that Val's talk tried to show how we can try to put things together in the presence of a lot of data when you take some slices through the problem, but not a lot of data or information when you take other slices through that problem space. So that is really important that we remember as well, that it is not just about massive data mining, but it is about how to integrate all of this information.
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From page 565... ...
565 pretty energized by this and I look forward to continuing to interact with many of you on these.
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From page 566... ...
566 Remarks on Data Integration and Fusion Alberio Grunbaum Mathematics has anmazing, almost poetic, ability to look at a problem in a certain area and then take a step back and sucicten~y realize that maybe we can use these tools for some other problem in a different area. For example, there are a number of people who have worked first in geophysics and then in mectica]
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