Classical and Quantum Subjectivity

Hello, I'm Dr. Stuart Nettleton from Atura.

And I'm Dr. David Boyle, also from Atura. And our talk is basically considering the movement from classical to quantum subjective logic. So, Stuart, do you like being asked yes or no questions?

Well, I don't, because a lot of the time I don't know.

Absolutely, I don't know. And so, in fact, about 250 years ago, Pierre Simone Laplace had the same thought. He realised that the Bayesian representation of an opinion like that is not adequate. You need to have some representation of, I don't know, in other words, some representation of uncertainty. So what we're talking about is in fact Auden Yozang's classical subjective logic goes some way to addressing Laplace's concerns, but doesn't operate well in the context of a larger Markov network. So what we've done is that we've actually generalised the approaches to handling uncertainty, particularly in the context of larger Markov reasoning networks, so that we can address Laplace's concern. And we do that through modifying the way that uncertainty is brought into these techniques. And that actually borrows from the principles of quantum mechanics, which I'll let Stuart talk to you about.

Well, everything in the world is quantum, so we say quantum subjective logic and quantum uncertainty. That's our thing. Thank you.

Good morning, welcome, and thank you for coming. My name is Dr. David Boyle and this is Dr. Stuart Nettleton and we're from a Toura in Australia and we're excited to be talking to you today about something we consider to be a really exciting development in the field of decision intelligence. Let me just digress at the start here by asking the audience a little poll question. On Wednesday night at the Q and A session, Stephen Wolfram was asked an interesting question by a person in the crowd. And I'm going to paraphrase and I'm going to be a little bit vague on purpose, but Stuart was asked, do you think that everything in the universe is deterministic, or do you think that those who say that some things are stochastic simply haven't gone into enough detail to try and look at how to model everything as being deterministic? And furthermore, the questioner then said to Stephen, I'd like a yes or no answer, which of course brought a fair bit of hilarity. What I want to do is pose that question to all of you. There isn't a right answer. But who would say that they believe that everything is deterministic? Okay, we've got a couple who would Say they believe that there's some true stochasticity involved. Okay, now there are a few of you who didn't raise your hands. Who would have preferred there to be a. I don't know. I certainly would. I don't know. It turns out that about 250 years ago, Pierre Simone Laplace, who co invented Bayesian statistics with the Rev. Bayes, had the same conundrum about his own invention. He was concerned that the Bayesian representation of an opinion was inadequate because it accounted for yes or no, but it didn't have an I don't know option. And he wasn't able to resolve that problem much more recently, Professor Auden Yozang and his colleagues developed what is known as subjective logic. And subjective logic is important because it actually partially addresses what was was Laplace's concern in that it explicitly handled uncertainty. But unfortunately, classical subjective logic, the Yozang subjective logic, is not really applicable to generalized Markov networks. So what Stuart and I have done is we've developed an improved technique for handling uncertainty that meets Laplace's vision. Using principles from quantum mechanics, we've generalized classical subjective logic for combining expert opinions into quantum subjective logic paradigm. And this allows us to enhance decision confidence through application to Markov reasoning networks. So I'll be talking to you about classical subjective logic, about Markov nets and about our eigenvector root mean square approach combining uncertainties. And Stuart will then explain how the use of quantum principles enables this through extension of the allowable belief space, which provides a practical and meaningful way to quantify belief assessments and fulfill Laplace's vision. So let's spend a little time, very little time, on classical subjective logic. Classical subjective logic is described by Yo Zang himself in his book Subjective Logic, A Formalism for Reasoning under Uncertainty. And there's a couple of words of interest here that I just want to dwell on. The first word of interest is reasoning. Now Professor Daphne Coller of Stanford states that reasoning is broadly the ability to take available information and reach conclusions about what might be true and then how to act. And here the word might is important because might implies probabilities, it implies uncertainties, and it implies what we may or may not know about a certain situation. So probabilistic approaches are inherent in reasoning. And the second word of interest here obviously is uncertainty. And while classical probabilistic techniques can account as what is known as first order or aliatary uncertainty, they don't account for what's known as second order uncertainty or epistemic uncertainty. That's the unknown unknowns about a situation or in particular about a model of a situation about which we want to reason. And in a business context that's of course very important. So this is where subjective logic comes in because it explicitly handles both probabilities and uncertainties. So skip over some of the definitional information here. But the main point to note here is that in the definition, the classical definition of beliefs and opinions in subjective logic, beliefs are a function of two independent variables, the projected probability and the uncertainty, which you can see here in red in equation seven. Quickly, about some of the tools that are important here is that there is a bijective mapping that exists between beliefs and de rich lay or beta PDFs probability density functions. And the reason this is important is because it allows us to use probabilistic techniques to be applied directly to opinions. So easiest way to look at this is by way of an example. If we have an example here where we've got, let's say there's a situation where you have a belief state which could be red, yellow or blue. So It's a multinomial 3 element opinion. And what we've set up here is a simple example where we have an opinion of the belief of state 1, red, yellow and blue, 1, 2 and 3 and an uncertainty with prior base rates, base rates, prior probabilities set up as non informative priors here. So we've got our beliefs of 0.1, 0.3, 0.3, an uncertainty of 0.3 and a non informative of prior set up here. And if we look at the resultant derivative distribution, it turns out that if I rotate this around and look at the marginal distributions, those marginal distributions which I've split out down below here and shown the mean of those means actually directly give you the projected probability from the classical subjective logic equations. So you can see Here you have 0.2 for belief, 1.4 for belief 2 and the complement of that in the DE reach layers, 0.4 for the final. So it's a trivial example, but you can just see here that the two techniques map directly to each other. So the resultant probabilities we receive out of classical subjective logic and the means of the Dirichlet distribution representation are identical. So as I said, that's important because it means we can use classical probabilistic techniques to handle operations on opinions. So we've spoken about subjective logic and we've spoken about the mapping there to reach layers. But if we talk about Markov networks, what we've done in developing Algorithms for decision confidence is we've, we've utilized reasoning models in the form of graphical Markov networks, but we've combined the principles of subjective logic and uncertainty combination to overlay that and meet Laplace's requirements, meet Laplace's conundrum. And so Markov networks very briefly represent a joint probability distribution over multiple variables, where the nodes themselves represent the probabilities and the variables and the correction. The nodes represent the variables themselves, and the edges represent the notion of some sort of probabilistic interaction between those, between those nodes. Now there are, as I said, graphical probabilistic modeling techniques which are available for computing the resultant probabilities across a network. These are meaningless numbers. They're just there as an indicator, a visual guide. Those processes are entirely non trivial, but they do exist. So computing probabilities across a Markov network is possible. But what happens if we want to actually inject some uncertainty, if we want to inject opinions the way we spoke about at the top of the talk? So we've got this Markov network here, which is a directed cyclic graph and undirected correction. What happens if down the bottom here we've got some input nodes, some input opinions, where we need to understand how we project how we flow the uncertainties associ of those opinions across the network. Well, it turns out that classical subjective logic techniques fall down in this way because they tend to select the minimum of all component uncertainties in a resultant multiplicative distribution. And that progressively has the effect of minimizing the uncertainty values across a larger network, particularly a very large, very interconnected network. And it also has the undesirable effect of reducing all of the resultant base rates across the network and wildly varying those, when in fact, if you started with a non informative network, you really want to maintain that non informative base rate across the network. Classical subjective logic, due to its constraints of requiring beliefs to never be negative, ends up manipulating those base rates in a way which is not really consistent and not desirable. So what we've done is we've applied what we call the eigenvector RMS uncertainty approach, and that measures the transitive influence of uncertainty at vertices, expressed as a weighted sum of the uncertainties of its neighbors. This is analogous to eigenvector centrality, which measures the influence of a node in a network and is used by algorithms like Google PageRank. So that meets the two criteria that should be satisfied for a consistent approach, namely that we're keeping the base rates consistent throughout a non informative network and we're supporting the assumption that the uncertainty is independent of the probability, as we mentioned at the top of the talk. Now this is just one of many potential approaches. There could be other approaches to combining uncertainty, but this is an approach that we've developed and that, that we favor because of those two reasons. So let's have a look at a couple of examples. This is a trivial example. Just to show you the relationship between the classic eigenvector approach. If we simply have a parent node here which has four child nodes with associated uncertainties and we want to combine those at the top to understand, you can apply, you can do it two ways here because it's trivial. You can directly apply a root mean square calculation to come up with a result. And that you can say that's about 0.463. But if you take the adjacency matrix of this graph and you multiply that by the vector of uncertainties that constitute the neighbors of the node you're interested in, it ends up being a classical eigenvector equation. And you can resolve that eigenvector to produce the same result. And so you can generalize that to a larger Markov network to achieve the result that we're after. And that's essentially what our approach is based on. Now if we have a look at a random combination of two binary opinions, binary being you've only got two states, yes and no, and your uncertainty. If we combine those two binary opinions and we, and we do a random probability, a random combination of those two. So we're randomly varying probability one and two in these two beliefs, but we're keeping the base rate as a non informative base rate of 0.5. You can see here that in fact across this simulation our RMS approach falls nicely in between the two classical subjective logic approaches of the multiplicative merging of uncertainty, the minimum and the maximum. And that's really as you would expect, so no problems there. But if we start to then vary the base rates that are occurring, you can see down here that there are areas, the green areas, where classical falls down. It actually it's not able to produce or compute a result because of its constraints, particularly that belief can't be negative. Whereas our RMS approach, eigenvector, RMS approach for uncertainty combination is able to compute the result. And so that's very important for us. So there are issues, particularly when you're requiring combinations of opinions that have very differing uncertainties, where classical subjective logic combination techniques do fall over. And there are lots of reasons that we've got problems with the classical approach, and I'll let you read all of those at your leisure. But the main point to note here is that as I've highlighted in yellow, there are regions in the belief space where the classical is not able to produce a result, and those regions correspond to where the belief goes, because the constraint is that the belief can't go negative. So in a final example here, just to highlight this more, if we combine two binary opinions once again, and we look at the varying uncertainties, what we've done here is I mentioned the fact that classical requires manipulations of the base rates across when you do this, and this occurs in particular when you get the combination of two opinions with vastly varying uncertainties. What we've done here is we've taken an approach where we use the classical technique, but we take a minimization which tries to minimize the difference between the a priori non informative base rates and the manipulations that are required by the classical technique to produce a result. And so what you can see here is that in the situation where the uncertainty of our two input opinions are identical, we get no difference between the classical and our eigenvector approach. But as we start to vary the differences between the uncertainties of these two opinions, we start to see some interesting behavior where the two approaches start to diverge. And in fact we're here at 0.2 and 0.5, you can start to see some strange behavior starting to creep in from the blue, which is the classical but minimized approach for base rate variation, where we get to this extreme case down here, where our two input combination opinions have uncertainties of 10% and 90%. And in fact, you'll note that not only is there a very strange shape that is produced by the classical approach, but it can only produce a result across a very small region of the resultant belief space. Whereas if we look at the eigenvector RMS combination approach, you can see that in fact we can produce a result across the entire belief space with the very important note that we do allow beliefs to go slightly negative in part of the space here. So this allows the calculation of both probabilities and uncertainties across a Markov network, and it fulfills Laplace's vision of including both probabilities and the I don't know, the uncertainty. And that's the important point we want to make here. So we've seen that using eigenvector RMS uncertainty allows beliefs to move into that slightly negative space, and we've seen how it removes those constraints on classical subjective logic. And allows the that paradigm to be utilized in large Markov reasoning networks. And that's what we've been using with our clients to great success. I'll hand over to my colleague Stuart because he can explain to you how the use of quantum methodologies actually facilitates this. The resultant quantum subjectivity has been implemented by us and is being taken forward with our clients to improve their decision confidence. Stuart.

Thank you Dave, for finishing within your allotted time. Okay. In quantum beliefs, as Dave said, you can have a negative belief. Paul Jurak said, well, of course you can have negative beliefs. Take money for example. You can have money a hundred dollars and you can have no money, $0, but you can have negative money as well. You can have negative $100 which is a loan. And that's like a negative yes state. Now there's Dave's painted two views of uncertainty. He, Dave at the start said there's epistemic uncertainty, which is the fact that we don't have full knowledge of the world and our models might be wrong. And as Joshua Box said, all models are wrong, but some are useful. And there's also first order uncertainty, which is measurement uncertainty. And the normal Bell curve we're interested in. There's other forms of uncertainty too. In, in quantum area, we know that if you try and measure a quantum conjugate variable, a quantum state, it, it drops down to zero or one with equal probability and you don't know which it's going to drop to. And entanglement is based on that. And so that's called ontological uncertainty. And there was a great battle of course, which hasn't been resolved between on the one hand, Einstein and Schrodinger who said that the only uncertainty is epistemic uncertainty, and between Heisenberg and Bell who said that ontological uncertainty exists as well. And of course there's a third, there's a fifth form of uncertainty that we're starting to learn about from Stephen Wolfram and his multipath ensembles. And that's an ensemble uncertainty. And so that's something that Stephen's just starting to formulate and release. So there's quite a few bits of uncertainty. But what we do know is that in the world everything is uncertain and everything is quantum. And the only reason we happen to see a classical reality is because we have limitations in viewing the quantum states. And quantum states are multi dimensional. They could be 3, 4, 12 hyperstates, Hilbert vectors. But when we look at everything now, we need to put the word quantum in front of it because quantum is the approach. So Born's rule states that all states are probabilistic. But the fact that states are probabilistic doesn't exclude the fact that Markov networks can exist and one state can influence the next state. So events are not completely random, but they're uncertain. In quantum. We're very used to standard physics looking at Heisenberg's uncertainty principle, where you can't understand the position, the momentum simultaneously of a variable. There's a very famous joke, of course, about Heisenberg and others being caught speeding. And the police say to the people in the car, you are measured doing 100 miles per hour at this zone. And Heisenberg disputes the fact. That's impossible. You can't tell me that we're here and we're doing 100 at the same time. Well, so what they're referring to is conjugate variables. And you can only understand one, one of a pair of conjugate variables at the same time. And for us, frequency and time is a good thing. If I click my fingers like that in a very brief second of time, it's very difficult for you to know the frequency of what I've just done. But if I continued to click my fingers over a longer period of time, you can measure the frequency, but that then becomes uncertain exactly when I did that. So we have conjugate variables. Davis described probability and uncertainty of a person's opinion, a psychological opinion, as being conjugate variables. And Dave has also described to you the almost inconceivable aspect that some of the beliefs are negative. So these are called quasi probabilities, and they represent probabilities in the same way as we have classical probabilities. And it's only in some cases they go negative. This is best described by the star child of all modern current distributions called the Kirkwood direct distributions, something that may be new to you because it's only really been realized that this is important in the last year. It's been around since the 1930s, and we'll quickly talk about what that means in a minute. But it's now being used in things like quantum tomography. It's now being used in quantum metrology. We're hoping to use it in everything from finding needles in haystacks. The important thing about a negative probability state is when you inject a probe, and that probe's designed to act on the negative probabilities rather than the positive probabilities, you can get a magnification of 100 times the quantities that you're trying to measure. So if you Have a very large. If you have a haystack and you're after the needle in the haystack, you can eliminate all of the haystack and search for the needle with great fidelity. So I'll try and. Whoops. Dave, I'm not.

You might want to use the mouse, Stuart.

I'll use the mouse. Yeah, great. Yes. There we go. So we have quantum operators in when we build quantum circuits. And we don't need to actually build a quantum circuit for me to display this because we quantum can be run in what is known as a. As a. Can be run in normal code. And so quantum operators here are represented within Mathematica by the Kronecker product. And so when we're talking about the distribution that is the Kirkwood Dirac distribution, it's basically one of the conjugate variables, the operator of that times the operator of one of the other conjugate variables with the dot product to the probability density function. And that arrives as a function of the probe we're using. And so that there, as simple as that, is the Kirkwood direct distribution. And we take the trace of that. In other words, we take the diagonal of the matrix. So this was invented by both Kirkwood and Dirac simultaneously, and they thought it was very useful. And it's only now been realized that that's the case. So I want to give you a quick example. Say we have an auto repair shop and we ask the manager, the foreman or the floor supervisor if he thinks he can complete the batch of cars coming in on a particular day. And he might say, I think that's true, but I'm only 50, 50 on that. On another day he says, I'm quite positive we can 60% probability. But, and I'm pretty sure of my estimate this time, 80%. So he's starting to give us probabilities and certainties that something can be done. Well, let's look at what that means. Now I have to change my resolution down to 50% for these diagrams. Oops. Right, so here we have the states of. Of probability over here. And we have, in this case it's probability bins. We have this state four is 100% probability. State one is on the magenta. There is 30% probability. And here's the probabilities. 10% of being in this state, 30% of being in that state, 40% of being in that state, and 20% of being in that state, and they add up to 100%. So we've got probability bins and the probability of being in the state and over here we've got the same for certainty. We take what the manager has said over the month and we've got the probability of being in a certain certainty state and we've got the certainty bins. And as you know, you can multiply those two things together and end up with a classical probability. And I'll show you what that is in a minute. But when, when we start to look at what that means in subjective logic terms, what Dave has just described, we take certain certainty states together with probability states. And Dave gave you a little formula that said belief equals probability minus the uncertainty and minus the uncertainty on two. If we've got a non informative prior. And so we end up with various beliefs and one of them is negative. When we're looking at stage one quantum that Dave mentioned that we're using with our clients at the moment, we tolerate that negative belief state there. We actually psychometrically remap that state so that we're not challenging the challenging clients with that. They've got to understand, you know, what the negative probability of something that they're dealing with is. Okay, so that's a negative belief state. So we're going to now talk more about how the, the quantum quasi probabilities, the Kirkwood direct probabilities interact with that. And so the first thing we do is we inject a probe into the system. We don't just use the normal expected value of probability times certainty, we inject a quantum probe into the system. And this quantum probe is a summation of the states that are designed. And these come from the Pauli spin electrons because they're orthonormal. So we're putting together basically a pure injected state. And we see that the identity that injection has a negative probability in it. The thing we'd normally be looking for. We can represent that as a Hilbert matrix and you just see that as a diagonal of orthonormal vectors. We can represent that as just the psi vector itself, which is just a four by one vector. And so then we can look at the A matrix and the certainty, the probability and the certainty matrix, which here are known as the A and the F matrices. And we put those in the Pauli matrix, Pauli spin electron form. And only we use that only to give orthonormal and but to go to the bottom line here, I keep going down the bottom line. The Kirkwood Dirac distribution provides us these negative probabilities. These, these red numbers within the classical distribution. Normally Kirk would direct negatives are only a small part and the distribution is mainly positive. And so the bottom line is that classical probabilities a 0.7 probability with a 0.47 expected uncertainty that was from the bins times the probability of the bins that we saw at the start gives a belief of 0.435 there. And if we take quantum quasi probabilities that we've got by injecting our probe into the system, we get a belief of 0.4. So there's about a 10% difference. And so there we have a 3D plot of the difference. And here we have. If we start with the quantum, the classical would suggest we have an increased probability and that's something very bad for our clients to have a suggested increased probability above what they think our clients are in the defence area. And decreased classical certainty that's. That's conservative you might say. But the offsetting is that We've got a 10% increase in our probability in our belief space and that's a significantly adverse negative for classical probability to be delivering something that's increased. So we're proceeding to implement stage two of quantum which is Kirkwood direct distributions in our. In our belief space. And then we will migrate as quickly as we can across the hybrid area where we're calculating those in on normal computers into calculating those Kirkwood directs in a quantum circuit and putting that out to IBM or one of the quantum machines for processing. So we have a fairly aggressive approach to. To quantum. We believe it's the future and that everyone needs to put quantum in front of whatever they're talking about. So thank you. That's our talk. We just have one minute for questions. Is anyone interested? Yes, if we did a Monte Carlo of probability and certainty of people's opinions. Yes, indeed. We often get the states where someone will say I have 0% probability of achieving that and I'm 100% certain. Now what that means is a negative belief of 100%. And we have those situations with our clients that the. I'm absolutely sure I can't do it. And another difficult. So that's one difficult part of logic. Another is what we affectionately know as the grads corner. What's that one, Dave?

That's basically it. So that's where you've got. So there's a couple of corners where you have a 0% likelihood and a 0% certainty and a 0% likelihood and 100% certainty of that. And our learnings are that the latter is actually a worst case. Because if you're absolutely certain that you can't do something, then that's, that's a problem. Whereas if you're not certain and that you can't do it, you're totally uncertain of it. Well, what does that really mean? And that's the corner that Stuart's actually referring to there. But yeah, the Monte Carlo is an interesting question, because if you stay purely classical, of course, you're just dealing with the probabilities. But a question for us moving forward would be how would we modify a technique that is like almost a quasi Monte Carlo technique that merges what we're doing here? And I think that could possibly be doable because we'd be able to use the techniques we've used here and apply the Monte Carlo multiple, if you know what I'm saying.

Well, our system is fully Bayesian in the sense that we are combining up to a thousand opinions through Markov networks, which are Bayesian networks, but undirected Bayesian networks.