クリティカルポイント・エピソード10: クリティカルポイントとは何か?

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作成者 Hans K. Leida, Doug Norris |  2019年3月4日

クリティカルポイントとは何か、また、なぜミリマンのポッドキャストではこの数学的概念にちなんだ名前を付けるのか?

Transcript

Announcer: This podcast is intended solely for educational purposes and presents information of a general nature. It is not intended to guide or determine any specific individual situation and persons should consult qualified professionals before taking specific action. The views expressed in this podcast are those of the speakers and not those of Milliman.

Jeremy Engdahl-Johnson: Hello and welcome to Critical Point, a podcast brought to you by Milliman. My name is Jeremy Engdahl-Johnson. I will be your host today. Today is not our first Critical Point podcast, but it is the first podcast we’ve done based on critical points. So with that in mind, we’ve got a couple of exciting guests today, Hans Leida and Doug Norris. Both of them are health actuaries. They are also both math PhDs. And it’s their fault that we have named this podcast Critical Point. When I was trying to come up with a name for it, we reached out to the folks that are deepest into the math weeds and asked them what is a good math term that we could use as the basis for our podcast and the overarching metaphor and what we came up with is Critical Point, so I’m going to put these guys on the spot.

Hans Leida: Sure, I can go first since I'm the one you emailed, Jeremy, I guess back when you said you needed a list of math terms that could be podcast names and this is the one we settled on out of the email that resulted.

Jeremy Engdahl-Johnson: We have you to blame for it.

Hans Leida: Yeah. Yeah. I’m the one to blame.

Jeremy Engdahl-Johnson: Like it or love it.

Hans Leida: I mean a critical point can mean different things in different fields of math or science, but maybe the place to start is in first-year calculus you learn about derivatives and you study the rate that things are changing and a critical point is a point where the derivative of a function vanishes. So it's a place where the slope of a line is horizontal and that sounds kind of abstract and maybe only mathematicians like Doug and I care about it. But in reality, everyone should care about it because this happens a lot where there's a maximum value or a minimum value of something. And a lot of times in business or science or engineering, that's your goal, is to make something as big as possible or as small as possible.

Jeremy Engdahl-Johnson: Doug, does that pass your peer review?

Doug Norris: I would say that in addition to that, a critical point is sort of where something becomes interesting or where that changes. So a function is increasing, increasing, increasing, and then all of a sudden, it's not. And we like those.

Jeremy Engdahl-Johnson: Right. So another thing you were saying I was trying to, you know, kind of wrap my head around this concept, and I think we've got a lot of helpful examples here, but it's like the magnet, right?

Doug Norris: Yeah. If you hold a magnet in one hand and you hold a nail in your other hand and they're sufficiently far apart, you probably won't feel any pull from the magnet on the nail and you can move your hands closer together and nothing will happen. Nothing will happen. Nothing will happen. And then all of a sudden you'll feel a pull. And then all of a sudden it'll snap together. That's a critical point. And you don't necessarily know when it's going to happen in advance. It's just something that all of sudden happens.

Jeremy Engdahl-Johnson: Right. Okay. Well, and so the reason you guys are here, it wasn't enough—I know one preceded the other but it wasn't enough just to have FSA behind your name. You also needed to get some PhDs along the way. And I understand you both come at the topic of critical points from maybe different directions. So do you want to tell us a little bit about the topology, abstract spaces, such things?

Hans Leida: Sure. Yeah, that's my PhD. So I was planning on being a pure mathematician before I somehow found myself as an actuary instead here at Milliman. And the field that I studied as you said, Jeremy, was called topology. And topology is all about studying the shape of things. Like a topographical map shows you the shape of the land, topology is really about mathematicians trying to understand what makes different shapes fundamentally the same or different from each other. So you spend a lot of time trying to imagine, you know, if I imagine that something is made out of silly putty and I'm able to stretch it and deform it in different ways. Can I transform one shape into another if I'm not allowed to tear it or do something kind of discontinuous? So the standard example that topologists give is that you can stretch, you know, a coffee cup and mold it into the shape of a donut but you can't make it into a ball because it's got a hole where the handle is. So that's a difference in topology.

And critical points come up in topology, like they do in many fields of math, when you're trying to understand where things change, as Doug said. So it may seem easy to think about the shape of things when you're picturing coffee cups and donuts, but a lot of the interesting spaces or mathematical objects that you want to study aren't as easy to visualize. They might be very high dimensional and you can't really picture them very easily. So one way to study these different types of spaces is by slicing them up into kind of thin slices that are lower dimensional that you might have a chance of understanding and when you're slicing them, the shape or the topology of those slices changes at critical points. So there's a whole theory of trying to understand these really hard to understand spaces that relies on critical points.

Jeremy Engdahl-Johnson: Wow. I kept thinking while you were saying that that at some point Weird Al Yankovic is going to probably spoof the, you know, Ed Sheeran “Shape of You” song with kind of a topology song. That—I think the world is ready for that.

Hans Leida: If he's out there, I really hope he gives me a call because I would like to be involved in that.

Jeremy Engdahl-Johnson: Yeah. I think you're maybe the only person who could do that - you and Lin-Manuel Miranda. Cool. Well, Doug optimization theory, what do you got for me?

Doug Norris: Well, I've wanted to be an actuary ever since the third grade. And somehow I went off the path and got my doctorate in optimization theory.

Jeremy Engdahl-Johnson: You lost your way.

Doug Norris: I lost my way. I came back to the fold eventually. But in the meantime, I became a math professor and tried that out for a while. When I think of critical points and optimization theory, you're trying to make things as large as possible or make things as small as possible. And there's a theorem I really like which speaks to critical points. When I think about it, it really kind of speaks to my heart and it's called Fenchel’s duality theorem. And what Fenchel’s duality theorem says is: if under the right conditions, the minimum of a function is satisfied at the exact same place and same value as the maximum of a related function in its dual space. So to give you an example—to give you a one-dimensional example. If you're trying to find the minimum distance between two convex sets, that minimum distance is satisfied at the same exact place and same value as the maximum of any parallel lines that are tangent to those sets. So the way you apply this in the real optimization theory world is you have a problem that's very hard to solve and you get to a point where you simplify it as far as you can and you can't get any further. What this theorem lets you do is pop it over to the dual space, make the minimization problem a maximization problem, and you still can't solve it, but you can simplify it a bunch more and get it back into its dual space, which brings it back into the original space, and you're minimizing again. And you keep bouncing it back and forth until you get it simple enough that you can actually solve it. And so, when I think about how the minimum in one spot is a maximum in the other spot, that speaks to critical points to me.

Jeremy Engdahl-Johnson: So you're both healthcare actuaries. And, you know, if I'm getting a little metaphorical about things, we've got a couple metaphors to deliver here. How do critical points apply to healthcare? I mean in some ways you can almost think of the Affordable Care Act, it's at least an inflection point. I don't know if we can mathematically with any mathematic integrity call it a critical point. But what do you guys think?

Hans Leida: Critical points also occur in science and physics. The term “critical point” is used when you have a phase transition and that is when a liquid becomes a solid or a gas or if you get the right temperature and pressure you can have water that's freezing, boiling, and liquid all at the same time. So it is where, yeah, the rules fundamentally change. And that is absolutely a great metaphor for the Affordable Care Act. For a lot of the healthcare system, the rules fundamentally changed in 2014 and honestly have continued to change at a pretty impressive clip ever since. So the significance of that, I guess, is that when you're talking to your client, if the rules have changed, the data that you had before may not be relevant anymore. Unless you can adjust it or model it in some way that makes sense under in the new phase you’re in on the other side of the critical point. And so that's been a lot of what Doug and I have both been doing here at Milliman over the past, I guess, half a decade or so since things changed in 2014.

Doug Norris: Yeah, I think of the joke about the insurance company being a car driving down the road with the actuary looking out the rearview mirror driving and helping guide the car. And that's no longer possible with a sea change like the Affordable Care Act. You have to think differently. You have to think bigger. You have to think strategically. And that's—I know that’s some of Hans’ favorite work and that's some of my favorite work as well.

Jeremy Engdahl-Johnson: Yeah. Well, somehow the idea of both boiling and freezing at the same time sounds like kind of the plight of the healthcare actuaries in certain years. So what else? What do our listeners need to know about critical points? Like what one thing could we not let them go home without knowing today?

Hans Leida: One thing that I found incredibly interesting when I was trying to do some prep for this podcast was that I found that actually in neuroscience, right now, some of the scientists trying to understand how brains work believe that they operate near a critical point. And I'm by no means an expert on any of this. But what I gathered was that there's at least a theory that your brain or even other animal brains operate right on the boundary between where there's sort of a runaway cascade of activity that kind of spirals into chaos or not enough activity that would sort of fade away. And really, there's sort of a critical point in the middle that's like a phase transition where things change from one to the other, and your brain and mine might operate kind of near that critical point in some way.

Jeremy Engdahl-Johnson: So we were spit-balling earlier and you were talking about critical points in social media. You know, we're not trying to stretch or flog this thing too hard, but you know, what do you think?

Hans Leida: Yeah, it definitely in a lot of cases, you know, there can be a runaway process. Right? In social media when something goes viral is a great example of that, where there's sort of a tipping point and things can grow exponentially. And below that point, something might end up in obscurity with very few retweets or views or eyeballs on it. So it's definitely another example of a system where at a certain point the behavior of the system changes fundamentally, depending on, in this case, how many eyeballs you get on something.

Doug Norris: You could get one celebrity scrolling past on their phone at just the right moment and retweet or something and all of a sudden, everyone's talking about it.

Jeremy Engdahl-Johnson: Yeah.

Hans Leida: Yeah.

Jeremy Engdahl-Johnson: And critical points really are—I mean that's—it's this moment before things kind of fall apart sometimes, right?

Hans Leida: Yeah. It's the boundary between orderly predictable behavior sometimes and chaotic behavior. If you study dynamical systems, which is a theory in math, you get all these pretty fractals that people like to, you know, put on their wallpaper, or maybe that's just me and my friends. But, you know, that's studying, absolutely, the boundary between order and chaos. And there's certain spots near that kind of criticality where you get these astonishing detailed patterns that look like nothing else. And are at the same time very ordered and very chaotic.

Jeremy Engdahl-Johnson: Kind of the border of chaos.

Hans Leida: Exactly. Yeah.

Doug Norris: So it's sort of the relationship between critical points and chaos. If you are—picture yourself on the beach this being-- recording this in mid-October and I'm thinking about how I missed the beach this summer and I'm yearning for it. And so you're on the beach and you're pouring sand on a sand pile and as you go, the sand kind of makes a nice orderly pattern, a conical pattern as you go. And then, all of a sudden - you don't necessarily know when it's going to happen - there's an avalanche of sand that comes down and makes an unpredictable chaotic pattern. And that's one of the things I think of when I think about the relationship between chaos and critical points. And at any point, sort of healthcare as an analogy, healthcare was largely somewhat predictable until the Affordable Care Act came along. And then we said, “Oh, for a few years this is going to be really complicated and we have to have boundaries like risk adjustment and risk corridors and things like that to make it more predictable and make it easier to wrap your brains around. But it'll be fine in a few years because we'll have data and we'll have information and experience.” And now we're past that point where it's supposed to be normal again, but now we have all these new post-ACA policies, you know, short-term major medical, association plans, people staying out of the market altogether, and it's still interesting and still chaotic.

Hans Leida: More grains of sand…

Doug Norris: More grains of sand…

Hans Leida: … each year as we go along. And you're kind of waiting to see when the avalanche might happen or will it just continue as it had before?

Doug Norris: And that's the fun part because that's when you get calls and people want to talk about things, and this is cool stuff to talk about.

Hans Leida: Yeah. The other thing that occurs to me, Doug, we years ago were kicking around the idea of power laws in healthcare. And for those that haven't run into power laws before, it's a theory of a system where—here’s, I guess, kind of the standard example for healthcare where, you know, the top few percent of the population in terms of their healthcare cost uses a disproportionate share of the overall dollars in the healthcare system. In other words, care is kind of concentrated in a few individuals. This occurs in a lot of different systems that we observe in the world, like income or populations in cities you have, you know, a few cities with enormous populations. You have a lot of really small towns that have a much smaller population. So there's a lot of debate over whether these different systems actually follow a power law or not. But Doug and I were really interested in the concept and not necessarily trying to answer the question “is this truly a power law?” Nothing in the real world is going to be exactly a model, at least in the kind of messy actuarial world we live in. But if you even just grant that we see this pattern of some people really using a lot of the healthcare services, why is that? Why do sort of-- why does cost kind of accumulate in these sorts of hot spots or particular individuals? And then what might you do about it?

Doug Norris: Why are 80% of the costs driven by 20% of the people?

Hans Leida: Right, right.

Doug Norris: It's always an impressively large percentage and an impressively small percentage, but 80/20 is roughly where it centers. Some of the examples I think of when I think about critical points in the future are like immunizations or the spreading of influenza. And the spreading of influenza is a function of the number of interactions, so if you have—we have four people sitting in a room right now, the number of interactions is going to be six. I just did that in my head and I hope that's right. But as the number of people grows, the number of interactions increases exponentially. And finding the right people to immunize and make sure that they're inoculated against the disease is a critical point and is a place to start looking because if you get the right people, then you can help hopefully control this before it gets out of…

Hans Leida: Before it goes viral.

Doug Norris: Yes.

Jeremy Engdahl-Johnson: Before the virus goes viral.

Hans Leida: We're in the business as actuaries in trying to predict the future and we try to do it with math. And the struggle that-- and also the kind of exciting part of it often happens in these systems that are chaotic and sort of resist prediction. And so at least for us, that's been a really exciting part. I think I can speak for Doug, too, of our career. It's not what I expected when I came to Milliman, and I didn't know much about being an actuary at that point. And I thought it was you calculate and things are relatively stable. And that hasn't been the career that unfolded, but I'm really happy with the one that did.

Jeremy Engdahl-Johnson: The critical point, I think, it's a good metaphor for our business in general and for Milliman. We've got a lot of these kind of oppositional forces. And a lot of times we're tiptoeing right on the border of chaos, and you know, trying to bring things back from the brink.

I wonder if anyone has ever done a podcast on critical points before. I definitely feel better at this point about the name that we chose. But I do want to hear from each of you—like why does this name resonate for you?

Doug Norris: So ultimately, the insurance industry kind of carries along and it's the same day, you know, Monday, Tuesday, Wednesday, everything goes along normally. And then all of a sudden there's a sea change or something that shocks the system, and that's when I would say healthcare work becomes interesting. And that's what I like most about working here at Milliman.

Hans Leida: Yeah. I tell my clients, you know, you don't need me or our team for the day-to-day, you know, day-to-day stuff. They usually have a handle on that. It's the difficult problems that are where we get called in and where we can really provide value to our clients. And that's the work, like Doug said, that's the work that I enjoy the most. And it's our privilege to get to do that kind of interesting vital work.

Jeremy Engdahl-Johnson: Yeah. Well, thank you both for joining us. You’re like the godfathers of the Critical Point podcast. Exactly. You've been listening to Critical Point presented by Milliman. To listen to other episodes of our podcast visit us at milliman.com. Or you can find us on iTunes, Google Play, Spotify, and Stitcher. We'll see you next time.

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