Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm one of your hosts, Kevin Knudson. I'm a professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance writer, usually based in Salt Lake City, but currently coming to you from Providence, Rhode Island.

KK: Hooray! Yeah, you're at ICERM.

EL: Yes. The Institute for computational and experimental research in mathematics, an acronym that I am now good at remembering.

KK: I’m glad you told me. I was trying to remember what it stood for this morning because I'm going next week. We'll be in the same place for, like, only the second time ever.

EL: Yeah.

KK: And the universe didn't implode the first time. So I think we're safe.

EL: Yeah.

KK: So the ICERM thing is visualizing mathematics, I mean, we're sort of doing like—next week is about geometry and topology, which since both of us are nominally that, that's just the right place for us to be.

EL: Yeah, it's it's going to be a fun semester. I'm also very excited because I recently turned in—it feels weird to call it a manuscript, but it is being published by a place that publishes books. It is the final draft of a page-a-day calendar about math. And I hope that by the time we air this, I will be able to have a link where people can purchase this and give it to give it to themselves or to their favorite mathematician.

KK: Yeah.

EL: So that's just, every day you can have a little morsel of math to start your morning.

KK: I’m looking forward to that. That’s really exciting. Yeah, that's that's great. All right, so we're continuing a tradition in this episode.

EL: Yes.

KK: So Christian Lawson-Perfect organizes this thing through the Aperiodical called the Great Internet Math-Off [Editor’s note: Whoops, it’s called the Big Internet Math-Off!] of which you were a participant in the first one but not this one, not the second go-around. And we had the first winner on. The winner gets named the World's Most Interesting Mathematician (among those people who Christian could round up and who were free in July). And so we wanted to keep this trend going of getting the most interesting mathematicians in the world on this podcast. And we are pleased to welcome this year's winner, Sophie Carr. Sophie, you want to introduce yourself, please?

SC: Oh, hello, thank you very much. Yeah, I'm Sophie Carr. I studied Bayesian networks at university, and now I own and run a data analytics company.

EL: Yeah, and you’re the most interesting mathematician!

SC: I am! For this year, I am the most interesting mathematician in the world. It's entirely Nira’s fault that I entered because he suggested, and put me forward.

KK: That’s right. Nira Chamberlain was last year's winner. And so when we interviewed him he was sitting in his attic wearing a winter coat. It was wintertime and it seemed very cold where he was. You look very comfortable. It looks like you have a very lovely home in the background.

SC: Yes, I mean, I am in two jumpers. Autumn has definitely arrived. Summer has gone, and it's a little chilly at the moment.

KK: I can only dare to dream. Yeah.

EL: Yeah, Florida and UK have slightly different seasons.

KK: Just a little bit. So you own a consulting company? That’s correct?

SC: Yeah, I do. I set it up 10 years ago now. There’s me and two other people who work with me. We just have an awful lot of fun finding patterns in numbers. I still find it amazing that we're still going. It's just the best fun ever. We get to go and work on all sorts of different problems with all sorts of different people. It's fantastic.

KK: Yeah, that's great. I mean, I'm glad companies are starting to come around to the idea that mathematicians might actually have something to tell them. Right?

SC: Yes. It really is. When you explain to them, you're not going to do magic and it's not a black box, and you can tell them how it works and how it can really make a difference, they are coming around to that.

KK: That’s fantastic. All right, so we're here to talk about theorems.

EL: Yeah. What is your favorite theorem?

SC: My favorite theorem in the whole world is Bayes’ Theorem.

EL: Yay, I'm so glad that someone will be talking about this! Because I know that this is a great theorem and—confession: I just, I don't appreciate it that much.

KK: You know, same.

EL: I need to be told why it's great.

KK: Yeah, I taught probability one time and I said, “Okay, here's Bayes’ theorem.” I kind of went all right. Fine, but of course the question is what's the prior, Mr. Bates? So tell us. Tell us, please.

EL: Yeah, preach!

KK: Preach for Reverend Bayes.

SC: You know, I don't think there's any any preaching needed. Because I always say this. I mean, there are two bits of statistics, there’s the frequentist and the Bayesian. And I always liken it to rugby union, and rugby league, which are two types of rugby in England. It's different codes, but it's the same thing. So to me, Bayes’ theorem, it's just the way that we naturally think. And it's beautifully simple, and all it does is let you take everything that you know and every piece of information that you have, and use that to update the overall outcome. And you're right, that the really big arguments come about from what the prior is. What is the background information that we have, and can we have actually genuinely have a true prior? And some people say no, because you might not have any information. But that's the great bit! Because then you can go and find out what the prior is. You have to be absolutely open about what you're putting in there. I think the really big debate comes around whether people are happy with uncertainty. Are they happy for you to not give an exact answer? If you go and you say, well, this is the prior, this is what we think the information is as well. And we combine these all, combine these priors, and this is the answer. Let's have a debate. Let's start talking about what we can have. Because at its simplest, you've got two things you’re timesing together. Just two numbers. Something that runs your mobile phone. I mean, that’s quite nifty.

KK: So can we can we remind our listeners what Bayes’ theorem actually says?

SC: Okay, so Bayes’ theorem takes two things. It takes the initial, or the prior distribution. Okay, and that's the bit where the argument is. And that might be just, what's the chance of something happening? What do you think the probability is of something happening? And you combine that with something called likelihood ratio. And it's real simple. The likelihood ratio is just a ratio of the probability of the information, or the evidence you have, assuming one hypothesis,divided by the probability of that information assuming another hypothesis. So you just have to have those two values. [And I say you just have to keep it.

And then all you have to do is times them together! That really is it, and when you start to say to people, it's just two numbers—Now, you can turn that into three numbers if you want. You can turn the likelihood ratio bit into its two separate parts. And you can show Bayes’ theorem very, very simply with decision trees, and that was part of the reason I used decision trees in the Math-Off, was just to show the power of something that is really quite simple, that can drive so, so far. And that's what I love about Bayes’ theorem. I always describe it as something that is stunningly elegant, but unbelievably powerful. And I always liken it to Audrey Hepburn. I think if it were to be a person, it would be Audrey Hepburn. Quite small! I'd say it's, it's this amazing little thing that has two simple numbers. But goodness me, getting those numbers, well, I mean, you can just have so much fun! I think you can.

And maybe it's just me that likes finding the patterns in the numbers and finding those distributions. Coming up with the priors. So come on, Kevin, you said, you sat there and your class said, “Well, what's the prior?”

KK: Yeah.

SC: What do you say? How would you tell people to go about finding a prior? Are they going to use their subjective opinion? Are they going to try and find it from data?

KK: Well, that that is the question, isn't it? Right? So, I mean, often, the problem with probability sometimes is that—at least, like, in political forecasting, right—people tend to round up probabilities to 1 or lop them off to zero. Right? So for example, when, you know, when Trump won the election in 2016, everybody thought it was a huge shock. But you know, 538 had it as, you know, Hillary Clinton was a two-to-one favorite. But two-to-one favorites lose all the time, right?

SC: Yeah.

KK: And and so the question then is, yeah, people like to think about one-off events. And then the question is, how do you estimate the probability of a one time event? And you have to make some guess, right, at the prior. And that’s—I think that's where people get suspicious of Bayes’ theorem, or Bayesian statistics, because how you make this estimate? So how do you make estimates in your daily work as a consultant?

SC: Okay, so we do it in a variety of different ways. So if we're really lucky, there’s some historical data we can go looking at.

KK: Sure.

SC: And often just mining that historical data gives you a good starting point. I always get slightly suspicious of flat distributions. Because if we really, really don't know anything other than that, I think maybe a bit of research before where you find the prior is always a good thing. My favorite priors are when we go and talk to people and start to get out of them their subjective opinion. Because I like statistics, I genuinely love statistics, because of the debate that goes on around it. And I think one of the things that people forget about math is that it's such a living subject. And there are so many brilliant debates—and you can call some of them arguments— people are prepared to go and say, “Look, this is my opinion and this is what I think the shape is.” And then we can do the analysis. Inevitably somebody will stand up and go, “Well, that bit is wrong.” Okay, so tell me why!

EL: Yeah.

SC: What evidence have you got for us to change the shape, or why do you think it should be skewed, or Poisson, or whatever we're using? And sometimes, if we haven't got time to do that we can start to put in flat distributions. We can say, “Well, we think it's about normal.” Or “We think on average, it'll be shoved a little bit to the right or a little bit to the left.” That's the three main ways we go about doing it. And I think the ability to be absolutely open and up front about what you know and what you don’t know helps you find that prior. And I don't really understand why people would be scared of running away from that. Why you would not want to say what the uncertainty is or what you're not sure about. But that might go a long way when people think that math is certain.

EL: Yeah.

SC: That when you say the answer is 12, well it’s 12. And not, “Well, it’s 12 because we kind of do it like this, and actually if something changes, that number might change.” And I think getting comfortable with uncertainty and being uncomfortable, is really the crux for developing those priors.

EL: Yeah. Well, I guess for me, it's hard to reason about statistics in a non frequentist way. Meaning—you know, I'm comfortable with non frequentist statistics to a certain degree. But just like what, as you were, saying, like, what does a 30% chance mean if it's not that we could do this 10 times that have it happen three times. But you can't have a presidential election—the same election—10 times, or you can't run Monday’s weather 10 times, or something like that. But it's just hard for me to interpret what does it mean if there isn't a frequentist interpretation?

SC: Yeah. One of the things we found that works really well is if you start showing patterns—and that's why I always talk about patterns, that we find patterns. It's when you're doing Bayesian stats with priors if you start to show the changes as curves, and I don't mean the distribution, but I mean, just as that rising and falling of numbers, people start to understand what's driving the priors, what assumptions are changing those priors. And then you start to see the impact of that, how the final answer changes. That can be incredibly powerful. Often people don't want that set answer. They want to know what the range is, they want to understand how that changes. And showing that impact as a shape—because I think most people are visual. When you show somebody a surface or, you know, a graph, or whatever it is, that's something you can really get a grip with. And actually I come from a Bayesian belief network. So I kind of found out about Bayes’ theorem by chance. I never set off to learn Bayes’ theorem. I set off to design [unintelligible]. That’s what I grew up wanting to do. But I ended up working on Bayesian networks. That’s the short version of what happened.

EL: So, how—was this a “love at first sight” theorem? Or what was your initial encounter with this theorem? And how did you feel about it? Since this is all about subjective feelings anyway!

SC: Well, my PhD was part-time. I spent eight years collecting subjective opinions. So I started a PhD in Bayesian networks, and there was this brilliant representation of a great big probability table. And this is a while ago now. And I’ve moved on a lot into [unintelligible]. But I've got this Bayesian network and supervisor said, “Here we go,” and I went, “Ah, it’s just lots of ovals connected with arrows”

And I went, “There must be something more to this.” And he went, “There’s this thing called Bayes’ theorem that underpins it and look at how it flows. It’s how the information affects it.” And I went, “Okay!” And so, as with all PhDs, you have this pile of reading, which is apparently going to be really, really good for you.

So I got my pile of reading. I went, “Okay.” And genuinely I just thought, “Yeah, it's just kind of how we all work, isn't it?” And I really had not liked statistics at university at all because I’d only really done frequentist statistics. And it’s not like I dislike frequentist statistics. I just didn’t fall in love with it. But when there was something I could see—and I genuinely think it’s because it's visual. I see the shapes move, I could see the numbers flow, I could see the information flow. I thought, “Oh, this is cool stuff. I understand this. I can get my head around this.” And I could start to see how to put things in and how they changed. And I think also I've got at times a very short attention span. So running millions of replicates never really did it for me.

EL: Yeah.

SC: So I had a bit of an issue with frequentist, where we just have to run lots and lots and lots lots of replicates.

EL: Right.

SC: Can we not assume it's kind of like this shape and see what happens? Then change that shape. Look, that’s great. That's much better for me.

EL: Yeah. So it was kind of a conversion experience there.

SC: I think, for people my age, probably. Because I don’t think Bayesian statistics, years ago, was taught that commonly. it's only really in the past sort of maybe decade that I think it's become really mainstream and been taught in the way it is now. Certainly with its its wide applications. That's what I think people just go, something that they've never heard of is now all in the AI world and it’s in your mobile phone, and it's in your medicine, and it's in your spam filters. And when it suddenly becomes really popular, people start to see what it can do. That's when it's taught more. And then you get all these other debates.

KK: So the other fun thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with Bayes’ theorem?

SC: So this caused a lot of debate in our household.

KK: It always does.

SC: Yeah. And I am going to pair Bayes’ theorem with my favorite food, which is risotto, because risotto only takes three things. It only needs rice and onions and a good stock.

KK: Yes.

SC: And Bayes’ theorem is classically thought with three numbers. And it’s really powerful and gorgeous. And risotto only takes three ingredients, and it’s really gorgeous.

KK: And also, the outcome is uncertain sometimes, right?

SC: Oh, frequently uncertain. And if you change those prior proportions, you will get a very different outcome.

KK: That’s right. You might get soup, or it might might burn.

SC: So, I am going to say that Bayes' theorem is like a risotto.

EL: And you mentioned Audrey Hepburn earlier so maybe it’s even more like sharing a risotto with Audrey Hepburn.

SC: That would be brilliant. How cool would that be?

EL: I know!

SC: I will have my Bayes’ theorem discussion with Audrey Hepburn over risotto. That would be a pretty good day.

EL: Yeah, you could probably get a cardboard cutout. Just, like, invite her to dinner.

SC: Yeah, I'll do that. I'll try and set up a photo, superimpose them.

EL: Yeah.

KK: But Audrey Hepburn should be breakfast somewhere right?

EL: But you can eat risotto for breakfast.

SC: Yeah, you can eat risotto any time of the day.

KK: Sure.

SC: There’s never a bad time for risotto.

KK: No, there isn't. Yeah. My wife actually doesn't like risotto very much, so I never make it.

EL: So is that one of your restaurant foods? So we have this whole like foods that you you tend to order at a restaurant because your partner doesn't like them. And so it's like something that you can—like I don't really like mushrooms, so my partner often will order a mushroom thing at a restaurant.

KK: Yeah, so for me, I don't go out for Italian food because I can make it at home.

EL: Okay.

KK: So I just have a generic I don't I don't eat Italian out. There’s kind of no point, I think.

SC: So you’re right that risotto is my restaurant food because my husband doesn't like it.

KK: Oh.

EL: Aw.

SC: It's my most favorite thing in the world, so yeah, every time we go out, the kids go, “Mom, just don't get the menu. There’s no point. We know what you’re getting.

EL: Yeah. So you said this caused a debate. Did he have a different opinion about what your pairing should be?

SC: Well, there were discussions about whether it was my favorite drink with [a bag of crisps?], and what things could be combined together. And I said, “No, it just has to be risotto.”

KK: Okay. Excellent.

EL: Yeah, we do make that at home. And actually the funny thing is I don't really like mushrooms, but I do like the mushroom risotto that we make.

SC: Oh.

EL: Yeah.

SC: So you've not got a flat prior. You've actually got a little bit of a skew on there.

EL: Yeah, I guess. I’m trying to figure out how to quantify this. Yeah, like my prior distribution for mushroom preference is going to depend on whether it is cooked with arborio rice or not.

SC: See, there we go and you don’t have to worry about numbers you just draw a shape.

EL: Yeah, nice.

KK: Cool. So we also like to give our guests a chance to plug anything they want to plug. Do you have things out there in the world that you want people to know about?

SC: So the only thing I think that's worth mentioning is I do some Royal Institution maths master classes, where we go out and we take our favorite bit of math, and we go and take it to students who are between the ages of about 14 to 17. And that's really what I'm doing coming up in the near future, and they are a brilliant way for lots of people to engage with maths.

EL: Oh, nice.

KK: That’s very cool.

SC: Yeah. They are really good fun.

KK: Have you been doing that for very long?

SC: I’ve been doing them for about two years now. And the first one I ever did was on Bayes’ theorem. And I've never been so terrified, because I don’t teach. And then you have this group of students, and they come up with just the best and most fantastic questions. Every time you do it, you go, “I hadn’t thought of that.”

KK: Yeah.

SC: “And I don't know how to answer that question straight away.” So it's brilliant, and I love doing them. So that's kind of what we've got coming up. And you know, work is just going to be keeping me nicely busy.

EL: Nice.

SC: Yeah.

KK: Well, this has been great fun. Thank you for joining us, and congratulations on being the world's most interesting mathematician for this year.

EL: Yes. Yeah, thanks a lot.

SC: Thank you. I’ve been so excited to do this. I've been listening to your podcast for quite a long time, and I couldn't believe it when you emailed.

okay, thank you very much.

Okay. Thanks.

On this episode, we had the pleasure of talking with Sophie Carr, a statistics consultant and winner of Christian Lawson-Perfect’s Big Internet Math-Off last summer. Here are some links you may enjoy as you listen to this episode.

As we mentioned at the top of the show, Evelyn’s math page-a-day calendar is available for purchase in the AMS bookstore!
Sophie Carr’s twitter account
The Big Internet Math-Off at the Aperiodical
Royal Institution Masterclasses
Sophie Carr is this year’s World’s Most Interesting Mathematician. We also had last year’s World’s Most Interesting Mathematician, Nira Chamberlain, on the show in January. Find his episode here.