Max: Welcome, everyone. Welcome. You have reached another Local Maximum. How you doing? I hope you enjoyed the three day weekend. I hope you had a great Memorial Day weekend. I know I did.

Still lots to do here in Connecticut, though, with the move and the unpacking, and so it's taking up a lot of my time. I do have some interviews coming up, which is going to be good, and we can kind of fill those in as we head to the episodes of the 280s here on The Local Maximum.

A little update on Local Maximum Labs, which is basically just what I call me working on papers, usually math and computer science papers that put out some of my ideas. But of course, maybe I can write about other things as well. But I actually went through this process of updating a paper called Fast MLE, for Fast Maximum Likelihood Estimate for the Dirichlet-Multinomial. I'm not going to get into all the details to what all those crazy words mean today, but I wrote that back in 2014. Really? I wrote it ten years ago in 2013? But then it took me a year to realize, yeah, it's going to take a while to get this published and I just better put it out on archive and move on because I've got other things to do with my life.

So what is it? The original purpose for that paper was when I was working at Foursquare and trying to figure out how people were reviewing different venues and trying to see how that generalizes. So, for example, if I'm looking at a restaurant and I see there are nine reviews and let's say five of them are positive reviews, two of them are like meh reviews, and how many do I have left? I have two left. Two were negative reviews. And it's like, okay, well, is that a large amount of data? Is that a small amount of data? How much can I take away from that?

And it turns out, well, it depends on the context. So maybe if you're thinking about a restaurant, maybe if you had talked to four random people who went there and they all said it's great, maybe in that case you would think, okay, this is probably a pretty good idea, maybe I should go. But if you had four people who all said it was terrible, maybe that's an indication that you shouldn't go.

Then again, if you're talking about like a random coin that you almost certainly think is a fair coin, if it lands on heads four times versus if it lands on tails four times, you are not going to have a different opinion on that coin just based on those four data points.

So there's like a fundamental difference in context between not a fundamental difference and just maybe it's more of just a difference in terms of degree, in terms of context. Whereas on one hand, you have a very weak prior where you want to take this data that you have and sort of lean into the data that you're getting. And in another case, you have the coin where you have the very strong prior.

And so you're like, okay, well, this data doesn't really change my opinion. Now, interestingly enough, you could have something called not called, but you could have these crazy priors in terms of the restaurant example, where maybe all restaurants in a certain area are either good or bad, and there really is not much in between, and everyone kind of agrees. In that case, you could take a single data point and generalize it to what the restaurant is, and you could do it very well. So there are cases like that as well.

So how do we distinguish between all of those? And that is the purpose of the paper. You use a Dirichlet distribution to do it. I came up with a good way to compute this based on a full data set, and so I put that out.

Over the years, I've noticed a few errors in the paper that I went ahead and fixed, and also it was hard to get it published at the time, but for the last ten years, people have been contacting me, asking about it, asking more information about it, to seeing whether they want to use it or not. And so if it's still relevant to people after ten years, I figure it's probably worth updating and making it better. So if you're interested in that kind of stuff, go to Local Maximum, go to localmaxradio.com/labs to see all the papers.

I also ended up doing a little addendum where I solved a linear algebra problem. I don't know how much you guys are into inverting matrices. Maybe not. Maybe some of you this is all going over your head with the math stuff. But if you're into linear algebra, if you're into inverting matrices, I sort of hand waved in the paper. Oh. We start with this equation, and then we end up with this algorithm. And the reason why I did that was because other papers in the field did the same thing with the same exact calculation. So I'm like, okay, I don't need to explain this.

Well, it turns out that not only did I get the algorithm wrong and that I quoted the wrong algorithm, so that was a problem. But also, it's not that obvious to people. It's like, hey, show your work. How did this math problem go out? So I did that, and it's basically, how do you invert a matrix that's a diagonal plus a hessian?

So anyway, if you're interested in that localmaxradio.com/labs, we've also talked about the Dirichlet distribution previously on the show. That would be in episode 265, the Multi-Armed Bandit. Okay, so that's some Bayesian inference. That's some mathematics.

Let's talk about something that's a little bit along the same lines, definitely along the kind of epistemological lines. How do we determine? How do we get at the truth? How do we know what's true? But this is something that I think everyone can wrap their head around, and it's a pretty disturbing kind of thing.

So there's this article going around about high school debates. It's from thefp.org. And before I get into it, I thought that the description of the bio of some of the judges in high school debate was actually satire. I was like, there is no way that this could be real. But as far as I can tell, it's real. And so imagine you're a high schooler. You're not an adult. You're in high school. And a lot of people in high school are driven. They want to do a good job. They care about getting into a good school.

So they're really kind of disturbed by the idea that an authority figure might kind of put a red light on that or kind of tag their resume or whatever. So imagine you're about to and these extracurriculars are really important to you.

So you're about to enter a high school debate, and you read the judge's profile, and it looks like this: Before anything else, including being a debate judge, I am a Marxist Leninist Maoist. I have realized as a result of this, I cannot check the revolutionary proletarian science at the door when I'm judging, and that's both impossible and opportunism. If you had me as a judge before, this explicit decision of mine does not change how you understand I evaluate rounds with one specific exception. I will no longer evaluate and thus ever vote for rightist capitalist, imperialist position arguments, meaning the argument positions which defend the bourgeoisie class, dictatorship, monopoly, capitalism, and thus imperial-

What is it with these people? First of all, I'm trying to read this person's bio, and they like to put all these isms in it. Like it's like, oh, you're not just a bad guy, you're a capitalist, an imperialist, a bourgeoisie, a monopolist, an imperialist. They say imperialist twice. That's interesting. All right.

From a right wingist, the politics, ideology, and practice of right wing of the bourgeoisie. Examples of arguments of this nature are as follows fascism, good. Capitalism, good. Imperialist war, good. Neoliberalism, good. Okay, look, these people just love to go on and on about different with different labels. And this is already just getting of course, there's a whole anti Zionism screed here, so this is pretty crazy.

Let's try to get to the point where this person is going to say what they're going to do to you, because this person is in sort of an authority position in the context of a debate round, by default, this will function through drop the argument, i.e., If you read an advantage or DA that represents the right wing-.

Okay, I'm sorry. It's horribly written, and I'm trying to read something that's horribly written. So not only does this person have terrible ideas, they're writing something horribly. But I'm not here just to make fun of it. We'll talk more after I'm done. I just can't get through some of these things sometimes. I won't evaluate that advantage if your whole strategy is rightist, capitalist, imperialist in nature. I won't evaluate, this only becomes drop the debater if you vehemently egregiously defend counter revolution.

All right, we better stop before we lose touch with reality. I'm sorry. This person is crazy. First of all, is this what Aaron has told me? This might be nutpicking. Are we taking, like, the thousands of debate judges here and just finding the nuts? And there's a little bit of that, but I want to see why it's still very alarming.

So what's really going on there? Have there always been nuts in the high school debate judge circuit? And are there nuts on either side, or are there just people who are, is it only kind of the Marxist sort of point of view that feels the need to do this? Is there kind of an opposing side that sort of talks this way?

Is there a libertarian who comes in and says, anyone who wants the government to do anything, I'm going to drop you from your if you defend these criminals in government, I am going to drop you from the debate? Are there MAGA people who are know if you say anything bad about Trump, I am going to give you bad marks in your debate? Of course not.

And, well, we wouldn't expect that. So what's really going on? So we'll look at this article that describes it, and then I'll give my point of view. The article is, At High School Debates, Debate Is No Longer Allowed. It's by James Fishback.

But first, seriously, what would you do if you're a student? I was slightly skeptical or counter narrative in high school, but not that much. And most people, even if you're going to be in high school, you're going to disagree with everything that the teachers, the people in authority say. It's hard to do it effectively when you don't have a whole lot of life experience. And honestly, you're also like, you're a minor, so you are not in a very good position to disagree with that.

Some of the stuff, so I might have said that I disagree with some of it, but you have to be resigned to the fact that you're playing by their rules. And I don't think having this judge would have changed my opinion on anything, but it would certainly affect my debate performance, and I think that's what's going to happen to the students here. So I think it'll lead to a lot of self censorship. I don't think it's true of what a lot of people are saying that it's actually going to convince young people to think this way. You're generally not convinced.

First of all, these arguments that this judge had in their bio, it's not a coherent argument. I don't think there are a lot of people who read this or they're going to be like, oh, I like this. This makes sense to the world for me. And so it's not going to cause people to become true believers, other than people who really want, like, power over someone else and want to find the magic words that they can say that could be like, oh, this is a way to tarnish people with these magic words.

But if a belief system doesn't really help you think through your day to day life, if you just want to shout, okay, imperialism, capitalism, fascism, it's all one big thing. And if it doesn't make a whole lot of sense to you in terms of figuring out how to live your life, you're not going to be indoctrinated for very long. It's almost like the idea, like if somebody is forced to agree with you, they probably really don't agree with you. They're just being forced to agree against their will.

So even if students adopt these judges' worldviews in a practical play to get ahead or keep their head down, no one is going to be defending it vociferously. So I don't think they're indoctrinating the next generation like they think they are, particularly because their worldview is not very coherent.

But let's read more from the article and see what this person has to say. My four years on a high school debate team in Broward County, Florida, taught me to challenge ideas, question assumptions, and think outside the box. It also helped me overcome a terrible childhood stutter, and I wasn't half bad. I placed 9th in my first time at the National Speech and Debate Association Nationals, 6th at the Harvard National, and was runner up at the Emory National after college. Between 2017 and 2019, I coached a debate team at an underprivileged high school in Miami. There, I witnessed the pillars of high school debate start to crumble.

Since then, the decline has continued from a competition that rewards evidence and reasoning to one that punishes students for what they say and how they say it. The author then goes on to the examples I gave above and raises the question of what this means for the students. He adds, how does that sophomore feel when as she walks into her debate round, how will knowing that information about the judge change the way she makes her case?

Traditionally, high school students would have encountered a judge like former West Point debater Henry Smith, whose paradigm asks students to focus on clarity over speed and reminds them that every argument should explain exactly how they win the debate.

In the past few years, however, judges with paradigms tainted by politics and ideology are becoming common. Debate judge Shubham Gupta's paradigm reads if you are discussing immigrants in a round and describe the person as illegal, I will immediately stop the round, give you the loss with low speaks, low speaker points, give you a stern lecture, and then talk to your coach. I will not have you making the debate space unsafe. Once students have been exposed to enough of these partisan paradigms, they internalize that point of view and adjust their arguments going forward.

And so I think, first of all, it does matter. Even if 1% of the judges are openly espousing these views like this, and especially if they're saying, oh, I'm going to stop the debate, and I'm going to chew you out, it's very scary to be chewed out by an authority figure, even as an adult. But as in a student, it's very hard to stand up to that. So even if you see that like, oh, there's only one judge who's going to do this, or 1% of judges are talking like this, it still has an effect, you're going to be very concerned about it. And so, yeah, 1% is a lot. A half percent is a lot. Even a handful is a lot.

And also you have to figure that if a very small number of judges and it's not like there's one there's a whole bunch are putting this stuff on their profiles, imagine how many are doing this and just not saying that they're doing it. Probably a good multiple of that. So a good percentage. You got to figure that if you're a student. And so even if it's a small number of people, it has a huge effect.

Just like at work, when there were a small number of people on Slack who were enforcing their political ideology at work, it's kind of like, well, why do I work here? Why isn't this being shut down? And the fact that this isn't being shut down in the debate organization shows that the debate organization kind of tacitly approves of this. Or if they don't want to do anything about it, then they think it's fine and they're going to allow it to continue.

Again, you don't need every judge doing this. You only need 1%, makes a huge deal. So, yeah, this is not just concerning. It's not perhaps surprising. I mean, this is just how really every institution has gone over the last five years or so. And so we need to rebuild our institutions, and we need to rebuild the norms, and the norms have to revolve around not a political ideology, but how to get at the truth in the end

And so that's why this particular story bothers me so much. And that's why this particular story I thought it was important to mention it because it's not just that people in authority have bad ideas. It's not just people have terrible ideas. It's not just that people have unclear ideas.

It's that they're going after teenagers in their formative years who are trying to find the truth. And they are attacking the idea of looking for truth itself. This kind of thing is an attack on truth and they're providing a disincentive for younger people to seek the truth. That, to me, is the most offensive thing about this whole thing.

And again, even just if you judge, this has a tremendous effect. So I would like to know what you think. If you go on the locals, maximum.locals.com, how do we fix this? And I'm thinking that ultimately the people in charge of these organizations have to shut this down and they have to have in any organization that you run, maybe you could think of it in terms of your company or your workplace or whatever it is that, hey, let's come up with a bunch of standards. Here are some lines that you do not cross in order to have kind of an open debate.

Maybe we have an organization that has a particular point of view and we can define the range for debate, but there should be rules against a kind of verbal bullying for people who are just seeking the truth. In fact, I can give some good examples of when I was an undergrad at Yale, I often went to Yale political union debates.

And what people would have to do is when they're giving an argument in one of those debates, they would have to go up and they would have to take questions from the audience and you could ask any question you wanted, really. And some of those questions were very illuminating. But it was never the case where, oh, you couldn't bring something up.

There was a certain amount of respect that you had to show to the speaker and to the people asking the question. But having that broad range of questions really helped people improve their arguments and also get at the truth in the end.

So again, I wish I could speak more eloquently about this and maybe I'll work on it in the future. And of course, I wrote about my Ode to the Podiverse, how the podcasting industry itself is diverse with people coming on each other's podcast. It does not suffer from the type of censorship and devastating kind of deplatforming that other platforms have, that other media systems have. And that's maybe why I like podcasting so much.

But I feel like we need to talk a lot about how the truth is not just how the truth is spread. It's not like, hey, I know the truth, I'm going to spread it, but how to seek the truth. And we've kind of lost that in our country right now. So very upsetting here. Maybe there's something we could do about the future. All right.

Narration: And now the probability distribution of the week.

Max: All right, it's the probability distribution of the week. Today we are talking about a really fascinating distribution. It's called the Cauchy distribution. Cauchy is a mathematician, as you know. But first, of all, I want to talk about the idea of the real number line and why it's so difficult to say, hey, I want to find a random number, even a random integer or a random real number. I'm thinking of a random number, right?

Well, what does that mean? Because even if you're talking about let's talk about random integers, like most integers are way higher than any integer that we can imagine, or way lower, or let's say positive integers, right? If I say, hey, give me a random integer, and you say, I don't know, a million, billion, billion, billion, trying to multiply that out, I'm sure it has a name.

But even if you say that, the person will be like, wow, that's a very low positive integer compared to the other positive integers out there, because it's an infinite set. And so the same thing can be said about real numbers. You can't normalize a probability distribution that's just flat and uniform across all real numbers.

Now, I'm okay with using improper distributions like that. They work very well with Bayesian inference if you just say, okay, let me just make my distribution function f of x equals one, and we're not going to normalize it, but that's okay. We could still deal with it in terms of Bayesian inference. And so I'm going to consider every real number to be equal. Sure, okay. But it sort of is a little bit jarring when you get into the mathematics of it.

Let's say if we look at it another way, look at the real number line in another way, so oftentimes we think of the real number line and we think, okay, you have this number zero. And then right next to it, one unit away, you have this number one. And then next to that, you have this number two. Same thing on the negative side, you could go infinitely out in both directions.

So you imagine that space extends infinitely to the right, infinitely in the positive direction, and space goes on infinitely to the left in the negative direction. Okay, so that's a pretty crazy object to imagine. Mathematicians do it all the time. Engineers do it all the time. That's what you imagine.

But if you think about it, the first time you were exposed to this, if you remember, and probably the first time children are exposed to this idea whenever they're exposed to it, I'm not sure exactly when. It's probably like, wow, you could really just extend that forever, or what does that mean? Or extend it as far as we want.

Okay, but then there's another way to look at this infinite object, and it's a way that you could view it all at once, and that is sort of the projected line, and you kind of imagine it as a circle. Okay, let's think about it this way. Imagine that you have a real number line, and then you have a circle sitting on top of that number line, okay, the bottom of the circle is at zero, and the top of that circle is somewhere else. It's above zero. It's just a line with a circle on top of it. And then you're going to draw a line from the top of that circle through some point on the circle and then extend that and see where it hits the line. That means that every point on the circle corresponds to some point on the line itself.

So if you draw the line from the top of the circle to the bottom of the circle, then you're looking at the number zero. Let's suppose you draw a line from the top of the circle and you go through the rightmost point of the circle. And then let's say you extend that. Let's say you get to the number one, okay? So that point represents number one.

If you go up in the circle a little bit, you're going to get higher and higher numbers. And so if you want it to go through the point a million, all right, fine. You draw a point at the top of the circle. You draw a point in the line at a million. You connect those two points, and it intersects with the circle at one spot, and you can label a million there. So if you project that line onto a circle, you could essentially see all of the numbers on the number line all the way up to arbitrarily large. Then you can kind of view it all at once in that circle. You don't need to extend your page. You don't need to extend space.

So that's pretty cool. And then if you get very, very high numbers on the right side, you're going to be going through just at the top of the circle, maybe a little bit to the right. And then if you have very, very low numbers and you're connecting to the left hand side, you're going to get just to the top of the circle, maybe a little bit to the left. And so that top of the circle itself could be considered an infinity.

Now, it's not positive infinity or negative infinity. It's just a projective infinity, which means that it's sort of like horseshoe theory. Both the extreme right and the extreme left connect at this one point. And so that's sort of a projective number line, so to say.

Now, last week when we talked about circular distributions, we discovered that it's perfectly normal to have a uniform distribution around a circle. So, okay, now we have a circle. Why don't we build a uniform distribution around that circle? What do we get? Let's take that uniform distribution, do another change of variables, bring it back to the real line. What do you get? That's an interesting question to ask.

Well, it turns out and you'd think that, well, now I have like a significant amount of the distribution up near infinity just as I do near zero. So it turns out that this is something called the Cauchy distribution. The form of its equation is actually pretty simple, one over one plus x squared. But who's counting? And as you can imagine, it's very fat tailed.

So the normal distribution is thin tailed. If you go on Python and you ask for something from a simple bell curve, normal distribution, and you ask for it, you say, hey, I want a mean of zero and a standard deviation of one. You'll get a lot of numbers that are between negative one and positive one. You might get some twos, a few threes, so on and so forth.

You can also ask Python through the NumPy through those libraries. You can ask it for samples from the Cauchy distribution. And you'll also get a lot of small numbers. You'll get zero, one, all, that kind of thing. And then every once in a while, you get something in like the hundreds or even the thousands. And so that's the fat tail. That means that you can get a really wild pitch.

Now, the Cauchy distribution is crazy. Even if the graph looks a little normalish, it looks like it's centered at zero. You know, you get some tapers off at very high, very low. You can get values that are very far off in that tail. And so, in fact, it's so fat tailed that you can't even say that it has a mean and its variance is infinite too. Now, you kind of want to think that the mean is zero. Well, if you look at it on the projective circle, you could see why it doesn't have a mean, because it's equal all around the circle.

Obviously, there are different parameters that you could set. So you can kind of center it at zero or you could kind of center it somewhere else. And so there's kind of a mode that you could set where the curve is highest and kind of think of that as the mean. But in terms of the mathematical definition of the mean of the average, it doesn't have that.

Another few interesting things about Cauchy distribution. It's actually the ratio of two normally distributed variables. So that means that if you take a normal distribution, that's the one that's usually very close to zero. And then you take another one and you divide them, you get something that is distributed, like the Cauchy distribution.

And now when you look at it that way, you could see from another sense why you sometimes get a wild pitch, because you have the numerator that's centered around zero. You have the denominator that's centered about around zero.

So the denominator could very easily be very close to zero. And when the denominator gets very close to zero, it kind of blows up like that. So that's how it works. It's actually pretty easy to draw from it. If you could draw from a normal distribution, you could draw from Cauchy distributions. This one is actually really fascinating. It's a really good thing to think about if you have data that doesn't behave very well.

And it's also used in something for those of you who are doing Bayesian inference or interested in Bayesian inference, it's using something called the horseshoe prior, which Bayesian statisticians use to induce sparsity. So a few weeks ago, we talked about the Laplace distribution, how that's kind of used in something called Lasso regression, which basically, if you have a bunch of parameters, you want a lot of them to go to zero. So your model is small.

There's kind of a more Bayesian way of doing this, and that uses Cauchy's distribution. I haven't actually dove into it as much as I feel like I should have, but I will post a paper of that on the website.

All right, so Cauchy distribution is a lot of fun. Glad we got to do it this week. I'm hoping I can have Aaron on next week. We'll see how it goes. And I'll be going to New Hampshire for Pork Fest in a few weeks. So for those of you who are going, please let me know. I'm actually going to be in New Hampshire for two weeks, so for those of you who live in New Hampshire, certainly let me know. Maybe we'll meet up or maybe we'll meet up in Lancaster. And yeah, definitely looking for my next batch of guests, so we'll get to that soon. All right, have a great week, everyone.

That's the show to support the Local Maximum, sign up for exclusive content and our online community at maximum.locals.com. The Local Maximum is available wherever podcasts are found. If you want to keep up, remember to subscribe on your podcast app. Also check out the website which with show notes and additional materials at localmaxradio.com. If you want to contact me, the host, send an email to localmaxradio@gmail.com. Have a great weekend.

Based in Sydney, Australia, Foundry is a blog by Rebecca Thao. Her posts explore modern architecture through photos and quotes by influential architects, engineers, and artists.

May 30 Episode 280 - How to Attack Truth Seeking

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