Live CEOing Ep 677: Language Design in Wolfram Language [Around, Min, & WFR Review]

Okay. Hello, everyone. Welcome to another Wolfram Language Design review, part of our Incremental Language Development series. And we have a variety of topics here. First topic, around and min and numerical order. So I guess the question here is, what should min do with around? Right. A min of an interval is the low element of the interval. Correct.

Interval of centered interval. It's different of centered interval. It's the value of all minimums over all elements of all intervals.

Explain that.

So the standard thing that we do for evaluation of a numeric function of center interval is that we give an interval containing all values of the function. So for the function mean, we give an interval that contains.

Interval of, for example, 5, comma 1. Is that. Okay, so if I say min of that.

If you get min of two intervals.

Okay, I see. Min of. Let's say centered interval here, comma. And then let's say centered interval, 6, comma 3.

Right. And that's the interval that contains all the means of two elements, where the first element is in the first interval and second element is in the second interval.

I'm a little confused. So a sentence interval. If I look at the distribution of possible values that the number could have, in the case of the first one, five plus minus one, it ranges from four to six. Is that correct?

Yes.

And the second one ranges from three to nine.

Yes.

So how the heck does it get a min? That is not. Why isn't that min? Yeah.

So the mean, the lowest min that you can get is then three.

Right.

And the.

Why does it have any centered interval stuff? I mean, isn't the minimum just like. What's the min?

It's the set. It's the set of all mins you get where you take one element from the first interval and one element from the second. So the highest min you can get is 6.

Right.

If you.

I see. If it. Because that was the thing that you got. No, what is that? That's the top of the first interval.

Top of the first interval. Yeah. So it's the lower bottom and higher. And lower top. That's what you'll get.

There is a graphical example of this particular case in the next section.

Okay.

If you scroll down, you see. So there you are taking the mean of the yellow and blue intervals, and you get the green interval.

So it's from the lower bottom to the lower top.

I see. So that means. Okay, so the true data, so to speak, lies somewhere in the yellow interval and somewhere in the blue interval. And you're saying, given all possible true datas. Yeah. Okay. I Understand, I think the logic of this. Okay.

I think this is consistent with how center interval works. The important conclusion from here is that mean of two objects doesn't have to be one of those two objects. I think that's important lesson to learn here.

An important lesson. It's a little bit bizarre, but it's an important lesson. I mean, just like min of a single interval isn't an interval, min of an interval. Right. Is just a pure number, if I'm not mistaken. Yes, yes.

So for historical reasons, actually.

So do you think that's wrong?

So if we consider mean a numeric function as it is and what interval arithmetic is supposed to do for a numeric function, then it's wrong.

Oh, gosh.

I wouldn't call it wrong. I think it's a different approach.

Well, if you do sine of interval, you get an interval. If you do mean of interval and mini is a numeric function, just like sine, you get a number. That's somewhat strange.

Okay. All right. Anyway, but our topic today is around.

Exactly. So round sits in between intervals, numbers and distributions. And this discussion is to show that these different approaches have different opinions on what to do.

Okay.

And so I tend to believe that numbers is the most important precedent to follow here. And I think this idea of mean of several arounds returning one of them rather than a new one that wasn't there is something that users will expect.

What did Sanders suggest?

So Sander's suggestion is to return one of them and to decide based on some sort of tolerance. Because together with what to return there is the question of order.

Right.

So we have to separate the question of numerical order from the question of mean. The numerical order is easier because we can just return true, false, or 1, 0. There are two main choices. One is, do we consider overlapping? Do we consider a tolerance like equal does, or do we just go straight to the centers?

All right, wait a minute. So the issue is you've got a bunch of arounds. They effectively overlap in the way that this one does, for example.

Exactly.

And one thing we could do is just say, I mean, there's several different approaches. We could take the look at the centers, take the smallest center, just return that now, I mean, presumably. Okay, this is only relevant in the one dimensional case anyway because men is a one dimensional story. Other analogs of men that we have to worry about, like ranked min and so on, that are more complicated stories, or if we nail min, we automatically get these other things sort of philosophically nailed.

I think the first thing to nail is Numerical order. And the mean can be based on numerical order. Once we decide the sort of thing we were discussing with Adam, do we return another one of the objects that was there? Some sort of constructed new object, but that's a separate thing.

Okay. Oh, Sanders comment on our livestream is the thing is that they always overlap. That arounds always overlap because they're tails. They're animals with infinite tails.

Yes, but the tail decays very fast. So there are cases in which things are clear.

No, I.

Presumably there's a case where, you know, you're far enough on the tail that it's.

It's sort of the. The probability of them overlapping is like the. Well, yeah, there's some analogy to equal. Right.

Where.

Because.

Because equal also has, you know, approximate stuff. Right, Right. Well, which is the statement. This could be treated like numbers and like precision of numbers, but. Okay, let me ask another question. What about asymmetric intervals? Asymmetric arounds? I mean, what you're showing here, clearly if they're symmetric around, there is an ordering that can be defined by their maxima, AKA their means, right?

Yep.

If they're asymmetric, that's less obvious. And then one starts worrying about, you know, should one think about the integrals or what should one do?

I still think the center is the preferred value, even if they are asymmetric, because that's. I mean, when you spread an asymmetric around, you are still thinking that the center is the value, the most probable value.

Yes. Okay, but if you've got. If you're comparing two things, and I mean, can somebody make up a version of this where it really has most of its support? The one who has the center to the right has most of its support somehow to the left of another one. Can't that happen?

Yes. So for example, if you go up a bit and change the definitions of the rounds. Okay, so create two arounds, one symmetric, one asymmetric.

Okay.

So do AA equals.

Okay, hold on. Saying AA equals round of what four.

Comma one, we can keep that one. And then AB could be around the five, list one, comma three. And now re evaluate that plot. So that's.

Heck is that.

So that's the sort of thing we are.

So what the heck was down here? What happened here? Well, it does.

Yeah. So what this represents is an around in which the. The probability is still half Gaussians, one to one side, one to the other one.

I didn't know it did that. I thought it. I thought it somehow. Well, I didn't really think about It. I thought it was some kind of distribution with some kind of ketosis thing or something. I didn't really think about it. I didn't realize that was the distribution, that it was a piecewise distribution like that. Okay, Well, so that still has the feature. Yeah, I mean, so that didn't affect. It's not the picture that I thought I was going to see here. So. Okay, I understand.

So this is a way of still having Gaussian so that we can do all computations.

Yeah, right, right.

The question is if you. If you say. Say a b is equal 5 and then make the left bound really large and the right bound really small, then it will look like the center will be to the right of the first center.

But.

But actually most of the distribution will be smaller than the first one.

Okay, hold on. Tell me what values to put in here.

Ten comma, one and a half, one, something like that. So the.

That is.

I think the asymmetric cases is. Comes later. We should decide what to do with.

The center case is half the probability still to each side.

Yes.

Right. Okay. What does. What happens? Does anybody know your average physics situation? What do people actually do? They say I've got this number and this number and I take the min of those two numbers. What do they actually do? Do I pull down. My gosh, these are getting very long in the tooth these days. The handy little physicist. Dear, oh dear. I've only got ones from 1994. Well, I'm sure this hasn't changed much. Okay. Particle physics, you know, information booklet has sort of practical. Oh, look, statistics. They've got a whole section on that. But I wonder whether they talk about. I mean, what would you expect them to say if anything? Or will they not say anything?

Well, the question is not very well posed. If you don't specify more things like to some confidence levels and things like that.

Yeah, so.

That'S an alternative here to impose some sort of threshold above which we are sure of things. And below we just return unevaluated. Or it's a bit what interval does.

Right.

So if you try to compare two overlapping intervals.

So interval actually has 100% threshold. Meaning if they are disjointed and ordered. And they are ordered. If they overlap, then. Then we return unevaluated.

Right.

Sorry, say that again. If what when the.

So if intervals are completely disjoint, they are ordered. If they overlap, then the order function return unevaluated.

Yeah, but the disjoint property depends on setting a confidence level.

Here. Yeah. Here around you would need a Confidence level in intervals. They are sort of sharp things and bound it.

All right, so look, what are our options here? I mean, is there a preferred proposal for what to do?

So I think my proposal here is to judge by centers and then have a separate function or several separate functions that take confidence levels. And so if you want confidence levels, you have to go to a more powerful function.

Right.

And I'm. And you know, I'm not, you know, if you have, you know, 1.0, you know, plus or minus 1 and 1.1 plus or minus 10. I'm not sure that saying that the first one is less than the second is such an improvement.

For numbers. So numbers have a threshold. They have some seven bits of tolerance, and you have to go through those thresholds in order to accept things.

What would this thing with confidence both look like? Do you have a specific proposal here for what that function would be?

So it would be something like the distributions have to overlap more or less than such number. And we can impose 75% has been suggested by Sander. We could use 95% to be more in what physicists use.

Well, okay, but so is this an option? Would this become an option to numerical order or what?

I think we would need separate functions. In the same way that we have things like vectorless, we could have an aroundless that takes another parameter.

Oh, gosh. Right.

And Right. Numerical order can't return unevaluated.

Right, right.

I mean, the less and equal could take tolerant, you know, could have an implicit tolerance. Numerical order cannot.

But it can return zero when it doesn't know what to do effectively. It's like considering that they are equal.

Right. Which is. Yeah, well, yeah, right. Except zero. Right. Zero is supposed to mean that they're equivalent, not that they're incomparable.

What are we doing right now with around with. What does the code do right now? Does it do something sensible? Does it do something crazy?

No.

Around basically either returns unevaluated or if it returns something is because some general functions by coincidence do something. So I think in this case, effectively it's just comparing the centers.

But this is what you're proposing doing in general.

Exactly. Yeah.

Right.

But that's an accident of the fact that min has the numeric function attribute and, you know, I'm not sure, like I said, right. If that's, you know, if that's 1.0 comma 1 and you know, 1.1 comma, you know, 2.3. I'm not sure that, you know, returning the first one is really.

Well, that's What Jose is proposing is.

He'S that I, I understand that that's what he's proposing. And I'm saying I, I don't agree with that proposal.

Well, okay, so do you have a different proposal?

I think a different possibility would be to have center for the result to be expected value of the minimum and error to be standard deviation of the minimum. We cannot compute it symbolically, but we possibly would be able to approximate it.

Yeah.

A question I have is whether or not around is built on the assumption of Gaussian distributions or it doesn't even specify a distribution.

It will work with Gauss. It is doing the simplified math that is possible because it's assumed that everything is Gaussian. We're now realizing piecewise Gaussian.

So if it assumes Gaussian or piecewise Gaussian, then we can do confidence levels. And the thing I remember from undergrad physics lessons is that the three sigma limit is basically what they consider to be decidedly different.

Yeah, but so what, I mean, so what's your point then? I mean that if it's, if the two arounds differ by. Well, I mean, that's another question is if we assume that we really are treating these as Gaussians, then there are functions we can imagine that ask, you know, questions like at the three sigma level, are these two arounds different?

Yeah, but three sigma is pretty strict.

Whatever, Whatever. Who cares? One sigma, whatever it is, it doesn't matter. But I mean, in. And that's just computing the integral of those distributions. But I mean, what is the question that. Okay, the first thing seems to me like anybody who is relying on the min of 2 arounds, it sounds to me like the definitions are sufficiently fuzzy that if you're relying on. I'm just, I don't know. I mean, I'm, I'm. It depends on what the meaning of the thing is. If the meaning of the thing is this around is representing a large number of experimental observations that you are summarizing with this around. Yeah. So you've got two different arounds that both correspond to large numbers of observations. Then you take the min of those two, then you would expect something that takes the min of individual observations, much like we're doing in sentimental. Well, okay, but Itai, do you have a different proposal?

Well, I mean, my, you know, my proposal is that, you know, men should be like less and just return unevaluated. And you know, if we want to say that if it's at a certain signal level, we, we resolve, then great, but.

Well, then we'd have to have an option to control that, wouldn't we? Well, how does less do that? Less doesn't have such an option. Correct.

Oh, there's another possibility that's completely different which is that if you have a distribution and a confidence level, the minimum of the distribution at that confidence level just gives you the mean value minus whatever with a lower bound on the confidence level. So if say that you had four or one, the min of around four or one could possibly be three.

Yeah, but that's the same thing. That's something similar to what interval does, which we're not even sure we like.

Yeah, but I mean the reason I would suggest that is because if you have around 4 and 1 and around 5 and 3, you can create a joint probability distribution from those two different things, which I don't think it'd really be in a round.

No, it isn't. That's the whole point. It's not an around. I mean centered interval has that feature that it's doing these joint. Effectively this joint distribution thing.

Yeah. So I think I'm comparing with numbers and numbers always return something with numerical order and mean less, etc. And they judge by the centers. I added an example in chat to compare arounds with numbers.

Okay. Sandra is commenting on the weirdness of this. That doesn't evaluate, which I think is fine.

Right. And this is why, why I'm saying that this is this is an accident of implementation of the fact that min has the numeric function attribute. If min did not have the numeric function attribute, this would not happen. And but my list that min is should be made more like less. And Jose is saying that the opposite direction more like. More like men. And you know, in particular my example is, you know, suppose it's, you know, rather than the two having a one there, it's a two and a five.

So.

The fact that those two are different happens with interval already. So it's not that weird. And I think my point is that people are going to expect that when they do computations with arounds. It's like numbers, they always work. If we start to return things unevaluated, all computations will get stuck at some point.

I kind of agree. But then we've got that less up at the top there. What do we think should happen with that less? I mean, give you an example. Okay, which is going to get fairly silly if we say this and then this says a round of 100 comma 1. I think that's fairly silly.

You're saying that that should return True. Right?

Yeah, yeah, Right.

But suppose that's 100 comma 200.

This is where we're getting Brad's, you know, kind of criteria.

If it's a hundred comma 30, that's no 100. 200. We don't even worry about that. A hundred.

What do you mean we don't worry about it? We gotta do something.

100.

Exactly.

Well, 150 is already a problem though. Or 130 is already a problem because three. Three times 30 is 90. Okay, so that one, maybe.

We either have to decide, as we're doing with less, that there's some numerical fuzz. We're deciding. We could decide on default confidence interval here.

Well, can I ask another question?

Yeah.

Min, usually when I've used it just returns a number. If it's just returns a number, then numeric order follows as a consequence of that pretty easily. But what the proposal is here to return from min, a number plus possibly a confidence interval. So is that what we want or is there a way to make the input to min have a strict output of being just a number?

So when you see a number, we are talking about numbers with finite precision. And then these problems appear there too.

Right.

Because you may have two very, very unprecise numbers.

Right. But we elided that by having things like less just magically use some particular tolerance level.

Right, exactly. But that tolerance level is typically also joined to the idea of using the centers to decide. That's why adding numbers. So if you have a very complicated case with a rounds add number and we see what their correspondent numbers will do.

Yeah.

And my point is that they always resolve to something. They never return unevaluated.

Okay.

Can you look at an example of men with bad precision numbers?

Yes. That's a case you have before, you see in which the 401 and 405, 3. Those two are going to overlap.

Comma 1 and N. Is this what you're saying?

No, two lines before, below.

Sorry.

So I think he's saying that that number thing will convert the around to a number with low precision.

Exactly.

Oh, right.

Except that. That's.

Oh, yeah, sorry, I'm a little bit slow on following your.

So, so those two numbers, although I mean that that.

So that's the first one. That's why I had the equal. And you can have them closer together. So change the 405 to 401, for example. So they totally overlap.

Yeah, I understand that's. Look, I mean the basic point is. Okay, so look, there's a default behavior that we can have with a round. It's, you know, we should make it be as sensible as possible and. But there's clearly many possibilities here, as Sandra is commenting, and I was going to basically say the same thing. We need some more persnickety way to deal with this in which you get to explicitly specify confidence intervals and things like this. Because I think without that, I mean, and potentially then we can even do that with actual distributions. We can say. I mean, I don't even understand what happens if we say, you know, if we say min of x comma Y, X is distributed. Like, I don't know exactly how this would be work, but if I say min of xy, how do I do this? Do I do say distribution? I'm trying to do this with this. X is transform.

Transform distribution.

Okay.

Of this and then X is distributed. So it's escape. Dist. Escape. And you put the distribution here.

So normal with normal distribution with some numbers.

Okay, let's pick numbers here. Okay, so let's say this is now going to be 2 comma 2. Now how do I plot this? Can I say PDF of this?

Yes, you can plot PDF of that.

But here we all that need to be a plot 3D.

We would need perhaps parameters.

X.

It's not plot 3D, it's. It's just this. Right? Because why is it taking forever?

These computations are hard integrals for various points.

Integral for which point?

Yeah. Why doesn't it compute the integral once and then cache it?

If you compute PDF first and then plot then. Yes, in the plot function, everything is computed for each X in the plot points.

We should figure out how to cache those things. Now that we have compiler technology that we have, we kind of know when. Magnificent. I don't know what this is showing us particularly, but I mean, if I, if I make these distributions, what should I be doing here? I make this one.

So I think this is supporting Adam's idea that the result is another round with computed center and computer. Computed.

Yeah, because I mean if I. How do I find the maximum of that distribution here, this mode? Or how do I, how do I compute? Can I compute?

If.

If that is not a Gaussian, then it may be breaking some assumptions about around.

But it's approximately.

We could take just expectance, expected value, expectation, expectation.

But that's not getting me what I wanted to get, which was I wanted.

The top highest point.

Find maximum or mean of the transform distribution.

Expectation. We'll find the. Well, you need to spec. You cannot compute just the expectation of PDF expectation works on the variables and the specified distribution. So it would have to be that distribution, not PDF.

Yeah, like that.

No, no, it doesn't understand that as a function you would have to multiply. So that would be the integral integrate x times the percent free 318x. And that would compute the expectation or.

Or to just put in the, the transform distribution there.

Or put the transform distribution. Yeah, no, no, no. That would. The integration that work that way.

How do I do it? I've got the transform distribution. So.

Okay, so expectation x comma X distributed as the transform distribution.

Couldn't you also just say mean of the transform distribution?

Yes, you could say the mean as well. Yes.

Yeah.

It's probably computing the exact value if you, if you make it somewhere inexact. And it will hopefully be faster.

Yes. So the, the, it fails to compute the formula for symbolic mean and, and means and standard deviation. So an expectation or you know, you know if you put, put n in front and then it's gonna go numerically be faster.

Okay. Okay. So that expectation value, which is not the maximum.

No, I mean close to. Because it's a unimodal distribution.

Right. Okay, listen guys, we clearly need to have something. Okay, point one. I suspect there is a criterion for around. That is some kind of sigma based criterion that we can use to decide whether we evaluate around, less than around, so to speak. Just like we use the fairly arbitrary thing for numbers with less, which has frankly served us well over the last three decades or whatever. That was a criterion put in at the very beginning, hasn't been changed. It doesn't come back to bite one very often. Then I think we need a much more precise thing that can take around treated as Gaussians and can decide. I mean, then what we want is a bunch of. I see. Is a bunch of results of these kind of piecewise functions. Hmm. I don't even know what would happen if we have piecewise, for example, which is asking for, you know, is the value in this range in this range, in this range with the rounds.

So we can pre compute all these probability distributions and then judge based on some confidence levels. But for those separate functions. But I still think we need something for the first case. And the main question for me is this example shows it very well. So you computed the minimum of something one and something two, something around one, something around two and we got something around 0.3. Is this acceptable? I find it very weird.

I think it's very weird.

I think we need a criterion to decide whether to return the One or around two, but not around something else.

Sanders sent a notebook. Do we have Sanders notes here? Did you put them at the bottom here?

Yeah, they're in the section right there.

Okay.

Yeah.

So these are the things we could pre compute and create these functions.

So what is this about here? So this is.

So for example, around less having around less with a confidence level.

Okay.

And this could return true or false depending on the confidence.

By the way, compute comparing just the center. Is this confidence level 50%? Exactly.

Oh, is that obvious even for asymmetric distributions?

For asymmetric, I didn't think about it. For symmetric it is.

Oh yes, because half the probability.

Well, I think for asymmetric this must be.

Huh. Well, so what this is saying, oh gosh, but we don't want to have an around min around this around that, do we?

I don't know. I mean we have the president of vector.

Well, okay, let's assess how many of these around style functions we need to have. I mean, look, first point is we're not going to be able to get away with. Okay, around itself with pure less is going to do something which hopefully will be as sensible as possible. Maybe it's the 50% confidence thing that it was nice because you can just say it's 50% confidence, aka it's just taking the centers. And then if you want to be more precise, use this thing that allows you to specify the confidence level.

We could do what equal does, which is having a variable that controls the tolerance.

Right.

For equal, it's internal backtick. Equal tolerance.

Ugh, horrifying.

We could promote it to system. For the case of.

I think that's totally horrifying. Just imagine somebody resets that and a giant blob of code just stops working. I don't think we ever want people to reset it, but not as a global thing.

Can somebody explain the 50% thing? Is that important? Because it's not. It's confusing to me.

So if you have at least. I'm pretty sure for symmetric things. So if you have two symmetric distribution that the probability then the first one is less than the second one is at least 50% if you just compare the centers.

Okay, but, but all right, look. Okay. It would be nice. Okay, wait a minute. We've got vector less. But we don't have vector min, do we?

No, because it's, it's not one dimensional.

Well, but okay, but then what about things like ranked min and so on for arounds?

Well, I think those would have to follow whatever min does. Right?

I know that. But if we have a new function around min, are we going to have an around, you know, ranked min? That will be horrifying. I think the options are to have an option for some of these functions like min, which we haven't had before, that specifies confidence intervals to use for arounds. I wish there was a way of having some kind of wrapper or maybe even putting it into an around that says, hey, if I'm compared to this is the competence interval to use. Because then you put the burden on instead of having to have a zillion new functions. If people really care about this, they set the option inside around to tell it. I mean, let's just talk about. Let's just say that, I mean, we.

Could have a wrapper. I think it could be a single.

Around compare and then we put the function that has to do the comparison.

Well, what do you mean? But min has a separate around 1, around 2, around 3.

No, it would be something like around compare of min, comma list of arounds and then parameters. And the first argument is.

Well, that's definitely messing up how min works. Then when we've got a completely different.

No, no, no, no. It will be around compare. It will take the parameters, it will be around compare of min, list of arounds.

Oh, I see, that makes more. That makes sense.

And then we can have the confidence.

That'S similar to what we did for the. For the thing that has correlated errors in a round. What's that called?

Yes. So you're talking about vector around.

No, I'm talking about. Is it vector around?

Well, we have min around, for example, and. Yeah, yeah, wrong replace. I think you're thinking about a wrong replace.

That's what I'm thinking about. Yes.

Yes.

How is this related or not to this around compare idea?

Yes.

I mean if, for example, we could hold mean of x comma Y, then we could have X goes to around, Y goes to around.

Yeah, with some competence interval. The problem is, I mean, are people ever going to use this if let's say we have a round compare. Yes, Sanders suggesting around resolve. And I agree that that's better than. I mean it isn't really compare, it's not. Sure, resolve is quite the right name, but.

We should still have something in equal and mean, etc.

Oh, of course, of course. It's got to do something, right? We got to have a default. Right? This seems like the most sensible idea that we have something here where. Now wait a minute. What functions can we support? Can we support an arbitrary function in the first argument, how would we do that? Don't we need some kind of weird automatic differentiation thing to support arbitrary functions there?

Well, we are getting almost to around replace for the general case. This would be like a simplified case of around replace.

Yes, but the only thing is around replace doesn't have. Around replace has the feature that it can have multiple occurrences of a single around.

Exactly. It can handle correlation, but it cannot.

Handle confidence levels because we put confidence levels into a round replace. It's not obvious that one could do that because I think the confidence levels have to be done on a function by function basis, don't they? I mean, how. Right.

I agree. I don't, I don't know of a general meaning for confidence level for a general function.

Well, I mean, is that clear or is it. That thing, the first thing is a predicate, you're saying, I mean, like, like, okay, you know what, if I do this, if I say to be crazy, if I do this, the heck does that mean? Or no, let me, let me give a. Let me, let me make it even more crazy. I mean, okay, those are integer functions, so it doesn't really count because that's overly crazy. Is there not an integer function which has that feature that you know what functions take reals and return booleans?

Yeah. Or same question for what functions take intervals?

Yeah. All right, listen, I think we've, I think that we've pretty much just discussed there's got to be some default for which is not remaining unevaluated for around less of a round. Perhaps it's based on the centers which we can consider 50% confidence interval. Perhaps it's based on some larger confidence interval. Then for the more pedantic, we're going to have to have some separate function. This seems like a reasonable possibility, but let's inventory what first arguments it could have.

Okay.

And I mean, like for example, it's unclear to me what unequal does. I guess, I guess what it is, is. Yeah, it is. Okay, so given a confidence interval there, if I'm not mistaken, you assume those are distributions. You assume they're Gaussian distributions, and you simply ask is the. You know, there's a true branch and there's a false branch, and you simply ask what's the integral of the true branch? What's the integral of the false branch? How do they compare with the stated confidence interval? And that gives you the answer, right?

Yep.

Okay, so I think this is a path. I mean, in other words, we compute. We imagine that we compute efficiently. Hopefully the. Yeah, I mean, we should be able to. In principle, that first argument there could be any boolean returning function. And worst cases, we use transform distribution to compute the support for true and the support for false there. Am I right? I mean, is that right?

Yeah, I think it's something like that, yes.

Is this what we want for equal though? Because I think it will always be false. Where the probability of two numbers being equal is zero. What if you have it's measure zero, the equality. If you have two numbers with normal distributions, then the probability are equal to zero.

Well, I actually figured out the way to think about this. I think of you have these two probability distributions. You take an outer product of them in a two dimensional space and then there's going to be a diagonal line going through them for greater than, less than. And if you take the integral of the area versus the area. Oh, the equal sign is just the integral along the particular line.

It will be which is line over to this set.

Adam. What?

So it's an integral of something that's true on the line and false everywhere else. So the line is measure zero.

Yeah, no, but I, I don't understand. And if I take these, this picture here, what I would have expected.

There'S actually.

There's an overlap here. There's a. And that. I would have expected that integral of that overlap. Right. To be compared to the whole interval integral. Does that make any sense? What's that?

That's how I would do it too. For equality and inequality, I would check the area of overlapping with respect to the total.

So in other words, the probability of them being of equal, you would return the complement of a probability of them being in some specified less or greater. So if they are not less or greater than, that's what you would return for equal.

Sorry, sorry, sorry.

No, that's still not true. I think.

Wait, it sounds like what you're suggesting is the integral of the product distribution.

Perhaps. Okay, listen, there's some math to do here. Okay, let's do the math and get back to this and you know, see if we can come up with a reasonable definition. Because roughly what we. I mean, what I would intuitively think this meant is I want the probability that the things are equal. Your point is the probability the things are actually equal is zero.

Yes. Mm.

Look, this is a question of this whole pedantry that people always use in talking about statistics. Maybe Gosha can help us out here about, you know, samples and whatever. I mean, we're asking the wrong question. It's, it's, it's not. Are Two given samples ever equal. It's some other, there's some other formulation of this that makes more sense.

Well, you can ask the question whether two samples come from the same distribution, which is not helping you here. But the other thing that comes to mind is the likelihood, the log likelihood, which computes the likelihood of a sample coming from a given distribution.

Right.

Which is the. For the single element, it corresponds to the PDF. I mean, it's computed through the PDFs.

We need some math down here. We. We have questions. Okay. We've got.

I'll happily type up some notes because I think I have a half of an idea what's going on here.

Good.

The other thing I'm still wondering about is if we should be trying to make an analogy between around an interval, because I think a lot of this stuff is more clear what to do with the intervals than it is with around.

Well, we think of interval as basically being a unit step type function.

And the unit step could be units of the standard deviation of around.

But I don't think around. Sorry. Intervals is the right model here. I think numbers is the right model. And for example, I, I just place in the chat a comparison that gives true. Even though the numbers are different. And I think that's what we should try to imitate.

Right.

And what do less and greater say there?

Well, there is tolerance, so they may also. Yeah, so less.

We've got more to figure out here.

Less than greater won't work because of that order example that I gave, Which looks broken.

All right. This is a surprisingly complicated area. I mean, you know, in a sense, around is trying to pack, you know, around a sort of, you know, it's like everything with these statistical kinds of things where we're trying to pack a lot of information into the. We're trying to say, you know, this around thing is trying to pack tons of information about an infinite collection of measurements and so on. And it's not clear, you know, we're always going to get bitten somewhere because it's trying to summarize something that we don't even know what the underlying thing is and so on. Right. Are we going to. Let's take a very quick look here.

So. Right. So less and greater give false for that example, which would seem to contradict Jose's claim that just looking at the centers would be doing what numbers do.

Oh, gosh. All right, I, I'm. We need to do some homework here. Okay. I mean, I don't think we can get further. Right now. We've defined a possible direction. Let's see whether we can make that work. And then, you know, and then it is my feeling that rounds that are far away from each other should evaluate with less directly and not try and be pedantic.

Right.

Okay. And, You know, I think somewhere we've got to have a. Less remains unevaluated just for single numbers. Right. If they're too close, is that true?

No, for numbers, they never less.

It has tolerance, so.

Yeah, exactly.

If they are close, then they are not strictly equal. I think it just gives false.

Yeah. If equal gives true, then less has to give false.

I see, I see, I see, I see, I see, I see. Well, let's try and work that through. All right, I think we should wrap up for today here. And I don't think we're. I think we've. We've set two. I don't think we're likely to be able to, By the way. These things here probably should become system functions at some point. They're sufficiently commonly used. It seems like they are these distances between distributions. Gaussia seems like that should be. You know, I just see that quite often. All right, I think we should probably wrap up for today here. Okay. And thank you all very much. Thank you to Sander for raising this admittedly incredibly naughty issue, which normally would have been in an LCC meeting, not an ILD meeting, but I guess we.

Well, it was supposed to be, but we. We didn't make it last week.

Okay, fine.

Yeah.

It wasn't the agenda last week.

Yeah. Okay. Well, it was a difficult topic. All right, sounds good. Thanks a lot. See you all soon.