 Hello, everyone. Today we'll discuss the philosophical essay on probabilities by Pierre Simon Lablas. This is a very important essay because it brings together the theory of probabilities and unexpected connections with moral philosophy, ethics, epistemology, the way of thinking correctly, et cetera. And the reason why we picked this, the anecdote why we picked this text to read actually was that a few weeks ago, the statistic community had a controversy to rename an important award in statistics called After Ronald Fisher. And people were protestants that Fisher had some eugenistic and racist views. And I was personally wondering, so some people argued that Fisher in the early 20th century was a man of his epoch, et cetera. So personally, I was just thinking that there are many thinkers who had views that are not necessarily based on the generally held opinion of their epoch or of their culture or of their region, et cetera. And I thought of Lablas because he wrote this essay. And I just went to the essay and searched for keywords related to race or slavery, et cetera. And found out that Lablas was arguing that commonality and frequently held beliefs, if something is commonly held or frequently held, that's not a valid moral or epistemic argument. And I was personally aware of this essay, thanks to Lay, but didn't read the last edition of it until recently. And it's interesting in the context of our reading group because, as I said, it discusses at least the three chapters we'll discuss today are applications of probability theory to moral philosophy, something that is highly important in the context of AI ethics, in particular. The second part we'll discuss is the application of probability theory in the judgments made in a court. And the last part we'll discuss is what Lablas called on illusions in estimating probabilities or in what 20th century psychologists would call probably cognitive biases. Maybe I can even backtrack a little bit and explain the context a bit of this essay. So a bit of the history of probability theory essay. So probability theory was probably really started around the 17th century with people like Kerma or Pascal. And then Bernoulli and so on. But before Laplace, most of the probability theories were, in a sense, deductive, meaning that we had an initial source of probability. And then we tried to see the consequences of this initial source of probability. So you start with axioms, which are the axioms of probability. Typically, heads or tails would be one half and half. And then you compute the consequences of all of this. And in 1776, Pierre Simon Lablas, so there was this guy, Thomas Beis, who did some work in the meantime in England. But it was like he did not publish it. He did not really believe it. In any case, the most fundamental initial work on inductive probability theory was by Pierre Simon Lablas in 1776. And he basically put forward what we know today as baseball, as this rule to go from the observations, the data. So we're going to go from this to general theories, for instance, to infer the laws of the universe from the observations that we make. Which, if my understanding is correct, Beis did that as an attempt to answer Hume's induction problem. So Beis, well, yeah. So it's related to a primary by Hume. So David Hume is a philosopher at the beginning of the 18th century, so a few decades before Lablas and Beis. And Hume asked this question, like if you see the sun rise every morning, is it sufficient to say that it will rise every morning from now on? Like is it generalization, like a rule, something that you can, is it a good way to think? And Hume already had this intuition that no, it's not exactly the right way to think. Instead, we should think in terms of probability. The fact that we observe the sun rising every morning increases the probability that it will be rising tomorrow. But Hume did not take this further. He did not formalize this idea. He did not relate this to the mathematics of probability theory. Beis did part of this work. But Lablas did most of the work, and especially Lablas not only solved this kind of small problem, I'd say. But he generalized this, and he had this very bold claim. And this essay, this philosophical essay from 1814, the first edition, and then 1840, the second edition, is really like the philosophical approach to probability. Like his 1776 essay was, Memoir was more mathematical. Though it has a bit of philosophy, of course, but it was more mathematical. And then Lablas thought of this probability course at the Ecole Polytechnique in France after a while, in the late 18th century. But probably he felt that people were too stuck too much to the mathematics and did not really see the philosophical importance of this work. And that's probably why he wrote this essay. And I think this essay is absolutely fantastic. I think this is not going to make a lot of friends by saying this, but I think this is the best philosophical essay ever written. Yeah. So this essay, like he does discuss a little bit of the mathematics of probability. But the main point is that there's this thing he calls a good judgment, bon sens in French. And he kind of argues that this is what bright people are endowed with in some sense. Maybe one important precision here about good judgments and bon sens. There is a lot of misunderstanding around that, often translated in common sense. It's not meaning common sense in the term intuition and the commonly held beliefs. Actually, Laplace is writing the last chapter where he mentions slavery as a commonly held belief that it's OK. It's not OK, of course. Actually, he's against common sense. The bon jugement, like good judgment, not common judgment. Bon jugement, some people translate it to the bon sens. And then bon sens becomes, which is common sense. And those are radically opposed things. It's clear. It's really clear, especially from the French version, that he's meaning good judgments and not common sense. And he's arguing against actually common sense and commonly held beliefs. And his point is that you have this common sense by held by most people. There's this good judgment held by some brighter people. And what he argues is that, like when you think longer, you get closer to the good judgment. But what he argues is that good judgment is still missing some of the important things. For one thing, it's not very quantitative. And what he argues is that probability theory is the ultimate way of thinking. That this quote, like he frequently in this essay discusses the fact that good judgment kind of leads us towards the right direction. But the computation of probability theory, the calculative probability, so probability calculus, is what gets us closer to makes us appreciate what's the exact and right way of thinking. So the essay is a lot about this. And it draws a lot of applications of this very fundamental and general principle. Like it's about how to think in general. So of course, it's going to have a lot of applications through all sorts of fields. And those we are going to discuss today are mostly related to moral sciences and lawsuits. So let's start with this first chapter that we read from the book. So why is a why our priority is important in discussions about moral philosophy? So the main argument of Laplace is simply that the world is extremely complex. And even if we take a long time to think and have the highest ability to provide good judgment, people will make mistakes at anticipating the effect of written laws on the world. So if we see for themselves, there are lots of crimes and we want to design a law to reduce the amount of crime. It can be done, obviously, but it will sometimes have side effects that are unpredictable. And that's why Laplace recommends that we should think of doing this kind of transformation of changes, but in terms of thinking about it, in terms of probabilities. So simply knowing that the effect of that law is uncertain and what we want is to be able to observe how this law is affecting the world and possibly change it if we see that the transformation is not what we expected. And this has been very common that laws are being changed over time as we see that they require improvement. One thing he discusses in this section is the fact that it's often the case that we see maybe part of the law that's never used or that has bad consequences in some points. And you may feel like we should remove this part of the law. And what Laplace argues is that it may be dangerous because we are not predicting well enough the consequences of the law. Just so that we understand which parts of the laws are important and which are not, we should not rely solely on our judgment, but also keep track. So there's this discussion like it's almost an invitation to do data science or to collect data or to have a good database, to have a data-driven writing of the law. And he really encourages people to keep track of all of the cases where the law was applied and for which reasons to better understand what makes a law good. I think this is not necessarily specific to probability theory. It's more about the complexity of the world. I think there's a bit of a computational complexity theory behind it all. And I think it has a lot of consequences to the way we think about safe algorithms, for instance. Algorithms are supposed to make judgment as well. And maybe parts of the algorithm is not going to be used. And you may want to just keep it because it's slow or something like this. But the way you should be doing this, according to Laplace, is that you should absolutely keep track of a lot of data and to have a data-driven approach to designing what a good judgment is. By the way, I'd just like to keep this discussion accessible. Let's not just mention algorithms because some people think it's something complex. Just decision-making procedures. Especially if we're thinking the era of what has to think of decision-making procedures. If you have a procedure to make decisions in a complex world where many data are missing and many phenomenons are interdependent in a complex way, in an untractable way, you can't track all the dependencies. Then this argument from Laplace holds, especially in the context of decisions made by machines and with lots of data that humans can process. But the argument is valid in human judgments, in courts, et cetera. Yeah, later Laplace compares to two ways of taking decisions. The first one is using your intuition and the best you can do according to your good judgment. And the second one is relying on collected data and writing some prohibited paper. According to your common sense, what's your good judgment? If you... It's a... It's separate. According to your here, you mean common sense or your intuition. But it can also be the best you can do to achieve good judgment. And the second thing to compare it with is using collected data and writing some computations of probabilities on a piece of paper and coming up with the results. And it's usually a difficult effort to make to accept that the computation done on a piece of paper is more stressful than the 10 minutes you spend thinking about an estimation in the general case. Yeah, it's a very general theme of the RGSA. And it's a bit more subtle than this because Laplace acknowledges the fact that most of the time you can't reduce things to computation. It's a very frequent concept in the RGSA. Like, he often says that we should try to reduce things to computation, but sometimes things are too complicated to be submitted to the computation. This computation is like a no record. It's like a computer you can imagine today. And if you can formalize everything, like the problem to it, then it will give you an answer. But more often than not, the problem is too complex for you to write it down and to ask the computer what do you think. And then Laplace argues that in this sort of situation you should then think in terms of analogy, but you should be careful about to which extent the analogy holds. And the analogy that Laplace currently discusses is like having this box with balls inside of it and you don't know what are the balls inside of it. And so he uses the thought experiment of drawing a ball and for instance observing that you drew a black ball. Then the question of Laplace is what is the probability that all of the balls inside are black or what is the probability that the next ball that you draw is black? And he's using this thought experiment that's very remote from the law or from everything. But somehow he constantly in the essay founds connections between these very simple thought experiment and actual problems that you face from something in the court of law. So it's one example he gives in the essay is the example of a testimony. So this is clearly very important in the law to have testimonies, but there's always the problem of how much you do trust the person who gives a testimony. And so, well, Laplace has all of this way really nice discussion, but essentially what he says is that they are like, if some event is extremely unlikely up here, like you like for instance, like a murder is like very unlikely up here, like most people don't murder another person. Then if somebody tells you that there was a murder, what you should compare is the probability up here of this murder with the probability that the person who gives the testimony is either lying or being mistaken. Now, this probability of a person lying or being mistaken can be small, but probably it has to be very, very small to be comparable to the probability of a murder. And so this is the kind of probability thinking that this essay is talking a lot about and it really answers also like some of the questions that David Hume raised earlier in the century. Yeah, a famous quote mentioned over this topic is that extraordinary claims require extraordinary evidence. Yeah. And this is something you can read if you look closely at the base rule where how much the probability you assign to some theories and some unknown theory will change is dependent on the probability of the observation and if you make extremely unlikely observations, it will change more the how much you, your beliefs in different theories. Yeah. Yeah. Another very interesting aspects of probability theory applied to the context of the law is the fact that when we rise to the law, like most laws are written as if the person is guilty, then do something. And if the person is not guilty, then do something else. And these kinds of principle of rule, this kind of algorithm requires perfect knowledge of whether the person is guilty or not. And yet in practice, we have to expect that we're only going to have limited data. We're not going to be able to have a mathematical proof of the fact that the person is guilty or not. We only have evidence, we only have data that will change what we believe, that will update our probabilities, but there may, and quite often, there is still a huge amount of uncertainty when the sentence has to be given. And so what Laplace argues is that the law, we should think more of the law as, or we should write more of the law, or maybe not write because this is difficult, but we should think at least of the law as more of something like if the person has a very high probability, a probability larger than 99% of, or 90% let's say, of being guilty, then we should give him this sentence. And maybe we can then have different levels of sentences depending on this probability. If the probability is between 50% and 90%, we have also harsh, harsh ruling, but not as harsh as it is if it were larger than 90%. Another illustration that's just like the introductory paragraph of the application of probabilities to the law and the court's ruling. I think the fact that we have this first instance, like the first tribunal and then you have the appeal. And then appeal, you go to a tribunal and Laplace argues that in the appeal, you need more judges and you need the majority vote, et cetera, because like the probability that an error was made in the first, so just like this in terms of probability thinking, this would just boil down to the wisdom of the crowd, like wisdom of the crowd, but not every crowd, the crowd of judges. Then he's making a probabilistic argument for the fact that if you go to appeal, you need to increase the level, the number of judges before you finish the procedure. To go back to the threshold that Leo was discussing about the fact that we can't be absolutely certain that someone is guilty, but we should still send that person to jail if there is a high probability that a person is guilty. This sounds quite hopeful because it means that with some frequency, we are going to put some innocent people in jail. And something else that is not desirable is that we release free some murderers that would kill other people. So there is this balance between a several undesirable outcome and because the system is not perfect and we can't have a perfect knowledge. So this algorithm should not even try to rely on perfect knowledge. Then we have to accept that the system is going to make mistakes. We can think of it as a, we can do our best to improve it, but there will be some mistakes. And choosing this probability of how sure do we need to be to send someone in jail? It could be a balance between the undesirable effects of putting innocent people to jail and the undesirable effect of releasing murder or free. Again, I'm just adding just all nuance here. So Laplace is not saying that in all cases it would be impossible to have close to perfect knowledge. Just arguing that many cases, perfect knowledge is hard. So we have to have, so then we go to appeal, et cetera. But then he says like in easy cases where it is easy to establish close to certain, like everyone in the village, so this person murdered this person and like even the judge saw the killer kill the victim, then you don't need to go to appeal. You don't need to do this sophisticated probabilistic thinking, just like just to close the door because sometimes when we bring in relativism, like this one that's like, we can't always know perfectly, et cetera. Some people interpret it in the wrong way and say, okay, then everything is relative. We can never know. No, no, Laplace is not closing the door to the easy cases. There are easy cases and in these easy cases, the simple, almost binary way of thinking is practical and is enough. So we're not ruling out simple and close to binary thinking. It's just that in complex cases where it is clear that no one has complex, like everyone has only partial knowledge. For example, evidence has been destroyed, for example, like the evidence was destroyed either by the guilty person or the likely guilty person or by someone who would like, someone who is really guilty and would like to make the accused person look guilty. So for example, those cases, those are complex cases where we need this relativistic thinking, probabilistic thinking, go to appeal, increase the number of judges. Laplace is not ruling out. So Laplace is not relativist for the sake of being relativist. And sometimes I read in some part of the literature, like people using Laplace, reasoning to say that, okay, knowledge of truth is always relative and it's like, and then they rule out close to certainty cases. Like there are cases close to sincerity is useful. I agree that it is a common mistake and it's good to mention it. Sometimes this mistake is described with the image that people think in black and white, so absolute certainty of false, absolute certainty of true. And this is the wrong way to think, obviously. But then when they realize that, oh, nothing is either black or white, things are gray, they make the mistake of having only one shade of gray. And thinking in terms of priorities, you should make your priorities go from as close to zero, as possible to as close to one, as possible, obviously in many cases, but also have priorities in the middle in for difficult cases that are uncertain. And so you should think of all, you should think with all the shades of gray from a white as close to one as possible and as close to black, zero as possible. Very dark gray for things that are extremely likely to be false. And there's some very nice quote in the essay, which early on in the essay, where he discusses the fact that what is probability theory, or maybe we can have another episode on this, but what is a probability? But essentially what he says is that a probability is a description of our ignorance and of our knowledge. So it's really both. Where we will discuss the introductory part of the book. So it's counterintuitive. Now we're discussing the final part of the book, the part of the book. So moral philosophy or law, et cetera. Then we will go back and discuss the introductory part of the book, why probability theory matters. I just like to close this part on relativism. So just like to make it short, we see like there's a lot of literature on the confrontation between binary thinking and relativism. And actually, probabilistic thinking uses both. Like there are cases where it's useful to be a relativist and to have nuances and to defer your judgment and delay it as long as possible. And there are cases where it's very useful and practical and fair to have close to binary thinking. So you should not rule binary thinking when it's useful. And you should be aware that you should be, like you should not use it always and you should be aware that complex cases do not, like are not solved by binary thinking. Yeah. Yeah, and so just to close the section on the law, there's also a nice discussion about, so let's say what we care about is actually this probability of the person being guilty and we want to make sure that it's larger than some high threshold so that we can convict the suspect. And then Laplace has this discussion about if you grow the size of the assembly or the number of judges to give the ruling. Like, should you demand that a larger fraction of these or a smaller fraction of these, but what is the fraction of these that need to say that the person is guilty so that we conclude that the person is indeed guilty? And, well, this nice question, like if you have a very, first of all, that's very close to 1 1⁄2 then if you have a small number of judges then it's very, very bad. But essentially, but the conclusion that Laplace comes to is that with a rough estimate is that out of an assembly of 12 people, maybe there should be something like nine judges that say that the person is guilty in order to convict the individual. And I think it's a nice way of framing the problem. Like you demand more than the majority not because, well, that's an arbitrary rule, but because you want to have a high probability to convict the person only if there's a high probability that the person is guilty. I think it's a nice way of thinking about this one. Yeah, and it's also seemed also the fact of accepting that mistake can be made that the jury will not be perfect. If the jury is perfect, either 12 will always agree or the 12 will always agree because they are perfect. This is not the case. So in the model that Laplace discusses, in the model that Laplace discusses, the jury are considered to be quite good, better than chance at deciding if someone is guilty or not. Maybe they get it right with a priority of 75%, something like this. And this is how Laplace run these computations. Yeah, now one caveat to Laplace's computation is that Laplace assumes in his model that the members of the jury are independent, like the opinions they have are independent. And unfortunately, we know by now that there's a lot of correlation of group polarization effects when you have an assembly. So this is a caveat to be given to this analysis of Laplace which would demand maybe even larger, but yeah, it's a complicated problem. Yes, and even if the jury will not polarize or have biases because they are shown the same data, it's surprising to expect that it would be like independent judgments. So one of the last sections of the essay, it's just called On the Illusions in the Estimation Probabilities. And it's absolutely fantastic. Like it's like 200 years ahead of its time, essentially. Well, he discusses the way people think poorly, I guess that other philosophers have noticed that people were not always thinking very clearly, but what's really nice is that now that he has this portrayed that poverty theory is the right way of thinking, then you can measure how people deviate from this right way of thinking. And in doing so, like he discusses essentially all the best known cognitive biases that we know of today, like for instance, the betters fallacies, like if you only see a stream of like, no, no, if you see a lot of red coming up in the roulette in Casino lately, then you might be tempted to say, well, the next one is not going to be red because it's come too often, something like this. But Laplace argues that this is an illusion. And then he discusses things that are probably closer to what we would know as we call today cognitive bias. Like familiarity bias, motivated reasoning. I think these are the two main that he really stresses in this essay. And he does this in a very, very compelling way. So I think this is really, really fascinating section. Yeah, one point that I'd like to write is that usually people underestimate how much of what they observe in the world happens simply due to randomness. So with the example of the lottery, a lot of people try to find out explanations of why this number came out. And one of the explanation is that some numbers come, the number 47 didn't come for two years and it's bound to happen at some point. So we bet on this one. Well, that sort of explanation is that there are people that would log all the numbers that come out of the lottery and find the numbers that come the most often, and then try to bet on these numbers because they have been observed to come more often. But, and Laplace discusses that he simply created a small model of generating lottery numbers and finds out that yes, we expect that in, if you observe past data, there will be some numbers that came out more than others. It's a normal thing simply due to the random process. And because you find such a simple explanation, uniformly for me randomly generating numbers to explain what have been observed, one should not think that there is a different processes for generating these numbers than the simple process that Laplace described and that builds actually the lottery. Yeah, and it's ready to this idea of what poker players call resulting bias. So that's like judging the decision of someone like whether he was wedged to play number five in the lottery based on the result. And you say, oh, I was stupid. I did not play the number five for instance. And poker player would say that this is a very, very, very, very bad habit, at least in poker because you give too much attention to things that are just noise and you're going to update your strategy based on this and you're not going to focus enough on your decision-making because this is what matter decision-making. So typically in poker players, professional poker players, there are these groups of poker players who just never discuss like the so-called bad bits, the way they lost in a tournament like the specific hand they lost on even though it was higher unlucky because what they care about is like the decision-making. What it is that you choose to do when you had this uncertainty and based on this uncertainty, whether what you did was good or not and not based on the result. You should judge based on the uncertainty and not based on the result. I think this was one of the, this is one of the greatest insights of probability theory. Yeah, and this is very hard to do in practice. I often reward myself for making decisions that ended up doing good and to finish myself from making decisions that ended up being bad. And I learned because of the result and today it's hard to do differently. Think about, so to illustrate this with the example of lottery. Lottery is well known to be a game with a negative utility, negative expectation of gains. So if you judge a decision process that decides to take the lottery or not, it's very easy to agree right now that the decision process that decide to play is making wrong decisions when the decision process that decide not to play is making correct decisions. But now if you imagine you see someone that decided to play and won, then it's very unintuitive to say that the decision to play for that person who won was a wrong decision. The decision pushed by a decision process that does not correctly maximize its expected utility. It's simply because of the result. And this can lead to difficult discussion. If you discuss with someone, they might tell you, you don't know whether I'm making a good or not decisions because we haven't seen the result yet. Yeah, yeah, that's not probably the thing I know. Like that's sort of like a confirmation bias. Confirmation bias and the example of live nets. So there are like, I don't know if the audience is familiar with that like some beliefs like in numerology, like people who believe in like the power of numbers if this number pops out and then there's, I don't know if the golden ratio in something, then there's something special about these objects. And this is something still common in these days. People like believing in miracles just because some sequence of numbers appeared, I don't know in the date of birth of some singer and then the date of release of her album or his album and then they will start like building up theories and the internet is very good in amplifying these theories that because the date of birth of the singer and her date of release of the album and then I don't know, 9-11 a year like happened and then the ratio between two. So this is something that sounds funny but even great minds were not immune to it. And he gives the example of live nets and Bernoulli also. And that's also not my like mostly live nets. Like Bernoulli and live nets computed this series like there's a series of number that gives some special results, et cetera. Done by Bernoulli and live nets that live nets use this result to argue with the Chinese emperor that God exists. God might exist for other reasons but not because the series is equal to one over two. So he's like, he said he, so live nets who were the tongue believer knew that the Chinese emperor loves mathematics. So he taught like, yeah, maybe like if this would convince him like Christian God and Christianity and then he sends him like a funny note on the results of a series and say, look, if you sum these numbers and then you obtain one over two or one over four, one over two, right? One over, yeah, I think it was one over two. Yeah, one over two. One over two and then look, this is like he creates something out of nothing and this is how God operates and this is a proof. And Laplace argues that like this like Laplace does not call it confirmation bias but today in light of what we know since the 20th century, all the work on cognitive bias this is the clear instance of cognitive bias. You believe something is true. So you believe God exists, you believe in Christianity or Islam or Judaism or whatever. So then you have a strong bias towards confirming like validating everything that comes from your religion or your ideology, communism or capitalism or whatever you want. And then he goes on with example like that. And so I believe this chapter on the illusions of computing probabilities if you want to rename it today in light of the developments we had in psychology can be called on cognitive biases actually. Like you can argue that what he calls illusions in computing probabilities are cognitive biases actually. So that's also I mentioned this. So the confirmation bias and the case of Leibniz who practiced like who's almost fall into numerology to argue for Christian God. But then there is the other example which might be I don't know a hybrid between confirmation bias and familiarity bias maybe more like familiarity bias in modern terms which is this the thing that it's not because slavery is commonly accepted that it is okay. So it's not like it's not because in some culture some practice is commonly accepted then this practice is morally good. So maybe we can even argue from this chapter that as people who learn probability theory we have a moral duty to go beyond the commonly held moral standards of our culture, of our time, of our era, of our region or and then for example you mentioned slavery but we can go on and make a case for like just moral progress. For example, like moral progress is debated in like moral progress versus moral relativism. So I'm not sure if I'm exact but like in moral relativism people would tend to tell you like you have to respect the moral standards of some culture or some region, et cetera. For example, let's say like there is a region where people don't let girls to school, for example. Like should you respect this practice because it is the commonly held moral standard of that region or should you like should you try to go beyond that? And if you read the chapter of Laplace you can come up like personally came up with the conclusion that when you learn probability theory you have to work with it and think with it and think harder and try to always go beyond the commonly held moral standards of your time and of your group, of your social group, whatever that social group. Yeah, so the connection with probability theory may be a bit loose in the essay itself. I think it's more like a point that well these are cognitive biases that people have but I don't think Laplace knew about this but there are actually strong connection with probability theory. So under confirmation bias for instance, a problem there's actually a theorem in Bayesianism that says that the expectation of the posterior is equal to the prior. So you should, before looking at the data you should expect to have in average the same opinion after looking at the data than prior to looking at the data. And intuitive is the reason for this is that like the data can make you go both ways. Like if you're surprised that the data is suggesting something like say like you assign a probability one half of Trump being reelected, I don't know, something like this then you expect that tomorrow you're going to have the same opinion. Tomorrow at the end of the day is going to be probably one half as well but this may evolve, it's not going to be exactly one half for sure. Even if you see a tumor. If tomorrow you see data that suggests actually the popularity of Trump is decreasing more than you expected then you should decrease your probability. But if you're surprised that it's maybe decreasing but not as much as you expected then you should increase your probability about the relection of Trump. So whatever happens, in average you should have the same so that's actually a theorem. Just to make the connection with probability theory even less loose. So for example, what Louis said, like many things are just due to randomness and the fact that you don't need sophisticated theory to explain them. You can see that as some form of Occam's razor. So you don't need sophisticated explanations. It's not because the moon is like that and the number of girls who were born that month and the number of boys who were born that month and the date of birth of your husband or your wife is like that that you would have a girl or a boy. You would have a girl or a boy just out of probably randomness, genetic randomness and how many white chromosomes the husband produces and et cetera. And it has nothing to do with the moon and the numerology and sophisticated computation of astrology, et cetera. And like a lay in his book makes a very good case for Occam's razor with probabilistic thinking. So Laplace in Laplace writing the connection is not straightforward but we can argue that many of the illusions he talks about Laplace talks about are some of them are due to a bad application of Occam's razor. So Occam's razor is this principle in the epistemology that tells you that out of many explanations you should always favor the simplest one the shortest one, the one that does not require a lot of additional assumptions. And like in the case of the illusions Laplace is mentioning a lot of these illusions involve additional assumptions with the moon and astrology or whatever numerology and a number of girls and boys that were born that month and I don't know the existence of Christian God. So those are like unnecessary assumptions. And Laplace is very well-known for actually his way of practicing Occam's razor. So in his book on the mechanic Celeste, right? Yeah, I don't remember the title, but yeah. Mechanic, like, it's like a... The celestial body emotions of the bodies. Yeah, motion of celestial bodies. There was this famous argument that he had with Napoleon, with Napoleon would tell him I don't see any mention of God in your book. And then Laplace just replies or is believed to have replied, sir, I just didn't need this assumption. This is an instance of Occam's razor. And we in this group argue that Occam's razor is just another way of being the user. What a component, it's not a subject. Yeah, but there is still something important to note concerning confirmation biases. So even though you might have a correct prior and apply Occam razor quite well, confirmation bias is something that happens at the moment where you look at evidence. And the problem is that sometimes people would look at evidence and no matter what is the evidence, they would change their belief in the same direction, which is what cannot happen because as Les said that when you look at the evidence in average, depending on what the evidence is, your change of belief should set up to zero in average. So that means that if the evidence points one way, you should change your belief one way. And if it points the other way, you should change your belief in the other way. A famously illustrative example for this was the condemnation of Dreyfus who was accused of something. And when people looked for proofs about this, they could find no proof. And unfortunately, finding no proof, they concluded that, oh, yes, he was guilty and good at hiding that he's guilty. So... Excellent, that's an excellent example. Let's elaborate on this example maybe to conclude. So just to put more context to the English-speaking audience, Alfred Dreyfus was a captain in the French army and he was blue. And the atmosphere was... So back then in France was quite anti-Semitic. So in an anti-Semitic context, Alfred Dreyfus was accused of intelligence, of being a spy, a German spy or an English spy, a German spy. I'm guessing German, yeah. A German spy. And the evidence against him was a note that was presented as a note Dreyfus writes to the Germans, so probably. And then Poincaré, so Hongri Poincaré, the polymath, one of the greatest last polymaths, as people say, used probabilistic arguments to prove the innocence or to argue for the innocence of Dreyfus. If you found no proof, and then you change your belief to increasing how much you think that that person is guilty, if you correctly apply a base law, it means that if you had found proof, you should have updated your belief in a direction that is not guilty. So when you observe A, you make an observation and you update your belief one way, it means that if you had made the opposite observation, you should update your beliefs in the opposite direction. And obviously, finding proof of someone guilty should increase your beliefs in the direction that this person is guilty. And it means that not finding proof should always make you update your belief in the direction that that person is not guilty because, or at least slightly, otherwise you are in the failure of confirmation bias. Yeah. Yeah, that's like, this is easy to say in theory, in practice it's always harder when you actually presented the evidence and you are always trying to find loopholes and explanations for why it go in your direction. So one way to better combat this tendency that we all have to motivate reasoning is to pre-commit. So ideally, you just apply base law, you just apply the laws of policy. But because we have limited, we have motivated reasoning, one way to combat it is to pre-commit, meaning that you're going to say, well, today I believe this, I believe that the right face has a 70% already of being guilty. And I know that they are going to look into this piece of evidence. And I'm going to predict what it is. And I'm going to say, well, I think that it's going to be something like that. And if it's, so let's say, for instance, it's the number of messages he sent to some general in Germany. And you say, well, probably he sent like five messages. And you're going to say, well, that's my prediction. And it means that if there are more messages than this, then you're going to increase your poverty that he's guilty. But if there are less messages than this, then you're going to decrease it. And you have to, well, one good way to do is to pre-commit. And so this is more generally a good habit of a Bayesian, which is to bet Bayesianism and betting have strong connections historically and still today. And betting is good because it forces you to explicit your prior and to pre-commit and to verify that you're not going to do motivated reasoning. So yeah, I think this is one of the important takeaway of cover is deep thinking. Yeah, yeah, I agree. Totally agree with that advice from me. Another advice I could give that is slightly less good, but maybe easier to do in practice also is to simply when you see yourself in the process of updating your bilis based on evidence. So if you are already doing this, I think it might be useful to ask yourself the question, how should I update my belief if I had observed the opposite evidence? In that case, it might help you detect when you are actually lying to yourself and doing confirmation bias and help you choose in which direction actually the evidence points to. So next week, we'll discuss a very important paper, Gender Shades by Joy Biolami and Thymine Guedru. That paper was important in showing empirical evidence that facial recognition is biased and it has strong biases that make it not ready to deploy. And thanks to that paper and the research follow-ups by these two researchers and a few others, now there is a moratorium on not deploying facial recognition by many companies. So IBM, then Microsoft, and many others follow, state that they will not deploy facial recognition and they will not use it especially for police and military use. So we'll discuss that paper, some of the follow-ups and what does it mean for today's technology is such as, so we'll focus on facial recognition that we'll probably discuss other aspects of biases that need more research and more work by people who work on artificial intelligence and computer science and moral philosophy. So thank you and see you next week. Bye.