Scientist Stories: Stefan Bauer, Neural Causal Models

Welcome all here in the lecture hall and also welcome at home remotely. So this is the NCT Data Science Seminar and also a joint event with Alice Live Heidelberg, and we are happy to have you here today, especially our guest speaker, it's Stefan Bauer. And welcome also to Ulrich Koethe, and he will do the introduction. Thank you very much.

Yeah, this is Speaking, I also welcome our speaker today, Stefan Bauer, in the name of the Alice Heidelberg Unit. So we are very glad to have him here talking about the news in causality research. So like 10 years ago, everyone thought that would be the next big thing, but then Stefan did not have a small influence showing that it's not that easy as was believed maybe 10 years ago. So showing some impossibility theorems, for example, and also some new ways of solving the problem. So Stefan graduated from the ETH Zurich with his doctoral thesis, which also won a prize, and then went on to Tübingen and learned Everything about causality in Bernard Tchaikovsky's school of thinking, then had a group leader position in Stockholm and now is in Munich. And the stage is yours. Thanks.

Thanks a lot for the very kind and nice introduction and for hosting me here. I already had some amazing discussions yesterday and wanted to thank Julio and Pablo and Sebastian for inviting me. And in the morning today, I was discussing with Olli Stegle and Kai, and it's pretty amazing the kind of research you do in Heidelberg. And I stay the afternoon till the evening today, so if you want, I would be more than happy to discuss more and meet more people. Unfortunately, we had last week a cyberattack on our IT system, so if you want to contact me, please use my former email address at KTH, or reach out on Twitter, or just talk to me after the talk. Unfortunately, I can't access my Helmholtz email account, so if you wrote me before I came here for a meeting or discussion, I unfortunately couldn't access and couldn't answer any of these. That was already from the introduction, and then I would like to start. And actually, I know it's an outdated slide, but I think I want to use it to illustrate a little bit some of the problems we have and why I believe it's still interesting. What you see here is a classical causation versus correlation plot. In this case, the example of chocolate consumption versus number of Nobel Prizes. What you see here is basically a strong linear correlation, and you see that in particular, you see that Switzerland has better chocolate than Germany and that it helps to be Swedish for winning a Nobel Prize. Admittedly, it's a little bit of an outdated chart, so some countries significantly moved up, but still this is a strong linear correlation we would have. And there were even afterwards additional studies. So there was a very nice Nature paper from Terence Sejnowski and a couple of Nobel Prize winners where afterwards they were asked, well, do you eat chocolate? Or what was the influence of consuming chocolate? And it's surprising, some credit their Nobel Prize to eating chocolate and some say Well, maybe they would have won too if they would have eaten more. So even there, there's not a consistent answer. But it's a fun article to read. And now you could argue, okay, well, what is even the problem to differentiate there? And what would change in the time of large language models? So we probably would go to the internet and then check for scientific articles. Here, for example, there's a double-blind placebo-controlled randomized trial of the effect of chocolate consumption a function on, let's say, neuropsychological functioning. And it's a surprisingly hard problem. So a large language model would probably be able to search the internet at some point, find these articles, but even there, it's difficult. So here, they, for example, they don't directly measure a significant effect. Part of the problem is that apparently it's still hard to find very good placebos for chocolate. So you realize it's not the right thing, so you stop eating it, and might affect the studies. But even in these trivial cases, the answer is actually non-trivial. So for example, you can ask ChatGPT and say, okay, does eating chocolate increase my chances of winning a Nobel Prize? And you can even spin it back and forth, but at least there the answer is that basically it's unclear. And why I pointed it out is that it's a surprisingly simple question in the sense that opposite to, for example, smoking cancer, where you can't do a randomized placebo-controlled clinical trial, in chocolate we can at least do it. Like, I can give half of you chocolate, I can give the other half of you chocolate placebo and see what happens. But even then, even if we allow for that, it's a surprisingly hard problem, which most likely won't even be solved by the fact that you might have access to all of the internet. And that is a very trivial example of a two-dimensional case, so a two-dimensional plot. And now I hope— oh, how does that work? Okay, so that was a two-dimensional plot, and on the right, I hope— okay, sorry. So it's a nice video. So basically what it does is showing— you can check the link afterwards. It's basically a video of someone cleaning a room. And then there are cyclists on a TV screen while he's cleaning the TV, and he lifts the TV up at the same time everyone in the TV screen basically crashes. So the question is a little bit, in higher-dimensional settings, it's surprisingly hard to find these examples which are convincing to humans. So this plot on the left, it's relatively easy to say, okay, there's probably an underlying factor like wealth of a country, ability to invest in education. Where you find and can debug correlation versus causation, but in higher dimensions it's surprisingly hard. I unfortunately don't know enough examples where we have these typical plots from the left in videos, for example. So if you know very good videos where there's this effect of correlation does not imply causation, I would be very happy if you share them. I think that would be a very interesting study. Okay, so now how does that again work? Okay, and now there's this— for causality, there has been an amazing success, like as mentioned roughly 10 years ago. So the first was Judea Pearl winning the Turing Award for his fundamental contributions to artificial intelligence and through the development of calculus for probabilistic and causal reasoning. In 2021, Guido Imbens and Angrist won the Nobel Prize in Economics for the methodological contributions to the analysis of causal relationships. The Nobel Prize in Economics was shared, but it's basically a Turing Award and a Nobel Prize for causality already, which is very amazing for the field. Yet a year, or at the same year where Guido Imbens won the Nobel Prize. He and Judea Pearl actually had a very interesting conversation in a couple of papers back and forth, and of course that slightly continued on Twitter and is still online a little bit, where they basically attack these two— and it's becoming apparent that causality is basically split into different communities. So one is focusing on cause-effect estimation where you assume a fixed model and are interested in parameter estimate, for example, and the other one is more based on the do-calculus from Perl, where you try to do causal discovery. So one is you want to learn the model, and the other one is basically I assume a model and want to get a good estimate. And here, for example, the criticism is that for causal discovery, in the Book of Why, we only illustrate very toy-like models similar to this chocolate versus Nobel Prize consumption where we have two variables. We understand there might be a hidden confounder and it's at most these three variables. And so there was— so this is a couple of interesting papers and there's even more criticism. And for this, I actually would reference you to very interesting interviews. I cite some of these quotations in the next couple of slides as well. And it's basically this— some of the superstars in causality giving answers to a scientific editor, I think, of where the field— how the field developed, what their experience was. And so it's very interesting from their perspective. And for example, on the left we see that causality won this Nobel Prize, and on the right, basically, Don Rubin, who's coming more from this potential outcome framework, he says that causal discovery is just a topic for cocktail conversation. So it's a very condemning, very clear statement that there are only toy models, it doesn't work, and it's not even worthy of a scientific discussion. And to some aspect, I think this illustrates this split somehow in this community, and to some extent you can indeed say that for causal discovery we've lacked a little bit of these success stories, and one of the big hopes now is that, for example, for genetic networks or these perturbational data, We can show some of the applications where this is indeed useful and where indeed it makes a difference. This is part of the motivation for today. What I do now a little bit is I give a brief introduction to causality, and then I present an approach for feature selection, which is not causal discovery. It's solving a significantly simpler problem, but it's one of the few approaches I know where basically a cause-effect approach, which was on one side of this divide, was used for structure search or for a subproblem of structure search, and where we can see very clearly what are frameworks when this is even possible from observational data, and I find that very interesting, and I think it's a very— it's a couple of recent papers in that direction which make that interesting, and then hopefully afterwards I will still be able to talk about experimental design, but I have to I have to see what my timeline for that is. And for both projects, I will mention it clearly, there are a couple of things we believe which are great now and work well and we have a good understanding. And we search for collaboration partners to bring that more into the biomedical field because we indeed lack a lot of the domain knowledge. And that is unfortunately very important for the validation of these approaches. Okay, so here, just as a quick primer, so for causality, what is the main question? It's basically what would happen if I do an intervention. So you are not interested in basically getting a prediction at the chocolate consumption within a certain value you already observe where you can necessarily plot the line, but you are very much interested what would, for example, happen if I change the chocolate consumption to, let's say, 150 kilograms per capita. So I move completely outside to the right of this chart. In order to answer that, you need to make a causal assumption. Otherwise, it's okay, and you have to basically say, I cannot answer that question, and that's completely acceptable. But if you provide an answer, you need to have an underlying causal model. So either you say that Nobel Prizes causes chocolate consumption, that chocolate consumption makes Nobel Prizes, or that there's a hidden confounder. Whatever you answer will depend on this assumption. And part of the problem is that all the options, so all your assumptions, are equally versi in this observational data, you can actually not differentiate between them. And that is why you actually have to mention them, because otherwise, I think, without stating these assumptions clearly, you shouldn't make a causal statement. And now what is the problem is basically what happens if you don't know the graph, but you need to learn or want to learn it from data. And what you see on the right is basically the number of DAGs and the number of possible graphs for the different number of nodes. So here it's only plotted for up to 10 nodes, and you see that basically for 3 nodes it's 25 directed acyclic graphs, and then for 10 nodes I can't even unfortunately pronounce the number because it's so huge. And searching this super exponential space, super exponentially growing space, is extremely hard. And in addition, it turns out that especially from observational data, I would argue that purely causal discovery from purely observational data is basically impossible. I think you need very strong assumptions, and it's very debatable if these assumptions are valid to do it from observational data alone. There's a big hope right now is that basically if you can move away from just observational data and get, for example, interventional data in addition, this becomes much more possible, and that with these recent advances of deep learning, you can actually search efficiently in the space. And for this, we gave recently an ICML tutorial with Rosemary Kay, and I put the link on the bottom of the slides, and part of it I actually take in this introduction. So for structure learning, you have this super exponentially growing space of DAGs, and there are two ways you can basically identify the causal graph. So one is you basically make very restrictive assumptions on the functional nature of the data. This might be graph assumptions, this might be functional relationships, on the subsets of these graphs, or you use this interventional data. Then you're actually able, using, for example, this interventional data to regularize your search space and actually hopefully converge much closer to these possible answers. This is something we see more and more now. Again, these assumptions which we make, so this I will illustrate in the next couple of slides, are extremely important to mention.. And this is another view of this topic. So on the left, we have a classical statistical model, which basically is a joint distribution over random variables, and it's one joint distribution. And on the right, you see an illustration of a causal model, which is not just a single joint distribution, but it's basically a poset of joint distributions, one for each possible intervention. So basically, an intervention indicates which joint distribution you should use. And how do you move from the left to the right? It's basically implied by these assumptions. So how you can move from a statistical model to a causal model is determined on these assumptions, which cannot be expressed by the joint distribution alone. And that is why it's so important to state them clearly. And there's a couple of problems with some of these assumptions. So here's again one of these quotations from these interviews. So this time it's Jamie Robins, who's a professor for epidemiology at Harvard.— and he has this quote: "Before you can pull a rabbit out of a hat, you have to put the rabbit in." So this references to this magician on stage who basically pulls the rabbit out. And somehow we all know that sooner or later there must be a rabbit put into the cylinder before you can pull it out. Otherwise the whole trick doesn't work. And here, for example, in this interview, he explains one of these algorithms, and in particular he mentions that basically it was for genetic networks, that one of these key assumptions is faithfulness. Faithfulness is kind of a technical assumption. I put some references for it below, but basically tells you when you can conclude from conditional independencies to infer basically dark constraints. So it's kind of a technical term linking both frameworks. And it's somehow necessary for a lot of these algorithms to assume faithfulness to perform causal discovery, but what he points out is basically that this assumption which seems reasonable or innocent at the beginning, potentially puts a lot of unsought or unwanted conclusions already in your algorithm by basically being the rabbit you pull out. So he mentions that they are able to, from complete ignorance of the gene network to complete knowledge seemingly by magic, and how is that possible? And that is basically part of this assumption, how he moved from the left, from a statistical to a causal model. So here, basically, there's a clear criticism of some of these assumptions, and there's a huge argument in the field when this is okay or when it's not okay. But what I want to mention is that some of these assumptions are really critical and potentially have unwanted effects. And another quote— so this is the final quote from these citations on these interviews. I think they are really, really good and highly recommend to read them. So he actually mentions that he's less critical now of this assumption of faithfulness, and the key reason is that he wants to evaluate most of these algorithms by a downstream task. So for example, drug target discovery or hit rate for recommending disease-causing genes, and basically if we have these algorithms, then we measure which algorithm performs very well in this hit rate, and then we can basically conclude that's a good algorithm. And right now, that is one of the key problems in our field, that we basically cannot directly say what is a good algorithm. So if you ask me what is the best algorithm, it's quite hard to say. Potentially, I would argue that if you're not interested in directed edges, it's probably LASSO, but it's hard to say. And one of the big problems and one of the big hopes we have for biology is basically that we can work towards getting access to data or towards these datasets and benchmarks where you can actually try and improve some of these algorithms, which has been extremely helpful in other disciplines. Like ImageNet significantly changed computer vision because there was something you can basically optimize for. There's some negative side effects as well, and crazy overfitting, but at least that definitely helped. And in our case, that's for sure still missing, especially in real-world datasets. And now I basically want to present an approach which goes in this direction, and combines cause-effect estimation and structure search. And there's one view I wanted to mention and one line of recent work I think is very interesting and provides a core strength interpretation. So it's a very particular interpretation of causality, but you could say it's a debiasing method. So before I provided this high-level view, and right now what I say is basically one view of causality is that you are interested in getting a debiased estimate. So you're interested, so if you have a parametric model, you're interested what is the ground truth or close to the ground truth estimate of a parameter theta, which gives you this effect of a certain variable. And for many prediction problems, you don't care about this ground truth parameter, but if you want to use it, for example, for decision making or for changes in distribution, you want to know the ground truth. And I think it's a less well-known or less general view, but thinking about it as a debiasing method where you do an estimation and then you want to get a de-biased estimate is one way to view causality because it likewise comes with a different trade-off. It tells you you're interested in de-biasing, not necessarily in getting a variance reduction. So it's very clear what you optimize for and that there's a side cost. And this has been more a trend in the statistics community. So there's a couple of papers, some of them are very highly cited already and are still ongoing research. So one of these directions is, for example, debiasing the lasso. So we know that the lasso, this L1 regularization, is very powerful. It's one of the best understood methods, but admittedly it introduces some regularization and some bias because you set low values to zero, which doesn't necessarily imply that this is true. Like, you do that for multiple reasons, partially because you have an underdetermined problem, but then in decision making you want to correct for this regularization term you introduced to solve your problem. And how can you do that? And there are some very nice statistical papers. They are not that easy to read in the sense that they are really written for statistics, for the statistics community, which you already see. Annals of Statistics are two papers referenced here. But I think it's a very interesting direction, and I will discuss this double or debiased machine learning a little bit more in detail. And interestingly enough, that is why I point the paper additional out. These papers are quite connected to conditional independence testing as well, and we know that conditional impendence testing is hard, so you shouldn't do that and use approaches based on conditional independence testing necessarily for causal discovery. And interestingly enough, I think it's kind of a rediscovery of methods developed in the '80s and '90s. So one of these earlier papers is this root n consistent semi-parametric regression. And what I present now is basically something very similar. I don't necessarily claim novelty in the method. Even that you can do that was basically mentioned both in at least two of these papers, in the double machine learning and the hardness of conditional independence testing. It was mentioned that you can use the same method for search, but it wasn't explicitly shown, and my guess is that even in the '80s they already mentioned it. But I find it nevertheless very interesting. Let me see. Okay, so here's basically the problem of double machine learning. So this is the paper I mentioned before. So assume that you have a problem in high dimensions and you have a target variable, let's say y, and then a particular variable which is called d where you're interested in the effect of d on your target. So let's say, for example, y is your income as a company, and then you want to understand what is the effect of a marketing action, or what is the effect of, let's say, an interest rate hike or something like that. You want to estimate this parameter which gives you this effect of this variability very well, while at the same time, you know, you have access to basically an Excel sheet where we have hundreds of variables which might all be correlated with your variability. And so what are possible solutions? What you can do? Well, one thing is you just ignore all the axes and you just do a regression for basically estimating one parameter, which is what is the effect of D on Y, but you know that you excluded multiple variables, this will be a biased solution. Another thing, and another solution is basically predict Y given D in the first step, then try to fit an additional model on the effect of X on Y and then iterate until convergent, but this again will give you a biased estimate. And one of these other solutions was what Viktor Chernokhryukhov and collaborators came up with, was this partially nonlinear model which they call double machine learning, which is very inspired from the semi-parametric statistics. So what you have here— so on the right you have basically assumptions on certain conditional independencies between your noise variables, but what you have here is you model it basically as a linear model in your variable of interest, which is d, and at the same time you model nonlinear models for the effect of these x's on the d and of the x's on the y. So it's a partially linear model which you try to fit. And now they provide a recipe of basically what to do, and the recipe is relatively straightforward. It's a very long paper. It's a technically difficult paper to read. It's 80 pages, and they basically show that the Neyman orthogonality condition holds. So I won't be able to go into the details, but it's It's about finding you an unbiased estimate of this parameter, of this parameter theta. And what they do is basically they first predict and use a machine learning model for this variable m, which is basically the effect of x on d. Then they train a machine learning model g, which is the effect of x on y. And then they do a linear regression of the residuals from the second step on the residuals from the first step. To get an estimate of theta 0. There's an additional step where you need to do cross-splitting, but what they basically show, and where you see multiple block entries on the right, is that in this procedure you get an unbiased estimate for your theta 0. And it's kind of an interesting problem formulation which is already highly picked up. And why is it called double machine learning? It's basically called double machine learning because M and G can be nonlinear machine learning based models. Potentially, I think in theory, only LASSO is valid for some of these assumptions, but potentially you— and what we use as well— is you can put in a random forest or neural network for estimating this, using somehow combining machine learning with classical statistical inference for an unbiased estimate. And that is quite an interesting method, but you are very interested only in one parameter estimate, given that you're somehow able to write down that model. And what is now interesting is that you can do causal feature selection with this method. And for causal feature selection, what you do is actually plotted on the left. So here you are not interested in full causal discovery. So you're only interested in the direct causes of your target variable y. So you have y and then you have these direct causes. In this case on the left, it's x1 and x2., and then you have an arbitrarily complex network afterwards, upstream, so to say. And why is that interesting? I think it's interesting for a couple of reasons. Multiple times, I think you're only interested in these direct causal parents. One of the reasons is that they have a really direct effect on your target variable. If something is far away of your target variable, it's pretty unlikely that if you intervene there, you will have significant effect on the target. So in principle, you would like to have these X1 and X2 variables because they are directly connected to this Y. And what is interesting about this problem is you are robust to hidden confounder on everything, basically, on all the— and all complexities you can imagine on this network between the covariates. So previously, you have very strong assumptions on causal discovery, for example, acyclicality, and many methods assume linearity. And here you only assume right now linearity in the direct causes on Y, and everything else can be arbitrarily complex, nonlinear, and so on. And that is kind of a very interesting setting. And what is interesting is that you can use DoubleML pretty much out of the box for feature selection for identifying these X1 and X2. And the interesting result is that if X1, if you basically do search through all possible features by basically doing two-stage regressions each time where you put X, your one of these Xs, in these regressions for M and G in this double ML. And what you can show is that a particular variable is a direct cause if and only if your conditional covariance is non-zero. So you basically run it for each variable, you check if the estimate is zero or not zero, and if it's zero, it's not a direct cause. So it's kind of relatively straightforward and scalable to implement, and you only need to do some form of normality test or t-test to decide if something is zero or not. And this is what you see here. So this scales extremely well because you only basically need to loop through your variables once, and you can do that for 1,000 variables relatively efficiently. And here again are some of these necessary assumptions on when this works. So some of these assumptions, what you need to have is basically that your noise is additive and exogenous and independent of these features. I think that is somehow a standard assumption in causality. I think that is not the main problem. Then you assume that there's a linear effect of these direct causes on your target variable. That is somehow a strong assumption, but that There are already papers now from Viktor Chernodchukov's group and so on that you can do that nonlinearly. And the strong assumption is that Y has no direct causal effect on any of the features. So there's no loop of Y back to the features X of these covariates. It's kind of a strong— it's a very strong assumption. It's hard to check directly. What you can see, some of these applications would be that, for example, Y is a phenotype and then the genes, which genes cause this phenotype, and then you would imagine that the phenotype doesn't cause genes. But that is a strict assumption, and it's part of where the magic happens, because you basically remove a lot of the cyclic relationships between y and the x's. Your problem becomes much simpler, and you trade— you make a trade-off for this, for basically allowing then acyclicities between these covariates, hidden confounders between these covariates, and you don't need any of these other additional assumptions. So this is kind of neat. But you can't do full causal discovery with that because your assumption that Y has no children implies that you just can't do it iteratively for each node. Because then if the next node is not allowed to have children, at some point you basically have a trivial graph where everything is independent of everything. So that you don't want to do. But I think for observational data, you can show that this is— for identifying these features, like these three assumptions I mentioned, are kind of necessary. So you can show that., and that is kind of interesting. And what you see here is basically that this works quite well. It's kind of an unfair comparison in the sense that we compare against a lot of methods which do full causal discovery, which is kind of a much harder problem in a more general algorithm, so you would expect it to suffer. But here you see basically that this works quite well across a massive number of simulated datasets and for different noise settings, for different sparsity settings of these features.. And it's very cool that you don't have these additional assumptions on, for example, acyclicity between covariates or nonlinear relationships. So this is why I'm relatively convinced that this works in these settings. And now where we search for especially collaboration partners is that what I showed before is that basically you can have restrictive assumptions on the functional nature of the data. So this is what we did. We basically said y is not allowed to have a child, and that makes this problem much simpler, and we even focus only on direct causes. So these are these assumptions what we basically put in such that this works. And the second aspect is that you can use interventional data. And now the interesting nature happens a little bit that you can get away and relax some of these restrictive assumptions by allowing for interventional data. So for example, right now we assume that basically here on the right, you see that everything basically happens to the left of Y, and we assume there's nothing to the right.. But if you allow an intervention on y, you can immediately say that you basically can remove everything to the right of y, because if you intervene, you basically see what changes, and then you can remove that from your set of covariates and then run this step again. So it's kind of an ill-posed problem because we have a method now which works quite well and we don't have— there are some problems where this is interesting, but if you have a real-world application where some of these aspects can be verified, where this is indeed, for example, a problem where you can either intervene on the phenotype, let's say, or you know that the phenotype Y has— or the variable Y has no children, this is kind of an interesting algorithm to play around because it's theoretically very clear. You have necessary conditions, you have convergence rates, you have valid confidence intervals. You have a lot of these things. You have doubly robust properties by just simplifying this problem a lot. So you don't do full causal discovery. You make one of these assumptions here that Y has no direct causal effect, and then you can show that for this specific subproblem, this is something which actually works. But it's a very specific subproblem. So now in the remaining time, actually want to— and for this, yeah, if you're interested in this problem, we are very excited about that approach and actually search for collaborators, especially in biomedicine, which would be interested to apply that. And now there are a couple of things I still want to talk about for full causal structure discovery. Before, we only used observational data. I think for observational data alone, full causal structure learning is a little bit out of scope without these assumptions. So you need to restrict your problem to, for example, feature selection, and then it becomes possible. And otherwise, you need interventional datasets. And generally, if you're interested in full causal discovery, there are these two different and competing approaches. Some are basically in this category of constraint-based approaches where you can perform a lot of conditional independence tests, which was already shown to be difficult and hard. And the other approach right now is score-based approaches, which is right now very popular, especially with these neural causal models. But there's another viewpoint how you can look at the problem, and this is basically a different separation. And the way I want to mention is that I think the supervised frameworks, they are slowly taking over across a wide range of problems. So on the left, we see these classical causal discovery problems. You basically have unlabeled data, You define a score, basically how well a graph fits the particular data, and then you basically go through iteratively, for example, or by some greedy heuristic, you go through all possible graphs and calculate the score, and then you pick the one which fits the best. And this is basically, I would argue, 99% of the algorithms out there if they are score-based. And now supervised frameworks become more popular, and this is slightly some of these relations again to these large pre-training models. So what performs incredibly well, and there are only two papers to the best of my knowledge, one from ETH and one from DeepMind, is that you basically simulate all possible graphs and you create a synthetic dataset. So you basically take, let's say if you have three nodes, you generate all possible 25 graphs, you simulate all possible noise scenarios, and then you have a dataset which basically covers your space of 3-dimensional graphs. And you train a neural network, basically, which gives you a mapping from the observation to the graph. So there are some technical details, and you can basically use different structures. But you can use a transformer, for example, train that, and then you have a supervised learning problem which gives you from observations to what was the graph which generated the synthetic dataset. And then at a new data point, when a new data point comes on, you already have a trained neural network And basically it's super fast to tell you the mapping, what was the closest ground truth graph which is most likely to generate you that dataset. And this is somehow interesting because now you can imagine, like for 3 nodes I mentioned, there are 25 graphs. That's doable. For 10 nodes, I already couldn't tell you the number, how many graphs there are, because it was so huge I didn't even know what it is. And that is only 10 nodes. Now, if you want to do that for 1,000 nodes, that's a huge dataset. But that we see not only in causal discovery as a winning approach right now, that basically you simulate for like 6 months on multiple GPUs and then you train this neural network. We see that across a wide range of problems. So people try that, for example, on tabular classification. You train a transformer model on simulated data, and then at inference time, you are very fast. It's kind of cheating a little bit in the sense that you try to be in distribution always. You try to cover your space with all possible simulated datasets, and then at inference time, you basically hope that you map it. And there's no theory for that, but it works surprisingly well. And it's kind of— it's a mixture between being frustrated that something like this works so well and you don't have theory, but empirically, that's a very strong method, and it's just getting better in a sense that you can keep— I mean, you can, if you have the industry resources, you can just keep up your GPUs running it and running it and running it. And for many problems, I would even argue that this is sufficient in a sense that we tried a lot of approaches, for example, for gene network inference, and tried to infer a 1,000-dimensional graph. But then when we give collaborators then this 1,000-dimensional graph, it's a little bit too tough to even interpret it. So at some point, I guess you don't even care anymore probably about the graph structure because it's so complex and huge that in any case you can't necessarily interpret it afterwards anymore. But I wanted to mention that explicitly because I think that is one of the trends we have right now where a lot of the field is working empirically that is surprisingly stable to train, it's surprisingly robust, and it works empirically very well. So if you're interested in one of these approaches, try some of these supervised frameworks out. And then I wanted to mention, so this is still one of these remaining overview slides, what are the hopes for combining deep learning with causal structure learning? And I think there are two. One is that basically you can avoid these discrete search spaces by using gradient-based optimization, and you get basically more complex relationships between these variables. So these functional relationships can be arbitrarily complex. And the second aspect is is very interesting as well. It's not clear yet how useful that is, but you basically learn a distribution over graphs. It's not necessarily always a posterior model, but it's some form of distribution over graphs, which potentially could be very helpful for experimental design or for exploration. Right now, even if you would get the true posterior, it's always hard to evaluate how useful these uncertainties or posterior models really are. And there are multiple alternatives like ensembles, like bootstrap models. So this is a whole open topic which will actually prevail, but this is an interesting aspect, I think, and what people hope about. And yeah, I think I mentioned these two slides. So why do these causal, neural causal models actually— how do they connect, or what is the underlying high-level heuristic for that? And one of these connections between causality, deep learning, and generalization is the insight that causality is a lot about factorizations of these joint distributions. And the insight is basically that the right factorization will lead to a faster adaptation. So this is again a heuristic, so we don't know if that is necessarily true, but we assume it's true. And this is illustrated on the right to some extent. So assume that, for example, you have the event that it rains and the event that you open the umbrella. Then the cause-effect, like the right factorization, is very clear.. But assume now that there's, for example, a change in the weather, then you would need to update your probability that it rains, but your mechanism, the mechanism which is I open the umbrella given that it rains, will stay fixed. So you will only need to update one factor, but if you have the wrong factorization, you will need to update both factors. Now it's a very crude heuristic that updating two factors takes more, sorry, takes more samples than updating one factor, and it's not even clear if that is in a range where measurable, but that is basically one of these assumptions. And what people tried now, and which was pioneered by Joshua Elkington in his lab, is basically use the speed of adaptation as a learning signal to identify the model. So if you have multiple interventional datasets, you basically sample a couple of graphs, you check which adapts faster, you keep that one, sample again an intervention, you check which of the possible solutions adapts faster, and so on. And with enough interventions, you clearly get to the ground truth model. Because if you have enough intervention, there's only one possible solution in the end if you intervene on everything. And this is basically one of these examples. There's now a wide range of possible models and there are a wide range of different scores, very different optimization strategies. But basically, you do this iterative loop where you sample graphs. You basically take as a reward the speed of adaptation or the robustness or different scoring mechanisms for checking which graphs you prefer. And then you update your graph sampling and you keep doing that until convergence and you get some distribution over these graphs. And before the purely supervised approaches came up, this was the state of the art on the simulated datasets. Now the supervised approach would lead on most of these datasets, I would argue, which are— on most of these synthetic datasets, we can evaluate it with some form of graph metrics.. And now why is that interesting? And so this is basically the remaining of the talk. What I want to talk about is especially experimental design for causal discovery. So right now we always assume that we have some observational data and we have these interventions given on which we can test it, but we didn't necessarily create the interventional dataset. And one of the frameworks to basically combine these models, which is very popular, is this framework of Bayesian optimal experimental design. Where you basically try to maximize the mutual information of potential outcomes given graph structures and parameters, and where you have some observation, some data, and then need to decide on the experiments, basically where to intervene and what to intervene on. And there are a couple of interesting problems. I think one of the interesting aspects is that these— it's not necessarily theoretically valid already, but for example, because you have a distribution for most of these neural causal models, you can somehow even if it's theoretically not clean, you can somehow just combine it in this Bayesian optimal experimental design by pretending that the distribution inferred is your true posterior. So somehow you can combine these frameworks and then have a pipeline for this experimental design. And there are a couple of problems with that. So you need a full posterior over these graphs, and then you need to optimize for this mutual information, which is an arbitrarily complex problem, and it's extremely hard. And for this, what I mentioned at the beginning and what Jamie Robbins likewise mentioned is we are very excited about biology to test these potential benchmarks in the real world for causal discovery methods. So if you have a causal discovery method, there are basically 3 natural downstream tasks. So one is cause-effect estimation. So given I have a structure, what is the effect of a particular variable? Another one is imputation. Assuming that data is missing but not missing at random, such that there's some underlying graph connecting it. And the third one would be experimental design, like where to intervene next to identify this underlying structure. And imputation is kind of an indirect downstream task. Cause-effect estimation is very particular. And so one of the big hopes of the community was that experimental design and the set-rate experiments would actually solve a lot of problems by providing these benchmarks. And one of these benchmarks we created is GeneDiscord, where we curated a couple of real-world datasets but the hope is— so it's clearly an insufficient dataset. So we hosted a challenge at ICLR, and some of these results are just a little bit disappointing. And what I want to show here is basically what part of the problem is. So I mentioned that there's this Bayesian optimal experimental design framework, and it's very related to active learning. So on the left, you see one of these leading approaches. It's a very cool paper. From Andreas Kirsch in Oxford with Jochen Galskrug, which is batch-balled. I think it's one of the first papers which uses batch experiments. And it's trained— so the plots you see here is trained on the extended MNIST dataset. So extended MNIST is basically MNIST, the numbers plus some handwritten letters. So it's more complex. And on the right, you see this dataset, so in small resolution. But it's surprisingly clean, right? Like for you, you should be able— like, you know, there's a U. Maybe that's a 5. Maybe that's an E. You're fairly confident in what these numbers are. And on the left, you basically see an active learning strategy. So you see the accuracy, you retrain after each sample, and then you sample new— so you get new datasets, you train a model on top on classification, and you basically see across this acquired dataset that the accuracy improves and improves. But what you likewise see, that on this dataset, on this very clean dataset, it's surprisingly hard to like even differentiate between a random strategy technology and kind of a leading approach from one of these leading groups in the world. Like, random is surprisingly close to these models. At least the confidence intervals are not overlapping. Yeah. Yeah, yeah, it's a consistent finding, and I think it's— it somehow opens this hope that at some point we should do much better because random shouldn't be the leading approach. But that's— so I think what is very successful here is that at least these lines are not overlapping. I think my consistent finding is more random, and these other lines here basically would all be together. And now this is basically the plot we see. So it's a different cost function, so it's reversed, but it's basically for these genetic datasets in this GeneDiscord with different acquisition functions. You see, like, if you move from extended MNIST to, like, these genetic datasets, like, all these acquisition functions are completely in the same range. It's of course just a hand-picked example, but that is consistent across these datasets. And again, some of these methods are computationally intense. It's super difficult to run them, it's super costly to run them, and Random is for sure included in this batch of very, very good methods. It's even better than some of these other runs in the median. So this is some of the problems we find, and some of these other problems we find is that if we want to do experimental design and Bayesian optimal experimental design, the first problem we need to solve is get a good posterior over causal graphs. And what we see here is some of these neural causal models applied to this reproclay dataset. There's multiple interventions. And unfortunately, we never really got that to work as of now. So somehow you would imagine it's a genome-wide knockout, it's thousands of interventions, you should be able to do it well. And here on this plot, it looks like we beat Scenic and all these other methods,, but it's just one cell line, and if you go to another cell line, it looks completely different. So somehow we don't get these consistent trends. Correlation performs quite well, and what seems worrying is that the more well-known and well-studied the cell line is, the more links are reported, and the more well-known and well-studied the cell line is, the better correlation-based methods perform. So it somehow seems that we do exploration by correlation and not necessarily by causation. Might be a problem of how we curate these datasets. So some of these open questions we are very interested in, and with which I would like to conclude, is basically how can we validate some of these causal models and some of these causal approaches. So we need some benchmarks and datasets, and biomedicine seems the closest for that. We tried a lot with robotics, but it's very hard there as well. And how can we actually consistently and significantly beat random acquisitions in experimental design? I think on both approaches we need to move forward. And with that, I would already conclude, but I'm around the whole day, so I guess some of these aspects were quite fast. But if you're interested in discussing in detail, please send me an email and I would be more than happy to meet you.