 Hello everyone, I'm Jan, and this talk will be about the package called TIP that we recently released. And it has to do with the Bayesian mixture modelling in clustering applications. So if you're not into Bayesian modelling or clustering, this might not be the talk for you. Now, Bayesian mixture models involves three steps usually. And the first step is a probabilistic generation of random artitions. And this could come from a probability distribution or stochastic processes. I mean, it's discretization. And then the second step is an allocation of observations into these blocks. And then the third step will be learning of these parameters. So here, the parameters could be those that are specific to each cluster, as well as those that are common across clusters that sort of guides the probabilistic generation of random partitions in step one. And each partition has a unique young diagram representation. So this partition corresponds to this young diagram. And the way to read this is each row corresponds to each block and the number of columns within each row corresponds to the number of balls within each partition. So that's why this and this is the same thing. And a prior partition essentially takes this young diagram and returns a probability for all yet possible young diagrams of sample size n, in this case n equals to six. Now, let's say that we know a priori that observation y1 y2 and y3 y4 should likely belong to the same cluster. And then let's say the map maximum of posteriori allocation looks like this. This from data for the posterior. So we learn something from data, but the role data may not be that much in some cases. And this chosen partition has a young representation young diagram representation as follows and other potential partitions that this and this and this. And the reason is because you know why one and why two and why three and why four has to be in the same partitions. So you get these squares. Now, and depending on the model and the hyper parameter you choose, it could be that you're, you're assigning a much higher prior probabilities on this partition, rather than these. And that would mean that a priori you're assigning a very high weight on this compared to these. And then, even if you try various allocations, even a posteriori your likelihood might be overwhelmed by the your prior. And you're just giving two high posterior probability on this partition. And then what you're doing is that, you know, you're comparing this allocation to, you know, this allocation, or others are the same. And you're picking this allocation as the maximum of posteriori estimate. So you're not really using data that much. You know, you're only using data to compare this to this. So you don't want to end up in a situation like this. And in order to prevent this from happening. You can generate all possible young diagrams and then compute these probabilities. And this is possible if you only have six observations or something because they're only 11 possible young diagrams. But let's, if you have like 100 observations, which shouldn't be that unusual, then there would be this many young diagrams so we can't do this approach. And this is where the feedback is comes into handy because we take an alternative approach and what we do is to consider computing a symmetric and additive functional over all possible prior partitions and one function we find particularly useful is called relative entropy. And it basically quantifies evenness of partition sizes of partitions that are kind of even. It returns a value will be written will get a value one, which is the maximum and the partitions that are uneven will get values that are close to zero like this and this. And this is how fit package can be used. So N is the sample size here we plug in six so we don't really need a fit package in K plus is the constraint that we have unfortunately put on and this is the length of the partition which corresponds to the role of the young diagram so here we are fixing that the three which means that we are only constrain these three partitions. And then, then we also supply the type this is the model here we supplied to just process mixtures and alpha is the concentration parameters of this model which we set to one. And then the mean is we get is two point eight seven. And this is a bit higher than the arithmetic mean of this and this and this, which means that the DPM with alpha equals to one assign slightly higher prior weight on this partition as opposed to these, but not too much so we're not likely in this situation at least. And we can do more advanced thing with a fit package like comparing DPM with the, it's discretized version dishes what you know me distributions with different by distributions on the number of clusters and partitions. But, and this is something advanced and I don't have a time to introduce. So, if you if you're interested in feel free to contact us and we can provide more details. Thank you for listening.