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Showing posts from March, 2008

Constrained MDPs and the reward hypothesis

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It's been a looong ago that I posted on this blog. But this should not mean the blog is dead. Slow and steady wins the race, right? Anyhow, I am back and today I want to write about constrained Markovian Decision Process (CMDPs). The post is prompted by a recent visit of Eugene Feinberg , a pioneer of CMDPs, of our department, and also by a growing interest in CMPDs in the RL community (see this , this , or this paper). For impatient readers, a CMDP is like an MDP except that there are multiple reward functions, one of which is used to set the optimization objective, while the others are used to restrict what policies can do. Now, it seems to me that more often than not the problems we want to solve are easiest to specify using multiple objectives (in fact, this is a borderline tautology!). An example, which given our current sad situation is hard to escape, is deciding what interventions a government should apply to limit the spread of a virus while maintaining economic
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Statistical Modeling: The Two Cultures

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Sometimes people ask what is the difference between what statisticians and machine learning researchers do. The best answer that I have found so far can be found in " Statistical Modeling: The Two Cultures " by Leo Breiman (Statistical Science, 16:199-231, 2001). According to this, statisticians like to start by making modeling assumptions about how the data is generated (e.g., the response is a noise added to the linear combination of the predictor variables), while in machine learning people use algorithm models and treat the data mechanism as unknown. He estimates that (back in 2001) less than 2% of statisticians work in the realm when the data mechanism is considered as unknown. It seems that there are two problem with the data model approach. One is that the this approach does not address the ultimate question which is making good predictions: if the data does not fit the model, this approach has nothing to offer (it does not make sense to apply a statistical test if th
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Bayesian Statistics in Medical Clinical Trials

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I came across a very interesting document. The document is titled " Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials ". It is a draft guidelines poster by the Center for Devices and Radiological Health of FDA, dated May 23, 2006. Why is this interesting? The job of FDA (the US Food and Drug Administration) is to make sure that the decisions in any clinical trial are made in a scientifically sound manner. Clearly, when following the Bayesian approach the choice of the prior and the model can influence the decisions. What does FDA do in this situation? The establish a process where they require a pre-specification (and agreement on) both the prior and the model, including an analysis of the operating characteristics of the design. This latter includes estimating the probability of erroneously approving an ineffective or unsafe device (the Type I error). This will typically be done by conducting Monte-Carlo simulations, where the Type I error is
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Curse of dimensionality

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I came across two papers today that discuss the curse of dimensionality. I thought this is just enough to write a short blog about the topic that definitely deserves attention. So, here we go: The first paper is by Flip Korn, Bernd-Uwe Pagel and Christos Faloutsos, the title is On the "Dimensionality Curse" and the "Self-Similarity Blessing" . This is a 2001 paper that won The paper is about nearest neighbor retrieval: You have $n$ datapoints that you can store and the task is to find the nearest neighbor among these datapoints of a query point. If the data lies in a $D$-dimensional space Euclidean space, it was a common wisdom to believe that the time required to find the nearest neighbor scales exponentially with $D$. The essence of the paper is that if the data lies on a low dimensional manifold then the complexity of search will depend only on the intrinsic dimensionality of the manifold. (The cover tree package due to Alina Beygelzimer , Sham Kakade , and John
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