From Newsgroup: comp.lang.mumps

<div>Dental calculus, both supra- and subgingival occurs in the majority of adults worldwide. Dental calculus is calcified dental plaque, composed primarily of calcium phosphate mineral salts deposited between and within remnants of formerly viable microorganisms. A viable dental plaque covers mineralized calculus deposits. Levels of calculus and location of formation are population specific and are affected by oral hygiene habits, access to professional care, diet, age, ethnic origin, time since last dental cleaning, systemic disease and the use of prescription medications. In populations that practice regular oral hygiene and with access to regular professional care, supragingival dental calculus formation is restricted to tooth surfaces adjacent to the salivary ducts. Levels of supragingival calculus in these populations is minor and the calculus has little if any impact on oral-health. Subgingival calculus formation in these populations occurs coincident with periodontal disease (although the calculus itself appears to have little impact on attachment loss), the latter being correlated with dental plaque. In populations that do not practice regular hygiene and that do not have access to professional care, supragingival calculus occurs throughout the dentition and the extent of calculus formation can be extreme. In these populations, supragingival calculus is associated with the promotion of gingival recession. Subgingival calculus, in "low hygiene" populations, is extensive and is directly correlated with enhanced periodontal attachment loss. Despite extensive research, a complete understanding of the etiologic significance of subgingival calculus to periodontal disease remains elusive, due to inability to clearly differentiate effects of calculus versus "plaque on calculus". As a result, we are not entirely sure whether subgingival calculus is the cause or result of periodontal inflammation. Research suggests that subgingival calculus, at a minimum, may expand the radius of plaque induced periodontal injury. Removal of subgingival plaque and calculus remains the cornerstone of periodontal therapy. Calculus formation is the result of petrification of dental plaque biofilm, with mineral ions provided by bathing saliva or crevicular fluids. Supragingival calculus formation can be controlled by chemical mineralization inhibitors, applied in toothpastes or mouthrinses. These agents act to delay plaque calcification, keeping deposits in an amorphous non-hardened state to facilitate removal with regular hygiene. Clinical efficacy for these agents is typically assessed as the reduction in tartar area coverage on the teeth between dental cleaning. Research shows that topically applied mineralization inhibitors can also influence adhesion and hardness of calculus deposits on the tooth surface, facilitating removal. Future research in calculus may include the development of improved supragingival tartar control formulations, the development of treatments for the prevention of subgingival calculus formation, the development of improved methods for root detoxification and debridement and the development and application of sensitive diagnostic methods to assess subgingival debridement efficacy.</div><div></div><div></div><div></div><div></div><div></div><div>calculus</div><div></div><div>Download Zip: https://t.co/aTv9qwoo78 </div><div></div><div></div><div>The Calculus exam covers skills and concepts that are usually taught in a one-semester college course in calculus. The content of each exam is approximately 60% limits and differential calculus and 40% integral calculus. Algebraic, trigonometric, exponential, logarithmic, and general functions are included. The exam is primarily concerned with an intuitive understanding of calculus and experience with its methods and applications. Knowledge of preparatory mathematics is assumed, including algebra, geometry, trigonometry, and analytic geometry.</div><div></div><div></div><div>You might have come across Judea Pearl's new book, and a related interview which was widely shared in my social bubble. In the interview, Pearl dismisses most of what we do in ML as curve fitting. While I believe that's an overstatement (conveniently ignores RL for example), it's a nice reminder that most productive debates are often triggered by controversial or outright arrogant comments. Calling machine learning alchemy was a great recent example. After reading the article, I decided to look into his famous do-calculus and the topic causal inference once again.</div><div></div><div></div><div>Again, because this happened to me semi-periodically. I first learned do-calculus in a (very unpopular but advanced) undergraduate course Bayesian networks. Since then, I have re-encountered it every 2-3 years in various contexts, but somehow it never really struck a chord. I always just thought "this stuff is difficult and/or impractical" and eventually forgot about it and moved on. I never realized how fundamental this stuff was, until now.</div><div></div><div></div><div>First of all, causal calculus differentiates between two types of conditional distributions one might want to estimate. tldr: in ML we usually estimate only one of them, but in some applications we should actually try to or have to estimate the other one.</div><div></div><div></div><div>This is perhaps the main concept I haven't grasped before. $p(y\vert do(x))$ is in fact a vanilla conditional distribution, but it's not computed based on $p(x,z,y,\ldots)$, but a different joint $p_do(X=x)(x,z,y,\ldots)$ instead. This $p_do(X=x)$ is the joint distribution of data which we would observe if we actually carried out the intervention in question. $p(y\vert do(x))$ is the conditional distribution we would learn from data collected in randomized controlled trials or A/B tests where the experimenter controls $x$. Note that actually carrying out the intervention or randomized trials may be impossible or at least impractical or unethical in many situations. You can't do an A/B test forcing half your subjects to smoke weed and the other half to smoke placebo to understand the effect on marijuana on their health. Even if you can't directly estimate $p(y\vert do(x))$ from randomized experiments, the object still exists. The main point of causal inference and do-calculus is:</div><div></div><div></div><div></div><div></div><div></div><div></div><div>Now the question is, how can we say anything about the green conditional when we only have data from the blue distribution. We are in a better situation than before as we have the causal model relating the two. To cut a long story short, this is what the so-called do-calculus is for. Do-calculus allows us to massage the green conditional distribution until we can express it in terms of various marginals, conditionals and expectations under the blue distribution. Do-calculus extends our toolkit of working with conditional probability distributions with four additional rules we can apply to conditional distributions with the $do$ operators in them. These rules take into account properties of the causal diagram. The details can't be compressed into a single blog post, but here is an introductory paper on them..</div><div></div><div></div><div>Ideally, as a result of a do-calculus derivation you end up with an equivalent formula for $\tildep(y\vert do(x))$ which no longer has any do operators in them, so you estimate it from observational data alone. If this is the case we say that the causal query $\tildep(y\vert do(x))$ is identifiable. Conversely, if this is not possible, no matter how hard we try applying do-calculus, we call the causal query non-identifiable, which means that we won't be able to estimate it from the data we have. The diagram below summarizes this causal inference machinery in its full glory.</div><div></div><div></div><div>The new panel called "estimable formula" shows the equivalent expression for $\tildep(y\vert do(x))$ obtained as a result of the derivation including several do-calculus rules. Notice how the variable $z$ which is completely irrelevant if you only care about $p(y\vert x)$ is now needed to perform causal inference. If we can't observe $z$ we can still do supervised learning, but we won't be able to answer causal inference queries $p(y\vert do(x))$.</div><div></div><div></div><div>I wanted to emphasize again that this is not a question of whether you work on deep learning or causal inference. You can, and in many cases you should, do both. Causal inference and do-calculus allows you to understand a problem and establish what needs to be estimated from data based on your assumptions captured in a causal diagram. But once you've done that, you still need powerful tools to actually estimate that thing from data. Here, you can still use deep learning, SGD, variational bounds, etc. It is this cross-section of deep learning applied to causal inference which the recent article with Pearl claimed was under-explored.</div><div></div><div></div><div>Users are permitted to make posts and comments linking their own content provided the content is relevant to r/calculus, and is not in violation of Reddit's site-wide policy on self-promotion.</div><div></div><div></div><div>Efficient C++ optimized functions for numerical and symbolic calculus as described in Guidotti (2022) . It includes basic arithmetic, tensor calculus, Einstein summing convention, fast computation of the Levi-Civita symbol and generalized Kronecker delta, Taylor series expansion, multivariate Hermite polynomials, high-order derivatives, ordinary differential equations, differential operators (Gradient, Jacobian, Hessian, Divergence, Curl, Laplacian) and numerical integration in arbitrary orthogonal coordinate systems: cartesian, polar, spherical, cylindrical, parabolic or user defined by custom scale factors.</div><div></div><div></div><div>I've been writing a lot of programs in the lambda calculus recently and I wish I could run some of them in realtime. Yet, as much as the trending functional paradigm is based on the lambda calculus and the rule of B-reductions, I couldn't find a single evaluator that isn't a toy, not meant for efficiency. Functional languages are supposed to be fast, but those I know don't actually provide access to normal forms (see Haskell's lazy evaluator, Scheme's closures and so on), so don't work as LC evaluators.</div><div></div><div></div><div>That makes me wonder: is it just impossible to evaluate lambda calculus terms efficiently, is it just a historical accident / lack of interest that nobody decided to create a fast evaluator for it, or am I just missing something?</div><div></div><div> df19127ead</div>
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