andrew_gelman_stats andrew_gelman_stats-2014 andrew_gelman_stats-2014-2311 knowledge-graph by maker-knowledge-mining
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Introduction: Mark Girolami points us to this paper and software (with Oksana Chkrebtii, David Campbell, and Ben Calderhead). They write: We develop a general methodology for the probabilistic integration of differential equations via model based updating of a joint prior measure on the space of functions and their temporal and spatial derivatives. This results in a posterior measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. We show how this posterior measure can be naturally incorporated within the Kennedy and O’Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration. . . . A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. We also demonstrate our methodolo
sentIndex sentText sentNum sentScore
1 Mark Girolami points us to this paper and software (with Oksana Chkrebtii, David Campbell, and Ben Calderhead). [sent-1, score-0.069]
2 They write: We develop a general methodology for the probabilistic integration of differential equations via model based updating of a joint prior measure on the space of functions and their temporal and spatial derivatives. [sent-2, score-1.239]
3 This results in a posterior measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. [sent-3, score-1.435]
4 We show how this posterior measure can be naturally incorporated within the Kennedy and O’Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration. [sent-4, score-0.782]
5 A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. [sent-8, score-1.58]
6 We also demonstrate our methodology on a large scale system, by modeling discretization uncertainty in the solution of the Navier-Stokes equations of fluid flow, reduced to over 16,000 coupled and stiff ordinary differential equations. [sent-9, score-1.618]
7 This looks interesting and potentially very important. [sent-10, score-0.061]
8 In any case it can be difficult because for big problems these differential equations can take a long time to (numerically) solve. [sent-12, score-0.744]
9 So there would be a lot of use for a method that could take random samples from the posterior distribution, to quantify uncertainty without requiring too many applications of the differential equation solver. [sent-13, score-1.004]
10 One challenge is that when you run a differential equation solver, you choose a level of discretization. [sent-14, score-0.628]
11 Too fine a discretization and it runs too slow; too coarse and you’re not actually evaluating the model you’re interested in. [sent-15, score-0.507]
12 I’ve only looked quickly at this new paper of Chkrebtii et al. [sent-16, score-0.069]
13 , but it appears that they are explicitly modeling this discretization error rather than following the usual strategy of applying a very fine grid and then assuming the error is zero. [sent-17, score-0.71]
14 The idea, I assume, is that if you model the error you can use a much coarser grid and still get good results. [sent-18, score-0.355]
15 This seems possible given that do apply any of these methods you need to apply the solver many many times. [sent-19, score-0.462]
16 As the above image from their paper illustrates, each iteration of the algorithm proceeds by running a stochastic version of the differential-equation solver. [sent-20, score-0.326]
17 As they emphasize, they do not simply repeatedly run the solver as is; rather they go inside the solver and make it stochastic as a way to introduce uncertainty into each step. [sent-21, score-1.023]
18 I haven’t read through the paper more than this but it looks very interesting and it feels right in the sense that they’ve added some items to the procedure so it’s not like they’re trying to get something for nothing. [sent-22, score-0.13]
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Introduction: Mark Girolami points us to this paper and software (with Oksana Chkrebtii, David Campbell, and Ben Calderhead). They write: We develop a general methodology for the probabilistic integration of differential equations via model based updating of a joint prior measure on the space of functions and their temporal and spatial derivatives. This results in a posterior measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. We show how this posterior measure can be naturally incorporated within the Kennedy and O’Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration. . . . A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. We also demonstrate our methodolo
Introduction: This is an echo of yesterday’s post, Basketball Stats: Don’t model the probability of win, model the expected score differential . As with basketball, so with baseball: as the great Bill James wrote, if you want to predict a pitcher’s win-loss record, it’s better to use last year’s ERA than last year’s W-L. As with basketball and baseball, so with epidemiology: as Joseph Delaney points out in my favorite blog that nobody reads, you will see much better prediction if you first model change in the parameter (e.g. blood pressure) and then convert that to the binary disease state (e.g. hypertension) then if you just develop a logistic model for prob(hypertension). As with basketball, baseball, and epidemiology, so with political science: instead of modeling election winners, better to model vote differential, a point that I made back in 1993 (see page 120 here ) but which seems to continually need repeating . A forecasting method should get essentially no credit for correctl
Introduction: Someone who wants to remain anonymous writes: I am working to create a more accurate in-game win probability model for basketball games. My idea is for each timestep in a game (a second, 5 seconds, etc), use the Vegas line, the current score differential, who has the ball, and the number of possessions played already (to account for differences in pace) to create a point estimate probability of the home team winning. This problem would seem to fit a multi-level model structure well. It seems silly to estimate 2,000 regressions (one for each timestep), but the coefficients should vary at each timestep. Do you have suggestions for what type of model this could/would be? Additionally, I believe this needs to be some form of logit/probit given the binary dependent variable (win or loss). Finally, do you have suggestions for what package could accomplish this in Stata or R? To answer the questions in reverse order: 3. I’d hope this could be done in Stan (which can be run from R)
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Introduction: Todd Schneider writes: Apropos of your recent blog post about modeling score differential of basketball games , I thought you might enjoy a site I built, gambletron2000.com , that gathers real-time win probabilities from betting markets for most major sports (including NBA and college basketball). My original goal was to use the variance of changes in win probabilities to quantify which games were the most exciting, but I got a bit carried away and ended up pursuing a bunch of other ideas, which you can read about in the full writeup here This particular passage from the anonymous someone in your post: My idea is for each timestep in a game (a second, 5 seconds, etc), use the Vegas line, the current score differential, who has the ball, and the number of possessions played already (to account for differences in pace) to create a point estimate probability of the home team winning. reminded me of a graph I made, which shows the mean-reverting tendency of N
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Introduction: Jeremy Fox asks what I think about this paper by David N. Reshef, Yakir Reshef, Hilary Finucane, Sharon Grossman, Gilean McVean, Peter Turnbaugh, Eric Lander, Michael Mitzenmacher, and Pardis Sabeti which proposes a new nonlinear R-squared-like measure. My quick answer is that it looks really cool! From my quick reading of the paper, it appears that the method reduces on average to the usual R-squared when fit to data of the form y = a + bx + error, and that it also has a similar interpretation when “a + bx” is replaced by other continuous functions. Unlike R-squared, the method of Reshef et al. depends on a tuning parameter that controls the level of discretization, in a “How long is the coast of Britain” sort of way. The dependence on scale is inevitable for such a general method. Just consider: if you sample 1000 points from the unit bivariate normal distribution, (x,y) ~ N(0,I), you’ll be able to fit them perfectly by a 999-degree polynomial fit to the data. So the sca
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Introduction: Mark Girolami points us to this paper and software (with Oksana Chkrebtii, David Campbell, and Ben Calderhead). They write: We develop a general methodology for the probabilistic integration of differential equations via model based updating of a joint prior measure on the space of functions and their temporal and spatial derivatives. This results in a posterior measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. We show how this posterior measure can be naturally incorporated within the Kennedy and O’Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration. . . . A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. We also demonstrate our methodolo
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Introduction: Martyn Plummer replied to my recent blog on DIC with information that was important enough that I thought it deserved its own blog entry. Martyn wrote: DIC has been around for 10 years now and despite being immensely popular with applied statisticians it has generated very little theoretical interest. In fact, the silence has been deafening. I [Martyn] hope my paper added some clarity. As you say, DIC is (an approximation to) a theoretical out-of-sample predictive error. When I finished the paper I was a little embarrassed to see that I had almost perfectly reconstructed the justification of AIC as approximate cross-validation measure by Stone (1977), with a Bayesian spin of course. But even this insight leaves a lot of choices open. You need to choose the right loss function and also which level of the model you want to replicate from. David Spiegelhalter and colleagues called this the “focus”. In practice the focus is limited to the lowest level of the model. You generall
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Introduction: Following up on our discussion from the other day, Angelika van der Linde sends along this paper from 2012 (link to journal here ). And Aki pulls out this great quote from Geisser and Eddy (1979): This discussion makes clear that in the nested case this method, as Akaike’s, is not consistent; i.e., even if $M_k$ is true, it will be rejected with probability $\alpha$ as $N\to\infty$. This point is also made by Schwarz (1978). However, from the point of view of prediction, this is of no great consequence. For large numbers of observations, a prediction based on the falsely assumed $M_k$, will not differ appreciably from one based on the true $M_k$. For example, if we assert that two normal populations have different means when in fact they have the same mean, then the use of the group mean as opposed to the grand mean for predicting a future observation results in predictors which are asymptotically equivalent and whose predictive variances are $\sigma^2[1 + (1/2n)]$ and $\si
Introduction: From August 1990. It was in the form of a note sent to all the people in the statistics group of Bell Labs, where I’d worked that summer. To all: Here’s the abstract of the work I’ve done this summer. It’s stored in the file, /fs5/gelman/abstract.bell, and copies of the Figures 1-3 are on Trevor’s desk. Any comments are of course appreciated; I’m at gelman@stat.berkeley.edu. On the Routine Use of Markov Chains for Simulation Andrew Gelman and Donald Rubin, 6 August 1990 corrected version: 8 August 1990 1. Simulation In probability and statistics we can often specify multivariate distributions many of whose properties we do not fully understand–perhaps, as in the Ising model of statistical physics, we can write the joint density function, up to a multiplicative constant that cannot be expressed in closed form. For an example in statistics, consider the Normal random effects model in the analysis of variance, which can be easily placed in a Bayesian fram
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Introduction: Mark Girolami points us to this paper and software (with Oksana Chkrebtii, David Campbell, and Ben Calderhead). They write: We develop a general methodology for the probabilistic integration of differential equations via model based updating of a joint prior measure on the space of functions and their temporal and spatial derivatives. This results in a posterior measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. We show how this posterior measure can be naturally incorporated within the Kennedy and O’Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration. . . . A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. We also demonstrate our methodolo
Introduction: Wayne Folta points me to “EigenBracket 2012: Using Graph Theory to Predict NCAA March Madness Basketball” and writes, “I [Folta] have got to believe that he’s simply re-invented a statistical method in a graph-ish context, but don’t know enough to judge.” I have not looked in detail at the method being presented here—I’m not much of college basketball fan—but I’d like to use this as an excuse to make one of my favorite general point, which is that a good way to characterize any statistical method is by what information it uses. The basketball ranking method here uses score differentials between teams in the past season. On the plus side, that is better than simply using one-loss records (which (a) discards score differentials and (b) discards information on who played whom). On the minus side, the method appears to be discretizing the scores (thus throwing away information on the exact score differential) and doesn’t use any external information such as external ratings. A
3 0.94242227 1626 andrew gelman stats-2012-12-16-The lamest, grudgingest, non-retraction retraction ever
Introduction: In politics we’re familiar with the non-apology apology (well described in Wikipedia as “a statement that has the form of an apology but does not express the expected contrition”). Here’s the scientific equivalent: the non-retraction retraction. Sanjay Srivastava points to an amusing yet barfable story of a pair of researchers who (inadvertently, I assume) made a data coding error and were eventually moved to issue a correction notice, but even then refused to fully admit their error. As Srivastava puts it, the story “ended up with Lew [Goldberg] and colleagues [Kibeom Lee and Michael Ashton] publishing a comment on an erratum – the only time I’ve ever heard of that happening in a scientific journal.” From the comment on the erratum: In their “erratum and addendum,” Anderson and Ones (this issue) explained that we had brought their attention to the “potential” of a “possible” misalignment and described the results computed from re-aligned data as being based on a “post-ho
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Introduction: Posted by Phil Price: A blogger (can’t find his name anywhere on his blog) points to an article in the medical literature in 1994 that is…well, it’s shocking, is what it is. This is from the abstract: In Tai’s Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method (less than +/- 0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin Yes, that’s right, this guy has rediscovered the trapezoidal rule. You know, that thing most readers of this blog were taught back in 1
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