text:
## ISBA 2016 [#7]

Posted in Mountains, pictures, Running, Statistics, Travel, Wines with tags , , , , , , , , , , , , , on June 20, 2016 by xi'an

This series of posts is most probably getting by now an imposition on the ‘Og readership, which either attended ISBA 2016 and does (do?) not need my impressions or did not attend and hence does (do?) not need vague impressions about talks they (it?) did not see, but indulge me in reminiscing about this last ISBA meeting (or more reasonably ignore this post altogether). Now that I am back home (with most of my Sard wine bottles intact!, and a good array of Sard cheeses).

This meeting seems to be the largest ISBA meeting ever, with hundreds of young statisticians taking part in it (despite my early misgivings about the deterrent represented by the overall cost of attending the meeting. I presume holding the meeting in Europe made it easier and cheaper for most Europeans to attend (and hopefully the same will happen in Edinburgh in 2018!), as was the (somewhat unsuspected) wide availability of rental alternatives in the close vicinity of the conference resort. I also presume the same travel opportunities would not have been true in Banff, although local costs would have been lower. It was fantastic to see so many new researchers interested in Bayesian statistics and to meet some of them. And to have more sessions run by the j-Bayes section of ISBA (although I found it counterproductive that such sessions do not focus on a thematically coherent theme). As a result, the meeting was more intense than ever and I found it truly exhausting, despite skipping most poster sessions. Maybe also because I did not skip a single session thanks to the availability of an interesting theme for each block in the schedule. (And because I attended more [great] Sard dinners than I originally intended.) Having five sessions in parallel indeed means there is a fabulous offer of themes for every taste. It also means there are inevitably conflicts when picking one’s session.

Back to poster sessions, I feel I missed an essential part of the meeting, which made ISBA meetings so unique, but it also seems to me the organisation of those sessions should be reconsidered against the rise in attendance. (And my growing inability to stay up late!) One solution suggested by my recent AISTATS experience is to select posters towards lowering the number of posters in the four poster sessions. The success rate for the Cadiz meeting was 35%.) The obvious downsizes are the selection process (but this was done quite efficiently for AISTATS) and the potential reduction in the number of participants. A medium ground could see a smaller fraction of posters to be selected by this process (and published one way or another as in machine-learning conferences) and presented during the evening poster sessions, with other posters being given during the coffee breaks [which certainly does not help in reducing the intensity of the schedule]. Another and altogether solution is to extend the parallelism of oral sessions to poster sessions, by regrouping them into five or six themes or keywords chosen by the presenters and having those presented in different rooms to split the attendance down to human level and tolerable decibels. Nothing preventing participants to visit several rooms in a given evening. Or to keep posters for several nights in a row if the number of rooms allows.

It may also be that this edition of ISBA 2016 sees the end of the resort-style meeting in the spirit of the early Valencia meetings. Edinburgh 2018 will certainly be an open-space conference in that meals and lodgings will be “on” the participants who may choose where and how much. I have heard many times the argument that conferences held in single hotels or resorts facilitated the contacts between young and senior researchers, but I fear this is not sustainable against the growth of the audience. Holding the meeting in a reasonably close and compact location, as a University building, should allow for a sufficient degree of interaction, as was the case at ISBA 2016. (Kerrie Mengersen also suggested that a few restaurants nearby could be designated as “favourites” for participants to interact at dinner time.) Another suggestion to reinforce networking and interacting would be to hold more satellite workshops before the main conference. It seems there could be a young Bayesian workshop in England the prior week as well as a summer short course on simulation methods.

Organising meetings is getting increasingly complex and provides few rewards at the academic level, so I am grateful to the organisers of ISBA 2016 to have agreed to carry the burden this year. And to the scientific committee for setting the quality bar that high. (A special thought too for my friend Walter Racugno who had the ultimate bad luck of having an accident the very week of the meeting he had contributed to organise!)

[Even though I predict this is my last post on ISBA 2016 I would be delighted to have guest posts on others’ impressions on the meeting. Feel free to send me entries!]

## ISBA 2016

Posted in Kids, Statistics, Travel, University life, Wines with tags , , , , , , , , , , on June 14, 2016 by xi'an

I remember fondly the early Valencia meetings where we did not have to pick between sessions. Then one year there were two sessions and soon more. And we now have to pick among equally tantalising sessions. [Complaint of the super wealthy, I do realise.] After a morning trip to San’Antioco and the southern coast of Sardinia, I started my ISBA 2016 with an not [that Bayesian] high dimension session with Michael Jordan (who gave a talk related to his MCMski lecture), Isa Verdinelli and Larry Wasserman.

Larry gave a [non-Bayesian, what else?!] talk on the problem of data splitting versus double use of the same data. Or rather using a model index estimated from a given dataset to estimate the properties of the mean of the same data. As in model selection. While splitting the data avoids all sorts of problem, not splitting the data but using a different loss function could avoid the issue. (And the infinite regress that if we keep conducting inference, we may have to split further and further the data.) Namely, if we were looking only at quantities that do not vary across models. So it is surprising that prediction get affected by this.

In a second session around Bayesian tests and model choice, Sarah Filippi presented the Bayesian non-parametric test she devised with Chris Holmes, using Polya trees. And mentioned our testing-by-mixture approach as a valuable alternative! Veronika Rockova talked about her new approach to efficient variable selection by spike-and-slab priors, through a mix of particle MCMC and EM, plus some variational Bayes motivations. (She also mentioned extensions by repulsive sampling through the pinball sampler, of which her recent AISTATS paper reminded me.)

Later in the evening, I figured out that the poster sessions that make the ISBA/Valencia meetings so unique are alas out of reach for me as the level of noise and my reduced hearing capacities (!) make impossible any prolonged discussion on any serious notion. No poster session for ‘Og’s men!, then, even though I can hang out at the fringe and chat with friends!

## dynamic mixtures [at NBBC15]

Posted in R, Statistics with tags , , , , , , , , , , , , on June 18, 2015 by xi'an

A funny coincidence: as I was sitting next to Arnoldo Frigessi at the NBBC15 conference, I came upon a new question on Cross Validated about a dynamic mixture model he had developed in 2002 with Olga Haug and Håvård Rue [whom I also saw last week in Valencià]. The dynamic mixture model they proposed replaces the standard weights in the mixture with cumulative distribution functions, hence the term dynamic. Here is the version used in their paper (x>0)

$(1-w_{\mu,\tau}(x))f_{\beta,\lambda}(x)+w_{\mu,\tau}(x)g_{\epsilon,\sigma}(x)$

where f is a Weibull density, g a generalised Pareto density, and w is the cdf of a Cauchy distribution [all distributions being endowed with standard parameters]. While the above object is not a mixture of a generalised Pareto and of a Weibull distributions (instead, it is a mixture of two non-standard distributions with unknown weights), it is close to the Weibull when x is near zero and ends up with the Pareto tail (when x is large). The question was about simulating from this distribution and, while an answer was in the paper, I replied on Cross Validated with an alternative accept-reject proposal and with a somewhat (if mildly) non-standard MCMC implementation enjoying a much higher acceptance rate and the same fit.

## An objective prior that unifies objective Bayes and information-based inference

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , on June 8, 2015 by xi'an

During the Valencia O’Bayes 2015 meeting, Colin LaMont and Paul Wiggins arxived a paper entitled “An objective prior that unifies objective Bayes and information-based inference”. It would have been interesting to have the authors in Valencia, as they make bold claims about their w-prior as being uniformly and maximally uninformative. Plus achieving this unification advertised in the title of the paper. Meaning that the free energy (log transform of the inverse evidence) is the Akaike information criterion.

The paper starts by defining a true prior distribution (presumably in analogy with the true value of the parameter?) and generalised posterior distributions as associated with any arbitrary prior. (Some notations are imprecise, check (3) with the wrong denominator or the predictivity that is supposed to cover N new observations on p.2…) It then introduces a discretisation by considering all models within a certain Kullback divergence δ to be undistinguishable. (A definition that does not account for the assymmetry of the Kullback divergence.) From there, it most surprisingly [given the above discretisation] derives a density on the whole parameter space

$\pi(\theta) \propto \text{det} I(\theta)^{1/2} (N/2\pi \delta)^{K/2}$

where N is the number of observations and K the dimension of θ. Dimension which may vary. The dependence on N of the above is a result of using the predictive on N points instead of one. The w-prior is however defined differently: “as the density of indistinguishable models such that the multiplicity is unity for all true models”. Where the log transform of the multiplicity is the expected log marginal likelihood minus the expected log predictive [all expectations under the sampling distributions, conditional on θ]. Rather puzzling in that it involves the “true” value of the parameter—another notational imprecision, since it has to hold for all θ’s—as well as possibly improper priors. When the prior is improper, the log-multiplicity is a difference of two terms such that the first term depends on the constant used with the improper prior, while the second one does not…  Unless the multiplicity constraint also determines the normalising constant?! But this does not seem to be the case when considering the following section on normalising the w-prior. Mentioning a “cutoff” for the integration that seems to pop out of nowhere. Curiouser and curiouser. Due to this unclear handling of infinite mass priors, and since the claimed properties of uniform and maximal uninformativeness are not established in any formal way, and since the existence of a non-asymptotic solution to the multiplicity equation is neither demonstrated, I quickly lost interest in the paper. Which does not contain any worked out example. Read at your own risk!

## O-Bayes15 [day #1]

Posted in Books, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , on June 3, 2015 by xi'an

So here we are back together to talk about objective Bayes methods, and in the City of Valencià as well.! A move back to a city where the 1998 O’Bayes took place. In contrast with my introductory tutorial, the morning tutorials by Luis Pericchi and Judith Rousseau were investigating fairly technical and advanced, Judith looking at the tools used in the frequentist (Bernstein-von Mises) analysis of priors, with forays in empirical Bayes, giving insights into a wide range of recent papers in the field. And Luis covering works on Bayesian robustness in the sense of resisting to over-influential observations. Following works of him and of Tony O’Hagan and coauthors. Which means characterising tails of prior versus sampling distribution to allow for the posterior reverting to the prior in case of over-influential datapoints. Funny enough, after a great opening by Carmen and Ed remembering Susie, Chris Holmes also covered Bayesian robust analysis. More in the sense of incompletely or mis-  specified models. (On the side, rekindling one comment by Susie and the need to embed robust Bayesian analysis within decision theory.) Which was also much Chris’ point, in line with the recent Watson and Holmes’ paper. Dan Simpson in his usual kick-the-anthill-real-hard-and-set-fire-to-it discussion pointed out the possible discrepancy between objective and robust Bayesian analysis. (With lines like “modern statistics has proven disruptive to objective Bayes”.) Which is not that obvious because the robust approach simply reincorporates the decision theory within the objective framework. (Dan also concluded with the comic strip below, whose message can be interpreted in many ways…! Or not.)

The second talk of the afternoon was given by Veronika Ročková on a novel type of spike-and-slab prior to handle sparse regression, bringing an alternative to the standard Lasso. The prior is a mixture of two Laplace priors whose scales are constrained in connection with the actual number of non-zero coefficients. I had not heard of this approach before (although Veronika and Ed have an earlier paper on a spike-and-slab prior to handle multicolinearity that Veronika presented in Boston last year) and I was quite impressed by the combination of minimax properties and practical determination of the scales. As well as by the performances of this spike-and-slab Lasso. I am looking forward the incoming paper!

The day ended most nicely in the botanical gardens of the University of Valencià, with an outdoor reception surrounded by palm trees and parakeet cries…

## O’Bayes 2015: back in València

Posted in pictures, Statistics, Travel, University life with tags , , , , , on September 11, 2014 by xi'an

The next O’Bayes meeting (more precisely the International Workshop on Objective Bayes Methodology, O-Bayes15), will take place in València, Spain, on June 1-4, 2015. This is the second time an O’Bayes conference takes place in València, after the one José Miguel Bernardo organised in 1998 there.  The principal objectives of O-Bayes15 will be to facilitate the exchange of recent research developments in objective Bayes theory, methodology and applications, and related topics (like limited information Bayesian statistics), to provide opportunities for new researchers, and to establish new collaborations and partnerships. Most importantly, O-Bayes15 will be dedicated to our friend Susie Bayarri, to celebrate her life and contributions to Bayesian Statistics. Check the webpage of O-Bayes15 for the program (under construction) and the practical details. Looking forward to the meeting and hopeful for a broadening of the basis of the O’Bayes community and of its scope!

## Cancun, ISBA 2014 [½ day #2]

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , on July 19, 2014 by xi'an

Half-day #2 indeed at ISBA 2014, as the Wednesday afternoon kept to the Valencia tradition of free time, and potential cultural excursions, so there were only talks in the morning. And still the core poster session at (late) night. In which my student Kaniav Kamari presented a poster on a current project we are running with Kerrie Mengersen and Judith Rousseau on the replacement of the standard Bayesian testing setting with a mixture representation. Being half-asleep by the time the session started, I did not stay long enough to collect data on the reactions to this proposal, but the paper should be arXived pretty soon. And Kate Lee gave a poster on our importance sampler for evidence approximation in mixtures (soon to be revised!). There was also an interesting poster about reparameterisation towards higher efficiency of MCMC algorithms, intersecting with my long-going interest in the matter, although I cannot find a mention of it in the abstracts. And I had a nice talk with Eduardo Gutierrez-Pena about infering on credible intervals through loss functions. There were also a couple of appealing posters on g-priors. Except I was sleepwalking by the time I spotted them… (My conference sleeping pattern does not work that well for ISBA meetings! Thankfully, both next editions will be in Europe.)

Great talk by Steve McEachern that linked to our ABC work on Bayesian model choice with insufficient statistics, arguing towards robustification of Bayesian inference by only using summary statistics. Despite this being “against the hubris of Bayes”… Obviously, the talk just gave a flavour of Steve’s perspective on that topic and I hope I can read more to see how we agree (or not!) on this notion of using insufficient summaries to conduct inference rather than trying to model “the whole world”, given the mistrust we must preserve about models and likelihoods. And another great talk by Ioanna Manolopoulou on another of my pet topics, capture-recapture, although she phrased it as a partly identified model (as in Kline’s talk yesterday). This related with capture-recapture in that when estimating a capture-recapture model with covariates, sampling and inference are biased as well. I appreciated particularly the use of BART to analyse the bias in the modelling. And the talk provided a nice counterpoint to the rather pessimistic approach of Kline’s.

Terrific plenary sessions as well, from Wilke’s spatio-temporal models (in the spirit of his superb book with Noel Cressie) to Igor Prunster’s great entry on Gibbs process priors. With the highly significant conclusion that those processes are best suited for (in the sense that they are only consistent for) discrete support distributions. Alternatives are to be used for continuous support distributions, the special case of a Dirichlet prior constituting a sort of unique counter-example. Quite an inspiring talk (even though I had a few micro-naps throughout it!).

I shared my afternoon free time between discussing the next O’Bayes meeting (2015 is getting very close!) with friends from the Objective Bayes section, getting a quick look at the Museo Maya de Cancún (terrific building!), and getting some work done (thanks to the lack of wireless…)


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format: plain


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Least Squares Approximation - Maple Help

Home : Support : Online Help : Tasks : Curve Fitting : Least Squares Approximation

Least Squares Approximation

 Description Calculate a least squares approximation using specified data points.

Least Squares Fit of Data by a Specified Curve

List of Data Points:

 > $\left[\left[{2}{,}{-}{1}\right]{,}\left[{4}{,}{3}\right]{,}\left[{6}{,}{-}{7}\right]{,}\left[{7}{,}{5}\right]{,}\left[{9}{,}{-}{2}\right]\right]$
 $\left[\left[{2}{,}{-}{1}\right]{,}\left[{4}{,}{3}\right]{,}\left[{6}{,}{-}{7}\right]{,}\left[{7}{,}{5}\right]{,}\left[{9}{,}{-}{2}\right]\right]$ (1)

Fitting Curve:

 > ${a}{{x}}^{{2}}{+}{b}{x}{+}{c}$
 ${a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}$ (2)

Independent Variable:

 > ${x}$
 ${x}$ (3)

Least Squares Curve:

 > ${\mathrm{CurveFitting}}_{}\left[\mathrm{LeastSquares}\right]\left(,,\mathrm{curve}=\right);$
 ${-}\frac{{6315}}{{5071}}{+}\frac{{5975}}{{10142}}{}{x}{-}\frac{{669}}{{10142}}{}{{x}}^{{2}}$ (4)

 Commands Used


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Submit Essay

Once you submit your essay, you can no longer edit it.

# New Megadonor in EA in 2026

### Question

Good Ventures is a philanthropic foundation managing the giving of Dustin Moskovitz and Cari Tuna which is a major donor to effective altruist aligned causes. They donated $219 million to such causes in 2020 and$274 million in 2019, for an average of \$246 million per year in these two years. This question asks if, in 2026, there will be another such foundation or individual donor in the effective altruism space.

Will there be another donor on the scale of 2020 Good Ventures in the Effective Altruist space in 2026?

This question resolves positively if, in 2026, an individual donor or foundation other than Good Ventures, sourcing its wealth from a single individual or family donates over 250 million dollars (adjusted for inflation), using an Effective Altruist framework to guide their donations (see fine print for details).

Grants should only count if they are publicly disclosed, as is currently done by Good Ventures or Survival and Flourishing, for example.

A donor will be considered to have donated the money if the money has been regranted to organisations outside their foundation to explicitly pursue direct work on improving the world (so, for example, donating to a Donor Advised Fund with the intention of investing the money for future charitable use would not count, but donating to an organisation doing cause prioritisation research would count).

An organisation should be considered EA aligned if out of 3 Metaculus moderators, 3/3 would consider the donor or the majority of the leadership of the organisation to be members of the EA community, or that the donations were motivated by Effective Altruist considerations. If moderators are divided or individually uncertain, I will make a best-effort attempt to poll 5 well-known members of the EA community who can credibly attest to being part of EA as of June 4 2021.

By way of a guide, this is intended to exclude, for example, the Gates Foundation, but include Open Philanthropy, Longview Philanthropy and The Life You Can Save as of 2021.

[EDIT] Sylvain 2021-09-28: clarified that Good Ventures doesn't count.

### Prediction

Note: this question resolved before its original close time. All of your predictions came after the resolution, so you did not gain (or lose) any points for it.

Note: this question resolved before its original close time. You earned points up until the question resolution, but not afterwards.

Current points depend on your prediction, the community's prediction, and the result. Your total earned points are averaged over the lifetime of the question, so predict early to get as many points as possible! See the FAQ.


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# L90 Warrior Mitigation Followup

Last week we discussed mitigation stat weights for warriors using a Monte-Carlo simulation.  While the results were interesting, the simulation was very limited, and left a lot to be desired.  Since then, I’ve been having detailed discussions with a number of protection warriors, both in the comments and on tankspot, to try and improve the modeling.

For starters, I’ve fixed a number of bugs in the initial simulation.  I was mistakenly using the wrong mastery conversion factor, generating block and crit block values that were far too low.  The Sword and Board proc rate was also too low, and Berserker Rage was being treated as on the global cooldown.  The mastery error made mastery much weaker than it should have been.  And all three of these errors reduced the average rage generation rate, which artificially propped up the value of hit and expertise.

The absence of Shield Barrier was another major point of discussion.  The original model only included Shield Block, whose cooldown mechanic put an effective cap on the usefulness of hit and expertise.  As a result, it was no surprise that hit and expertise performed very poorly once it neared the theoretical maximum of Shield Block uptime.  While the results we obtained are correct for that model, it’s a fair criticism that the model doesn’t do a great job of modeling a real situation.  So one of the major improvements I’ve made to the model is to implement Shield Barrier.

I’ve also made some minor improvements to the queue code.  We now have independent choice of generator and finisher priority.  The generator code handles Shield Slam, Revenge, Devastate, Thunder Clap, Battle Shout, and Berserker Rage, and we have two choices for our generator priority.  The first is a “long” scheme designed for steady-state single-tank fights.  It prioritizes SS>Rev>BS>TC>Dev, and uses Berserker Rage off-GCD whenever it’s available and won’t over-cap rage.  The second priority is a “short” scheme, designed for tank swap fights.  It assumes you’ll only be tanking for 20-30 seconds, and that your co-tank will be keeping up Weakened Blows.  As such, it prioritizes SS>Rev>BS>Dev, and ignores TC entirely.  It uses the same Berseker Rage logic as the long version.

For finisher priority, we have several choices.  The first is the one we used previously, a “Shield Block only” scheme.  We also have a complementary “Shield Barrier only” scheme.  Neither of these need much explanation.  The third scheme is a combination queue that uses Shield Barrier to bleed excess rage.  The logic is pretty straightforward: use Shield Block if we can (i.e. we have rage and it’s not on cooldown).  After trying to cast Shield Block, if we still have more than 100 rage, use Shield Barrier to bleed some off.

These improved models should give us a much better idea of how the stats perform in realistic scenarios, where Shield Barrier usage gives an outlet for the excess rage that hit or expertise provide.  Now let’s take a look at the data.

“Long” generator queue, “Bleed” finisher queue:

N=50, tau=10000, stat=1500

|          |  dodge |  parry |    hit |    exp | mastery |
|     mean | 0.0078 | 0.0078 | 0.0057 | 0.0058 |  0.0102 |
|      std | 0.0006 | 0.0005 | 0.0005 | 0.0006 |  0.0006 |
| std_mean | 0.0001 | 0.0001 | 0.0001 | 0.0001 |  0.0001 |
|          |        |        |        |        |         |
|     mean | 0.7594 | 0.7589 | 0.5568 | 0.5668 |  1.0000 |
|      std | 0.0553 | 0.0447 | 0.0445 | 0.0538 |  0.0598 |
| std_mean | 0.0078 | 0.0063 | 0.0063 | 0.0076 |  0.0085 |

This is the realistic “steady-state” simulation.  Again, the first three rows reflect the raw difference in damage taken, and the second three rows give the results normalized to the mastery value.  There are a few big things to notice here.  First, the corrections to the sim have vaulted mastery ahead of dodge and parry as the top stat.  That’s not too surprising – we expected it to get at least 2.2x better due to the conversion rate, and that’s before the extra rage income from critical blocks is considered.  So it’s fairly safe to say that mastery, not avoidance, will be the king stat for warriors.

More importantly, note that hit and expertise have a fairly significant TDR value now.  They’re a little over half as good as mastery, but lag behind dodge and parry a bit.  The corrections to the sim have put us above the “rage cap” (at which Shield Block uptime nears its theoretical maximum) even at zero hit and expertise.  So apart from a small smoothing benefit, all of this value is from Shield Barrier usage.  This also means that there’s no intermediate point at which hit and expertise drop in value – they should maintain roughly this level of effectiveness all the way up to their hard-caps.

“Long” generator queue, “Shield Barrier Only” finisher queue:

N=50, tau=10000, stat=1500

|          |  dodge |  parry |    hit |    exp | mastery |
|     mean | 0.0101 | 0.0099 | 0.0050 | 0.0050 |  0.0060 |
|      std | 0.0006 | 0.0007 | 0.0007 | 0.0007 |  0.0006 |
| std_mean | 0.0001 | 0.0001 | 0.0001 | 0.0001 |  0.0001 |
|          |        |        |        |        |         |
|     mean | 1.6633 | 1.6429 | 0.8347 | 0.8274 |  1.0000 |
|      std | 0.0956 | 0.1139 | 0.1218 | 0.1222 |  0.1048 |
| std_mean | 0.0135 | 0.0161 | 0.0172 | 0.0173 |  0.0148 |

A Shield Barrier priority shifts things around quite a bit. First of all, mastery loses significant value thanks to the lack of Shield Block uptime.  It still gives block, crit block, and resource generation.  But without all of the guaranteed blocks from Shield Block, it just can’t keep up with avoidance, which vaults ahead in the TDR race.  hit and expertise also gain ground on mastery, though in this case they don’t eclipse it.  The boss damage we’ve chosen is high enough that it should favor Shield Block over Shield Barrier in terms of finisher use, which also means that dodge and parry will retain strength over hit and expertise.

For a boss that hits more weakly or a boss with a significant amount of unavoidable damage we’d see a reduction in the dodge/parry/mastery values and a significant jump in hit and expertise.  The sim doesn’t support unblockable/unavoidable damage yet, but it doesn’t take data to recognize the dominance of Shield Barrier in those situations.

“Long” generator queue, “Shield Block only” finisher queue:

N=50, tau=10000, stat=1500

|          |  dodge |  parry |    hit |    exp | mastery |
|     mean | 0.0071 | 0.0071 | 0.0004 | 0.0003 |  0.0079 |
|      std | 0.0005 | 0.0005 | 0.0005 | 0.0004 |  0.0005 |
| std_mean | 0.0001 | 0.0001 | 0.0001 | 0.0001 |  0.0001 |
|          |        |        |        |        |         |
|     mean | 0.8957 | 0.9008 | 0.0556 | 0.0438 |  1.0000 |
|      std | 0.0626 | 0.0675 | 0.0677 | 0.0544 |  0.0580 |
| std_mean | 0.0088 | 0.0095 | 0.0096 | 0.0077 |  0.0082 |

The “Shield Block only” queue is essentially the model we used last time.  I’m only including it here to provide contrast and confirm our earlier results with the more accurate mastery modeling.  And indeed, we see again here the weakness of hit and expertise, and the dominance of mastery/dodge/parry for TDR.  We also see the residual value of hit and expertise is next to nothing in the absence of Shield Barrier.

“Short” generator queue, “Bleed” finisher queue, front-loaded 30 sec sim:

N=1000000, tau=0.5, stat=1500

|          |  dodge |  parry |    hit |    exp | mastery |
|     mean | 0.0068 | 0.0066 | 0.0069 | 0.0070 |  0.0111 |
|      std | 0.0787 | 0.0787 | 0.0783 | 0.0780 |  0.0777 |
| std_mean | 0.0001 | 0.0001 | 0.0001 | 0.0001 |  0.0001 |
|          |        |        |        |        |         |
|     mean | 0.6116 | 0.5950 | 0.6231 | 0.6306 |  1.0000 |
|      std | 7.0751 | 7.0736 | 7.0364 | 7.0122 |  6.9847 |
| std_mean | 0.0071 | 0.0071 | 0.0070 | 0.0070 |  0.0070 |

The “short” simulation runs a million 30-second encounters, and front-loads the initial conditions so we start with 120 rage.  The idea is to simulate a tank swap situation and see whether the short time frame and unusually-high Shield Block uptime changes the stat priorities at all.

And as we see, it does have a pretty significant effect.  Because we start with both Shield Block charges and full rage, we can keep Shield Block up for 27 of those 30 seconds.  Extra rage generation doesn’t have a large effect on Shield Block uptime; in fact, in the short simulations I ran using a “Shield Block only” finisher queue, hit and expertise were nearly worthless.  But it does help generate more Shield Barrier uses with rage bleed, and as a result we see very similar hit and expertise stat weights to those we obtain from the steady-state simulation.

What change significantly is the dodge and parry valuations.  The high uptime of Shield Block means that our dodges and parries are less effective – we’re turning blocks or crit-blocks into full avoids, which is a significantly smaller damage savings than turning a full hit into an avoid.  As a result, the dodge and parry TDR values fall behind hit and expertise.  Being able to get up another Shield Barrier is just more valuable under these conditions.

This is an interesting result for another reason.  It’s usually the case that “RNG” stats like dodge and parry are better for raw TDR than “control” stats like hit and expertise.  But in this specific case, the control stats give you the best of both worlds.

Damage Intake Mapping

This is a feature I’ve had implemented for quite a while, initially intended to look at damage intake patterns in my paladin modeling.  The idea is to try and quantify the benefits of “control” stats by analyzing how it affects the “smoothness” of damage intake.  I still intend to do that, but this warrior modeling has taken more time than I expected, so I haven’t gotten to it.  However, I think there’s something to be gained in this article by including some basic results.

The code calculates the moving average of damage taken per second over 5 attacks (7.5 seconds) for the entire encounter.  This 5-attack moving average represents a potential spike window.  The assumption is that “control” stats smooth the damage intake patterns and reduce the representation of high-DTPS values in the moving average, shifting the distribution to lower-DTPS values.

What I want to address here, though, is the disparity between Shield Block and Shield Barrier.  Under certain conditions, a Shield Barrier cast will prevent more overall damage than a Shield Block cast, even against blockable damage..  Some have suggested that this will lead warriors to simply spam Shield Barrier as a finisher.

I don’t know whether that’s true or not, but my gut feeling is that it won’t be optimal even if it is better for TDR.  And the reason is simply spike damage and control.  To explain, let’s look at the damage intake pattern for the “Bleed” finisher strategy.  We’ll use a raw boss damage of 250k per attack, which is around 93k after mitigation (armor, spec, Weakened Blows).  That attack size with a 1.5-second swing timer gives about 67k Vengeance AP, or a 150k Shield Barrier absorb per cast.

The first plot to consider is a histogram of hit size:

Hit size histogram for the “Bleed” finisher strategy

The statistics on this plot require a bit of explanation.  The stats for avoids, crit blocks, blocks, and full hits are just what you’d expect – the percentage of each of those event types.  Beneath that, I’m reporting some absorption information, which gives the (absolute) percentage of each type of attack that is absorbed.  In other words, 0.54% of all attacks were rolled as hits but partially absorbed, and 0.95% were hits that were fully absorbed.  So while 22.3% of all attacks were rolled as hits, only 22.3-0.54-0.95=20.81% were actually taken as full sized hits.

However, note the distribution of hit sizes here.  There are very few full hits, and a lot of partial mitigation events.  That’s the point of damage smoothing: whittle as many of those full hits as possible down into smaller-damage chunks.  This idea is further reinforced if we plot is a histogram of the 5-attack DTPS moving average:

Histogram of 5-attack moving average DTPS for the “Bleed” finisher strategy

You can see that this is a very smooth distribution.  We have a lot of events in the middle of the plot, in the 50% average DTPS range.  Remember that all of the DTPS values in this sim are normalized, so “0.5” or 50% average DTPS means half of the DTPS you’d get if the boss hit you for 93k on all 5 attacks.  Most importantly, you’ll note that there are very few events at the upper end of the spectrum, above 90% DTPS.  That’s the advantage of a smoothing strategy – you minimize the number of high-DTPS periods you’ll experience, reducing the chance you’ll die to a spike and making you easier to heal.

Now let’s see how this changes if we use the “Shield Barrier only” strategy:

Note that most of the events have moved to the extremes.  We have very few events in the middle, which are the small, slow hits.  Instead, we have about 40% of our events doing zero damage (avoids and full absorbs) and another 40% at the high end doing full damage.  The binary nature of this will generally mean that damage intake is spikier, even though we’re actually taking less damage overall (note that DTPS drops from about 53% to 49% by switching to this strategy).

If we plot the 5-attack DTPS moving average histogram, we see a similar story:

Even though the mean of this distribution is smaller, the standard deviation is considerably larger.  We have a much wider distribution of damage spike sizes.  And while we have more near zero intake, we also have more at the extreme upper end.  Rather than 1-1.5% of our events falling above 90%, we have closer to 4% of our events in that range.  That’s a 2x-3x increase in dangerous spikes that you or your healers have to react to.

Now, you could make the argument that a smart player will use Shield Barrier on alternating attacks to make sure they don’t take two full-sized hits in a row.  That’s somewhat limited by rage income, which is where the events above 90% intake come from.  However, you could make the same argument for the “Shield Block bleed” strategy.  A good player will be trying to minimize those upper 10% of events in either strategy.  But the player using the Shield Block strategy has fewer of them to deal with to begin with.

Conclusions

It’s perhaps unsurprising that we’re seeing some parallels to paladin gearing strategies in both the stat and finisher choices here.  There’s the high-TDR Shield Barrier strategy that gives spikier damage profiles, and the lower-TDR “Bleed” strategy that smooths the damage profile at the expense of about 10% increased damage intake.  You could make a similar argument for hit/exp/mastery gearing vs. dodge/parry gearing.  The inclusion of Shield Barrier in the sims has given a new level of validation to hit and expertise as reasonable defensive stats.

All things considered, I think that Blizzard has done a pretty good job of balancing the stats for warriors.  All of the different stats are within spitting distance of one another, and there’s some valid reasoning for choosing the lower-TDR stats.

This entry was posted in Tanking, Theck's Pounding Headaches, Theorycrafting and tagged , , , , , , , , , , . Bookmark the permalink.

### 30 Responses to L90 Warrior Mitigation Followup

1. Mythor says:

The 4p bonus (Reduces the cost of your Shield Block ability by 5 Rage and increases the damage absorbed by your Shield Barrier by 5%) will fit very nicely in the ‘bleed’ strat.

The spikes in Sbar only fight still make me worried about those kind of encounters but for sblock/sbar fights I like the results.

2. Button says:

Excellent analysis! Much more thorough than your last, and the best DR vs smoothing comparison I’ve seen yet.

3. tigerlol says:

Awesome stuff and greatly appreciated!

4. Coreus says:

This was a great read! I’m also really impressed with what the devs have done with Warrior stats overall. The strength of Mastery in a tank-swap fight is especially interesting.

Worth noting the confirmation bias though. I do agree with you, but it was hard not to notice your expectation of proving Shield Block as superior.

• Theck says:

I’m not sure what you mean by that comment about confirmation bias. The tests were run independently of bias. The text might sound biased, but keep in mind that I also had the advantage of *knowing the answers* while writing it.

If you’re saying I’m biased towards smoothness over TDR, well, sure. Have you read any of the other posts on the blog? 😛 I’m pretty consistently biased towards damage smoothing, but most good tanks are.

5. tigerlol says:

While mastery seems to be the strongest stat, how do you think we should go about customizing our gear? I am returning to the game after a pretty long break (before mastery was introduced). Back then it was EH all the way. Now when choosing between mastery, parry, dodge, hit and expertise, can we expect some values where X mastery becomes less favourable than X of another stat?

I mostly want to understand if and why it would be better to ignore hit and expertise and only go for mastery when you can, or if balance is king. At this point it seems to me like you should gem mastery, reforge hit and expertise to mastery if you can, and maintain a good ratio between dodge and parry. But are there situations where reforging from hit and expertise to dodge or parry can be more favourable?

Awesome job on this blog and thanks for doing some work for us warriors.

• Theck says:

I think that whether you use hit and expertise comes down to a damage smoothing argument. Hit and expertise might not give you the highest TDR, but they give you more Shield Barriers to use in the gaps between SB casts, which smooths your damage intake. It’s much the same rationale as the “Bleed” vs “Shield Barrier Only” comparison in the blog post.

I’m in the “damage smoothing” camp myself, so I see no problem with aiming for hit and expertise caps (soft or hard, depending on preference) before stacking more mastery.

6. Booi says:

Theck,

You mentioned that your sim uses a 1.5 second swing speed. I don’t have access to the beta so I can’t verify this. But Teng suggests that 2 seconds is closer to the norm:

• Theck says:

I’m sure there will be some non-standard bosses, but we’ve been told that the “default” swing speed for MoP should be 1.5 seconds. It was in a blue post somewhere along the line where they were explaining why they chose 3 seconds for Shield of the Righteous’ buff (and I assume the same reasoning holds for Shield Block being 6 seconds). They wanted an even multiple of the default boss swing timer so players wouldn’t feel obligated to try and game the abilities to get an extra block/mitigate out of them.

7. Jackinthegreen says:

Theck, were you using Barrier at 60 rage with this, or was it anywhere between 20 and 60?

• Theck says:

60 rage only.

• Jackinthegreen says:

Some tests run on EJ’s warrior forums (http://elitistjerks.com/f81/t126901-mists_pandaria_all_specs/p20/) show less spikiness when using barrier at various rage levels.

• Theck says:

Yeah, I’d expect that. Your spike period would generally fall during the SB downtime period. So using any SBr in that period, even a low-rage one, would lower spikes.

In fact, it’s probably more effective to scatter 20-rage SBrs and cover more of those periods. I didn’t think about coding the logic for that, but a simple (use SBr if rage>20 and SB unavailable) might approximate it pretty well. My worry is that it would accidentally lower SB uptime because of rage starvation, but it may still be a smoother result.

• Siliconlife says:

Awesome analysis here!

I’m interested in SBlockSBar discussion, as i still dont know which one is better. Some people say SBar all the way. Your conclusion on “Shield Barrier” only shows it’s pretty spikey, but mitigates more damage. I would like to see a 20-rage SBars simulation to see the impact on smoothness.
Also, what do you think about targeted SBar’s; Will saving up Rage for high spike damage put SBar > SBlock in respect to smoothness?

• Gregasaurous says:

As we saw in Dargon Soil, pooling resources to survive a Tank-Killing-Laser-of-Doom is sometimes an unavoidable strategy. I’m not sure if we’ll be facing many similar situations in MoP (haven’t acquainted myself with boss mechanics just yet) but at the end of the day we all do what we have to too stay alive.

8. nickstaroba says:

Thanks for this Theck! Very interesting read and not too difficult despite my lack of education in higher mathematics. I’ve referenced this article in my thread over at askmrrobot.com: http://forums.askmrrobot.com/index.php?topic=4088.msg27290#msg27290 (which you’ve already posted in). Just trying to get some updated stat weights ready for raid on Tuesday. Would appreciate any further input you may have.


text:
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 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2014, Volume 17, Issue 2, Pages 280–287 (Mi adm471)

RESEARCH ARTICLE

Subpower Higson corona of a metric space

Jacek Kucaba, Mykhailo Zarichnyiba

a Faculty of Mathematics and Natural Sciences, University of Rzeszów, Al. Rejtana 16 A, 35-959 Rzeszów, Poland
b The Ivan Franko National University of Lviv, 1 Universytetska Str., 79000 Lviv, Ukraine

Abstract: We define a subpower Higson corona of a metric space. This corona turns out to be an intermediate corona between the Higson corona and sublinear Higson corona. It is proved that the subpower compactification of an unbounded proper metric space contains a topological copy of the Stone-Čech compactification of a countable discrete space. We also provide an example of a map between geodesic spaces that is not asymptotically Lipschitz but that generates a continuous map of the corresponding subpower Higson coronas.

Full text: PDF file (125 kB)
References: PDF file   HTML file

Bibliographic databases:
MSC: 46J10, 54D35
Revised: 28.06.2014
Language:

Citation: Jacek Kucab, Mykhailo Zarichnyi, “Subpower Higson corona of a metric space”, Algebra Discrete Math., 17:2 (2014), 280–287

Citation in format AMSBIB
\Bibitem{KucZar14} \by Jacek~Kucab, Mykhailo~Zarichnyi \paper Subpower Higson corona of a metric space \jour Algebra Discrete Math. \yr 2014 \vol 17 \issue 2 \pages 280--287 \mathnet{http://mi.mathnet.ru/adm471} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3287934}


text:
# In Bayesian estimation, when can regression coefficients and scale parameter be jointly identifiable? When not?

Exercise 14.2 in Koop, Poirier and Tobias's book (i.e. Bayesian econometric methods) talks about the case that in probit model, the regression and scale parameter are not jointly identified.

I wonder, in general, is there any rules or heuristics that can help us to judge when regression coefficients and scale parameter can be jointly identifiable and when not?

• Please add self-study since this is related with an exercise from a textbook. – Xi'an Jul 21 '17 at 8:30


text:
Boson Sampling Tests Quantum Computing

### New Book Reviews!

 Boson Sampling Tests Quantum Computing
Written by Mike James
Wednesday, 02 January 2013

Despite all the hype, we still don't know if we can build a quantum computer that is worth anything. Now we have something that might provide a shortcut to the test of a full machine - boson sampling.

We have made a lot of progress with quantum computing, but there is still the nagging doubt at the back of every researchers mind that there might be a "no-go" principle in operation. The problem with building a quantum computer is keeping the quantum effects operating properly as the device grows larger. Roughly speaking you can build a quantum computer that works with a few bits, but extending things to enough bits to do some thing that goes beyond what can be achieved using a classical computer is a tough practical problem.

This is fine as long as it is a tough practical problem and there is no basic theoretical result which limits what a quantum computer can do. If it is just practical then we need to push on and refine our technology until we reach the point where a quantum computer can beat a classical computer. If there is a "no-go" theorem then we might as well give up.

To be clear about this - no quantum computer to date has computed anything that isn't well within the ability of a classical computer.

The quantum computers of today have done things like factoring two-digit numbers in a time that can be easily beaten by a classical computer.

If you can increase the number of bits that the quantum computer works with then they could in the same time factor numbers that would take a classical computer the rest of the life of the universe to factor.

You can see that it would not be unreasonable for there to be a deep principle that says something like - "no quantum computer can ever compute something that is beyond the reach of a classical computer".


text:
BoundaryMesh with ghost cells

223
views
0
6 months ago by
Hi,

In my code I use the jump operator, therefore I set
parameters["ghost_mode"] = "shared_facet";

However, I get the following error from the BoundaryMesh:

*** Error: Unable to create BoundaryMesh with ghost cells.
*** Reason: Disable ghost mesh.
*** Where: This error was encountered inside BoundaryMesh.cpp.
*** Process: 1

If I disable the ghost_mesh, my jump function in the weak form does not work in parallel. I Believe I did not have this problem before. Is something changed lately? I am running dolfin-2017.1.0.

Thanks,
Murtazo
Community: FEniCS Project
Could you post an MWE and tell which version it works with?
written 6 months ago by Jan Blechta
What is MWE? I think it worked with 2016.x versions, if I am not wrong.
written 6 months ago by Murtazo Nazarov
MWE: Minimal working example
written 6 months ago by Stein

Ok, here it comes MWE you said:

Test.ufl:

element = FiniteElement("Lagrange", triangle, 1)

u = TrialFunction(element)
v = TestFunction(element)
n = FacetNormal(triangle)

Then, main.cpp looks like:

#include <dolfin.h>
#include "Test.h"

using namespace dolfin;

int main()
{
parameters["ghost_mode"] = "shared_vertex";

// Create mesh and function space
auto mesh = std::make_shared<UnitSquareMesh>(32, 32);
auto V = std::make_shared<Test::FunctionSpace>(mesh);

// Define variational forms
Test::BilinearForm a(V, V);

// Create a matrix and assemble
Matrix A;
assemble(A, a);

// Create a boundary mesh
BoundaryMesh boundary(*mesh, "exterior");

return 0;
}

I have tried them with dolfin-2017.1.0 and dolfin-2016.2.0, both give

mpirun -np 2 ./demo
terminate called after throwing an instance of 'std::runtime_error'
what():

*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at
***
***
*** Remember to include the error message listed below and, if possible,
*** include a *minimal* running example to reproduce the error.
***
*** -------------------------------------------------------------------------
*** Error:   Unable to create BoundaryMesh with ghost cells.
*** Reason:  Disable ghost mesh.
*** Where:   This error was encountered inside BoundaryMesh.cpp.
*** Process: 0
***
*** DOLFIN version: 2017.1.0
*** Git changeset:  7f333cc629d5f12d2a9649f314e4ab02b2296ea6
*** -------------------------------------------------------------------------

terminate called after throwing an instance of 'std::runtime_error'
what():

*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at
***
***
*** Remember to include the error message listed below and, if possible,
*** include a *minimal* running example to reproduce the error.
***
*** -------------------------------------------------------------------------
*** Error:   Unable to create BoundaryMesh with ghost cells.
*** Reason:  Disable ghost mesh.
*** Where:   This error was encountered inside BoundaryMesh.cpp.
*** Process: 1
***
*** DOLFIN version: 2017.1.0
*** Git changeset:  7f333cc629d5f12d2a9649f314e4ab02b2296ea6
*** -------------------------------------------------------------------------

If I disable the ghost mesh, then I get another error:

mpirun -np 2 ./demo
terminate called after throwing an instance of 'std::runtime_error'
terminate called after throwing an instance of 'std::runtime_error'
what():

*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at
***
***
*** Remember to include the error message listed below and, if possible,
*** include a *minimal* running example to reproduce the error.
***
*** -------------------------------------------------------------------------
*** Error:   Unable to assemble form.
*** Reason:  Incorrect mesh ghost mode "none" (expected "shared_vertex" or "shared_facet" for interior facet integrals in parallel).
*** Where:   This error was encountered inside AssemblerBase.cpp.
*** Process: 1
***
*** DOLFIN version: 2017.1.0
*** Git changeset:  7f333cc629d5f12d2a9649f314e4ab02b2296ea6
*** -------------------------------------------------------------------------

what():

*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at
***
***
*** Remember to include the error message listed below and, if possible,
*** include a *minimal* running example to reproduce the error.
***
*** -------------------------------------------------------------------------
*** Error:   Unable to assemble form.
*** Reason:  Incorrect mesh ghost mode "none" (expected "shared_vertex" or "shared_facet" for interior facet integrals in parallel).
*** Where:   This error was encountered inside AssemblerBase.cpp.
*** Process: 0
***
*** DOLFIN version: 2017.1.0
*** Git changeset:  7f333cc629d5f12d2a9649f314e4ab02b2296ea6
*** -------------------------------------------------------------------------

written 6 months ago by Murtazo Nazarov
I believe this never worked, it does not work in 2016.1.0, the check is there for a long time: https://bitbucket.org/fenics-project/dolfin/commits/62bd2ae1420a5f927d8bcf0c0df717dc1aedf7bd. Contributions are welcome.
written 6 months ago by Jan Blechta
I have double checked my code, it seems it works in dolfin-hpc, a branch developed at KTH. I thought it also did here, apparently not.
written 6 months ago by Murtazo Nazarov


text:
What is the domain of sin and cos?

What is the domain of sin and cos?

all real numbers
The domain of the sine and cosine functions is all real numbers. The range of both the sine and cosine functions is [−1,1]. The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.

What is domain of Sine?

In the sine function, the domain is all real numbers and the range is -1 to 1. Here is the graph of the cosine function: This has the same domain and range as the last graph. Again, the domain is all real numbers, and the range is -1 to 1.

What is the restricted domain for cosine?

The restriction that is placed on the domain values of the cosine function is 0 ≤ x ≤ π (see Figure 2 ). This restricted function is called Cosine.

What is domain of Sinx?

The graph of y=sin(x) is like a wave that forever oscillates between -1 and 1, in a shape that repeats itself every 2π units. Specifically, this means that the domain of sin(x) is all real numbers, and the range is [-1,1].

What is the domain of Arctan?

Domain and range: The domain of the arctangent function is all real numbers and the range is from −π/2 to π/2 radians exclusive (or from −90° to 90°). The arctangent function can be extended to the complex numbers, in which case the domain is all complex numbers.

What is the domain of Arcsin?

Additionally, the domain of arcsin x = range of sin x = [−1, 1] and range of arcsin x = domain of sin x = [− π 2 , π 2 ]. Note: arcsin(x) is the angle in [− π 2 , π 2 ] whose sine is x.

Is SEC the inverse of cos?

The secant ( sec  ) (\sec) (sec) The secant is the reciprocal of the cosine. It is the ratio of the hypotenuse to the side adjacent to a given angle in a right triangle.

What is the domain for Arcsin?

Is arctan undefined?

If x=0 then arctanyx is undefined, but you may be able to find a limit as x approaches 0. When converting from rectangular to cylindrical, the statement that θ=arctanyx is a bit sloppy. You need to start with if x=0,θ=±π2 depending on the sign of y because the arctangent never returns ±π2.

What’s the domain of arctan?

How is the sine related to the cosine?

The relationship between the cosine and sine graphs is that the cosine is the same as the sine – only it’s shifted to the left by 90 degrees, or π /2. The trigonometry equation that represents this relationship is Look at the graphs of the sine and cosine functions on the same coordinate axes, as shown in the following figure.

What is the difference between sine and cosecant?

is that cosecant is (trigonometry) in a right triangle, the reciprocal of the sine of an angle symbols: cosec, csc while sine is (trigonometry|mathematics) in a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse. cosecant. English. Noun.

Is the domain all real numbers?

Domain: The set of all possible input values (commonly the “x” variable), which produce a valid output from a particular function. It is the set of all values for which a function is mathematically defined. It is quite common for the domain to be the set of all real numbers since many mathematical functions can accept any input.

What is the domain and range of a sine function?

The sine and cosine functions have a period of 2π radians and the tangent function has a period of π radians. Domain and range: From the graphs above we see that for both the sine and cosine functions the domain is all real numbers and the range is all reals from −1 to +1 inclusive.