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Simply put, this article concerns the problem of bounding the probability of an event when we know very little about the probability distributions and other functions involved. Mathematically, we write this event as [*g*(*X*) ≤ 0], where *g* denotes the response function and *X* denotes its random inputs. In “the real world”, the output *g*(*X*) being negative might be some undesirable outcome such as a bridge collapsing under an earthquake, a power plant failing to produce enough electricity, a disease advancing beyond the initial site to become a genuine epidemic, &c. The last example is a good one for this paper, because the exact functional relationship *g* between the initial conditions *X* of the outbreak and the outcome *g*(*X*) are only partly understood: we have some historical data, but we would encounter severe legal and ethical problems if we proposed starting new pandemics just to learn more about the function *g*!

Following the approach begun here, this article approaches this problem from a mathematical perspective: how can we bound the probability **P**[*g*(*X*) ≤ 0] of the event [*g*(*X*) ≤ 0], despite having significant uncertainty about the probability distribution **P** of *X* and the function *g*, and no reasonable way to extend our limited knowledge of them? How can we find lower and upper bounds on this partially-known probability that are as tight as possible given the available information? For example, “**P**[*g*(*X*) ≤ 0] is between 0% and 100%” is a true statement, but far too loose to be useful, whereas “**P**[*g*(*X*) ≤ 0] is exactly 52.3937225%” is very precise and informative, but is likely both wrong and more precise than the available evidence really justifies.

Abstract.We consider the problem of providing optimal uncertainty quantification (UQ) — and hence rigorous certification — for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.

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This is an interesting point, although I think that it is more relevant to, say, the dynamics of a throw in Aikidō than drawing the sword in Iaidō. While I don’t doubt that using sayabiki to exploit the conservation of angular momentum does allow a more powerful draw, I think that the primary reason for sayabiki is simple mechanics: if one does not retract the scabbard, then the sword will not come cleanly out of the scabbard mouth — if the sword comes out at all, then its cutting tip will damage the scabbard and indeed the Iaidōka’s hand!

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- we ought to be able to write
*x*as a function of*y*, i.e.*x*=*f*^{−1}(*y*) for*y*near*f*(*x*_{0}), - and, moreover, the slope of the inverse function
*f*^{−1}at*f*(*x*_{0}) should be^{1}⁄_{s}.

The “visual proof” of this statement amounts to sketching the graph of *f*, observing that the graph of *f*^{−1} (if the inverse function exists at all) is the graph of *f* with the *x* and *y* axes interchanged, and hence that if the slope of *f* is approximately ^{Δy}⁄_{Δx} then the slope of *f*^{−1} is approximately ^{Δx}⁄_{Δy}, i.e. the reciprocal of that of *f*.

Recall that the derivative of a function *f*: ℝ^{n} → ℝ^{m} is the rectangular *m* × *n* matrix of partial derivatives

whenever all these partial derivatives exist. With this notation, a more careful statement of the Inverse Function Theorem is that if *f*: ℝ^{n} → ℝ^{n} is continuously differentiable in a neighbourhood of *x*_{0} and the square *n* × *n* matrix of partial derivatives D*f*(*x*_{0}) is invertible, then there exist neighbourhoods *U* of *x*_{0} and *V* of *f*(*x*_{0}) and a continuously differentiable function *g*: *V* → ℝ^{n} (called a **local inverse** for *f*) such that

- for every
*u*∈*U*,*g*(*f*(*u*)) =*u*, and - for every
*v*∈*V*,*f*(*g*(*v*)) =*v*.

An interesting question to ask is whether one really needs continuous differentiability of *f*. For example, Rademacher’s theorem says that whenever *f* satisfies a **Lipschitz condition** of the form

for some constant *C* ≥ 0 it follows that *f* is differentiable almost everywhere in ℝ^{n} with derivative having norm at most *C*. Is this sufficient? It turns out, courtesy of a theorem of F. H. Clarke, that the Inverse Function Theorem does hold true for Lipschitz functions provided that one adopts the right generalized interpretation of the derivative of *f*.

The (set-valued) **generalized derivative** D*f*(*x*_{0}) of *f*: ℝ^{n} → ℝ^{m} at *x*_{0} is defined to be the convex hull of the set of all matrices *M* ∈ ℝ^{m×n} that arise as a limit

for some sequence (*x*_{k}) in ℝ^{n} of differentiability points of *f* that converges to *x*_{0}. One can show that, when *f* satisfies a Lipschitz condition in a neighbourhood of *x*_{0}, D*f*(*x*_{0}) is a non-empty, compact, convex subset of ℝ^{m×n}. The generalized derivative D*f*(*x*_{0}) is said to be of **maximal rank** if every *M* ∈ D*f*(*x*_{0}) has maximal rank (i.e. has rank(*M*) = min(*m*, *n*)).

- for every
*u*∈*U*,*g*(*f*(*u*)) =*u*, and - for every
*v*∈*V*,*f*(*g*(*v*)) =*v*.

It’s very important to note the maximal rank condition in Clarke’s Inverse Function Theorem: we need *every* matrix *M* in the generalized derivative to be non-singular. So, for example, the absolute value function on the real line ℝ does *not* satisfy the hypotheses of Clarke’s theorem at *x* = 0, even though it is Lipschitz with Lipschitz constant 1, since its generalized derivative at 0 is

which contains the non-invertible derivative matrix [0]. It is hardly surprising that the Inverse Function Theorem cannot be applied here since the absolute value function is non-injective in any neighbourhood of 0: both +*δ* and −*δ* map to +*δ*. On the other hand, the function *f* defined by

has generalized derivative at 0 given by

which is of maximal rank. The local (in fact, global) Lipschitz inverse of this function *f* is, unsurprisingly,

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This paper was a real team effort, with everyone bringing different strengths to the table. Given the length of the review process, I think that our corresponding author Houman Owhadi deserves a medal for his patience (as does Ilse Ipsen, the article’s editor at *SIAM Review*), but, really, congratulations and thanks to all.

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call

Optimal Uncertainty Quantification(OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we developOptimal Concentration Inequalities(OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ.

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What DO we tell the children?

Do we tell them evil is a foreign face?

No. The evil is the thought behind the face, and it can look just like yours.

Do we tell them evil is tangible, with defined borders and names and geometries and destinies?

No. They will have nightmares enough.Perhaps we tell them that we are sorry.

Sorry that we were not able to deliver unto them the world we wished them to have.

That our eagerness to shout is not the equal of our willingness to listen.

That the burdens of distant people are the responsibility of all men and women of conscience, or their burdens will one day become our tragedy.Or perhaps we simply tell them that we love them, and that we will protect them. That we would give our lives for theirs and do it gladly, so great is the burden of our love.

In a universe of Gameboys and VCRs, it is, perhaps, an insubstantial gift. But it is the only one that will wash away the tears and knit the wounds and make the world a sane place to live in.

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- reading the series, and
- not attempting to do so while eating, drinking, or sitting on a chair.

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O wad some Pow’r the giftie gie us

To see oursels as ithers see us!

It wad frae mony a blunder free us,

An’ foolish notion:

What airs in dress an’ gait wad lea’e us,

An’ ev’n devotion!

The trouble with the senior waiter’s fine philosophy is that, from this customer’s perspective, the most notable thing about him is that his neat appearance (black shirt, black trousers, black apron) is totally undercut by the fact that his trousers are hanging down below his hips, exposing his light grey underpants to the world.

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