The value y usually depends not only on the values x1, ..., xn, but also on values of other quantities which we do not measure. As a result, even when we have the exact same values of the quantities x1, ..., xn, we may get somewhat different values of y. It is often reasonable to assume that for each combinations of xi values, possible values of y are normally distributed, with some mean E and standard deviation s. Ideally, we should predict both E and s, but in many practical situations, we only predict a single value Y. How can we gauge the accuracy of this prediction based on the observations?

A seemingly reasonable idea is to use crisp evaluation of prediction accuracy: a method is accurate if Y belongs to a k-sigma interval [E - k * s, E + k * s], for some pre-selected value k (e.g., 2, 3, or 6). However, in this method, the value Y = E + k * s is considered accurate, but a value E + (k + d) * s (which, for small d > 0, is practically indistinguishable from Y) is not accurate. To achieve a more adequate description of accuracy, we propose to define a degree to which the given estimate is accurate.

As a case study, we consider predicting at-risk students.

The excess width is caused by quadratic (and higher order) terms in the function f. It is therefore desirable to come up with an estimation method for which the excess width decreases when the maximum of this quadratic term decreases. In the Mean Value method, while the excess width is bounded by O(Δ²), we cannot guarantee that the excess width decreases with the size of the quadratic term. In this paper, we show that, by using Grothendieck inequality, we can create a modification of the Mean Value method in which the quadratic term is estimated accurately modulo a small multiplicative constant -- i.e., in which the excess width is guaranteed to be bounded by 3.6 times the size of the quadratic term.

When we process the data, then the use of boxes instead of the original exact values leads to uncertainty. In this paper, we find the (asymptotically) optimal subdivision of data into boxes, a subdivision that provides, for a given statistical characteristic like variance, covariance, or correlation, the smallest uncertainty within the given level of privacy.

In areas where the empirical data density is small, boxes containing k points are large in size, which results in large uncertainty. To avoid this, we propose, when computing the corresponding characteristic, to only use data from boxes with a sufficiently large density. This deletion of data points increases the statistical uncertainty, but decreases the uncertainty caused by introducing the privacy-related boxes. We explain how to compute an optimal threshold for which the overall uncertainty is the smallest.

To properly perform data and model fusion, we need to know the accuracy of different data points. Sometimes, this accuracy is not given. For such situations, we describe how this accuracy can be estimated based on the available data.

We take an alternative approach based on using intervals to model uncertainty in security games. We present a fast polynomial time algorithm for security games with interval uncertainty. This provides the first viable approach for computing robust solutions to very large security games. In addition, we introduce a methodology for approximating the solutions to infinite Bayesian games with distributional uncertainty using intervals to approximate the distributions. We show empirically that using intervals is an effective approach for approximating solutions to these Bayesian games; our algorithm is both faster and results in higher quality solutions than the best previous methods.

Similar interval bounds are possible for probability density function, for moments, etc. The problem is that when we transform from one of such representations to another one, we lose information. It is therefore desirable to come up with a universal representation of imprecise probabilities in which we do not lose information when we move from one representation to another. We show that under reasonable objective functions, the optimal representation is an ellipsoid. In particular, ellipsoids lead to faster computations, to narrower bounds, etc.

In this paper, to implement the corresponding fuzzy versions of the Bayesian formulas, we use straightforward computations of the related expression -- which makes our computations reasonably time-consuming. Computations in the traditional (non-fuzzy) Bayesian approach are much faster -- because they use algorithmically efficient reformulations of the Bayesian formulas. We expect that similar reformulations of the fuzzy Bayesian formulas will also drastically decrease the computation time and thus, enhance the practical use of the proposed methods.

In such situations, a natural idea is to come up with characteristics tailored to specific application areas: e.g., select the characteristic that maximize the expected utility of the resulting risk-informed decision making.

With the new characteristics, comes the need to estimate them when the sample values are only known with interval uncertainty. Algorithms originally developed for estimating traditional characteristics can often be modified to cover new characteristics.