Probabilistic answer

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

This leads us to the other hand, which, it should be obvious, is that we feel that Chemometrics should be considered a subfield of Statistics, for the reasons given above. Questions currently plaguing us, such as How many MLR/PCA/PLS factors should I use in my model , Can I transfer my calibration model (or more importantly and fundamentally How can I tell if I can transfer my calibration model ), may never be answered in a completely rigorous and satisfactory fashion, but certainly improvements in the current state of knowledge should be attainable, with attendant improvements in the answers to such questions. New questions may arise which only fundamental statistical/probabilistic considerations may answer one that has recently come to our attention is, What is the best way to create a qualitative (i.e., identification) model, if there may be errors in the classifications of the samples used for training the algorithm ... 

The conclusion from all this is that the variance and therefore the standard deviation attains infinite values when the reference energy is so low that it includes the value zero. However, in a probabilistic way it is still possible to perform computations in this regime and obtain at least some rough idea of how the various quantities involved will change as the reference energy approaches zero after all, real data is obtained with a finite number of readings, each of which is finite, and will give some finite answer what we can do for the rest of this current analysis is perform empirical computations to find out what the expectation for that behavior is we will do that in the next chapter. 

This probabilistic or statistical result is always observed in experiments where individual photons are counted. As the wave theory cannot do any better, we might ask whether there is some other theory, which can predict the exact behaviour of each photon, and tell us where and when it will arrive. The answer is No . We do not have such a theory, and the results of other, more sophisticated experiments, make it seem very unlikely that it could ever work. Statistics, rather than deterministic certainties, seem to be inescapable aspects of the dual behaviour of light. This conclusion was disturbing to the physicists who developed the quantum theory, and Einstein for one was never able to accept it. We shall return to these difficulties in Section 2.4. 

What are the key sources of uncertainty in the exposure assessment This question can also be posed as Which exposure factors contribute the most to the overall uncertainty in the inventory This insight can be used, in turn, to target resources to reduce the largest and most important uncertainties. There are various ways to answer this question, including various forms of sensitivity analysis. For example, in the context of a probabilistic uncertainty simulation for an overall exposure assessment, various statistical methods can be used to determine which input distributions are responsible for contributing the most to the variance of the output. 

An uncertainty analysis gives the assessor the opportunity to re-evaluate the scenario, model approaches and parameters of the analysis and to consider their influence in the overall analysis. The practical impact of uncertainty analysis is illustrated within the annexed case-studies, which also clarify how uncertainty analyses follow a systematic methodology, based on a tiered approach, and consider all possible sources of uncertainty. The first step in uncertainty analysis consists of a screening, followed by a qualitative analysis and two levels of quantitative analysis, using deterministic and probabilistic data. The assessor should be aware that an uncertainty analysis cannot answer all the questions, which, moreover, may lead to new questions. 

Let us now consider how we might go about simulating the stochastic time evolution of a dynamic system. If we are given that the system is in the state n (t) at time t, then essentially all we need in order to move the system forward in time are the answers to two questions when will the next random event occur, and what kind of event will it be Because of the randomness of the events, we may expect that these two questions will be answered in only some probabilistic sense. 

The answer lies in the probabilistic nature of quantum mechanics. Recall that the position of electrons in an atom cannot be pinned down. Knowing this, the German-born physicist Fritz London developed the concept of what he called dispersion forces. These forces, now known as London forces, arise from fluctuations in the electron density of a molecule or atom. They are temporary—arising, reversing, and vanishing in an instant. 

This is exactly what Richard Feynman thought (and all those who followed) when he asked "Can a quantum system be probabilistically simulated by a classical universal computer . .. the answer is certainly, No " Put differently by Carlton Caves as "Hilbert space is a big place." And echos of these ideas exist anonymously in contemporary culture as "Size matters."... 

Classical mechanics is a deterministic theory, in which the time evolution is uniquely determined for any given initial condition by the Newton equations (1.98). In quantum mechanics, the physical information associated with a given wave-function has an inherent probabilistic character, however the wavefiinction itself is rmiquely determined, again from any given initial wavefunction, by the Schrodinger equation (1.109). Nevertheless, many processes in nature appear to involve a random component in addition to their systematic evolution. What is the origin of this random character There are two answers to this question, both related to the way we observe physical systems ... 

This chapter describes some of the formal foundations for data integration with uncertainty. We define probabilistic schema mappings and probabilistic mediated schemas and show how to answer queries in their presence. With these foundations, we show that it is possible to completely automatically bootstrap a pay-as-you-go integration system. 

Fig. 4.2 The running example (a) a probabilistic schema mapping between S and 7 (b) a source instance Ds, (c) the answers of Q over Ds with respect to the probabilistic mapping...

We define query answering under both interpretations. The first interpretation is referred to as the by-table semantics, and the second one is referred to as the by-tuple semantics of probabilistic mappings. Note that one cannot argue for one interpretation over the other the needs of the application should dictate the appropriate semantics. Furthermore, the complexity results for query answering, which will show advantages to by-table semantics, should not be taken as an argument in the favor of by-table semantics. 

In the probabilistic context, we assign a probability to every answer. Intuitively, we consider the certain answers with respect to each possible mapping in isolation. The probability of an answer t is the sum of the probabilities of the mappings for which t is deemed to be a certain answer. We define by-table answers as follows ... 

Finally, we discuss queries with aggregate operators COUNT, SUM, AVG, MAX, and MIN based on results from Gal et al. (2009). We consider three common extensions to semantics with aggregates and probabilistic information the range semantics returns the range of the aggregate (i.e., the minimum and the maximum value) the expected-value semantics returns the expected value of the aggregate, and the distribution semantics returns all possible values with their probabilities. Note that the answer under the former two semantics can be derived from that under the last semantics in other words, the distribution semantics is the richest one. We next formally define the three semantics. 

Even if we introduce probabilistic schema mappings, none of the listed mediated schemas will return ideal answers. For example, using M prohibits returning correct answers for queries that contain both hPhone and oPhone because they are... 

Expressive power A natural question to ask at this point is whether probabilistic mediated schemas provide any added expressive power compared to deterministic ones. Theorem 8 shows that if we consider one-to-many schema mappings, where one source attribute can be mapped to multiple mediated attributes, then any combination of a p-med-schema and p-mappings can be equivalently represented using a deterministic mediated schema with p-mappings, but may not be represented using a p-med-schema with deterministic schema mappings. Note that we can easily extend the definition of query answers to one-to-many mappings, as one mediated attribute can correspond to no more than one source attribute. 

To complete the fully automatic setup of the data integration system, we consider the problem of consolidating a probabilistic mediated schema into a single mediated schema and creating p-mappings to the consolidated schema. We require that the answers to queries over the consolidated schema be equivalent to the ones over the probabilistic mediated schema. 

The main reason to consolidate the probabilistic mediated schema into a single one is that the user expects to see a single schema. In addition, consolidating to a single schema has the advantage of more efficient query answering queries now need to be rewritten and answered based on only one mediated schema. We note that in some contexts, it may be more appropriate to show the application builder a set of mediated schemas and let her select one of them (possibly improving on it later on). 

We ask What is the state function t+) the instant after the measurement To answer this question, suppose we were to make a second measurement of position at time t+. Since differs from the time t of the first measurement by an infinitesimal amount, we must still find that the particle is confined to the region (7.103). If the particle moved a finite distance in an infinitesimal amount of time, it would have infinite velocity, which is unacceptable. Since t+)p is the probability density for finding various values of x, we conclude that t+) must be zero outside the region (7.103) and must look something like Fig. 7.6b. Thus the position measurement at time t has reduced from a function that is spread out over all space to one that is localized in the region (7.103).The change from t-) to " (x, t+) is a probabilistic change. 

In a chemical sense, what relates molecular subunits with their assembly product To answer this question precisely is to define self-assembly. From a qualitative standpoint, we can state that the two sets of molecules are related by a complex series of intertwining equilibria some of them dead-ends, some of them leading as directly to the assembly product as possible. As a step toward analyzing this type of complex system, we recently described a general, semiquantitative analysis of self-assemblies. This probabilistic approach" focuses on the options available to each molecular subunit as the assembly occurs and engenders a common frame of reference by which the efficiency of all assemblies leading to discrete species can be compared. 

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