Irreducible uncertainty

Aleatory uncertainty The kind of uncertainty resulting from randomness or unpredictability due to stochasticity. Aleatory uncertainty is also known as variability, stochastic uncertainty. Type I or Type A uncertainty, irreducible uncertainty, conflict, and objective uncertainty. 

An idea that is believed to be original with the present theory is that the second law, expressed here in the form of the stable equilibrium postulate, implies that systems may be found in mixed states characterized by irreducible uncertainties, i.e., uncertainties that cannot be represented by a mixture of pure states. These uncertainties are associated with the particles or,... 

Representation of State. By virtue of its features, the density operator 0 is the index of the measurement statistics of quantum physics. It will be seen from Postulate 4 below that p is also the seat of causality for certain types of changes of state. In addition, it will be shown that, for an ensemble of identically prepared replicas of a system, 0 is irreducible i.e., the ensemble cannot be subdivided into subensembles each of which would yield upon measurement statistics different from the statistics of 0. Alternatively, for an ensemble of identically prepared replicas of a system, p corresponds to irreducible uncertainties of measurement results. 

It will be shown that the stable-equilibrium postulate restricts application of the theory to states defined by irreducible uncertainties. 

In engineering risk analysis a distinction is commonly made between aleatory (stochastic) and epistemic (knowledge) imcertainty see e.g. Apostolakis (1990) and Helton Burmaster (1996). Aleatory imcertainty refers to variation in populations epistemic uncertainty refers to lack of knowledge about phenomena, and usually translates into uncertainty about the parameters of the model used to describe the variation. Whereas epistemic uncertainty can be reduced, aleatory uncertainty cannot and for this reason it is sometimes called irreducible uncertainty. 

This type of uncertainty is sometimes referred to as the "irreducible uncertainty" because it is caused by the stochastic and unpredictable nature of events. Even the knowledge and opinions of the subject matter experts (SMEs) cannot be expected to reduce aleatory imcer-tainty however, their knowledge may be useful in quantifying the uncertainty. 

Random perturbations that are irreducible in principle, such as Heisenberg s Uncertainty Principle. 

Note that uncertainties in the ignition conditions of the degenerate star leads to some irreducible diversity of the explosion kinetic energy, peak luminosity, nickel production for the same initial configuration. Modeling of light curves of SN la turns out to be a powerful tool to check the SN la explosion models (see recent calculations by multi-group radiation hydrocode STELLA [21]). 

Step 1. Derive an irreducible (i.e., with natural uncertainty) model from a probable set of data. 

The interrelation between irreducible quantal uncertainties and the maximum energy that can be extracted adiabatically from a system represents a radical departure of the present work from other statistical theories, classical or quantum. 

The criteria for unambiguous preparations given above provide operational means for distinguishing between dispersions of measurement results that are inherent in the nature of a system and those that are related to voluntary or involuntary incompleteness of experimentation. The former represent characteristics of a system that are beyond the control of an observer. They cannot be reduced by any means, including quantum mechanical measurement, short of processes that result in entropy transfer from the system to the environment. For pure states, these irreducible dispersions are, of course, the essence of Heisenberg s uncertainty principle. For mixed states, they limit the amount of energy that can be extracted adiabatically from the system. 

The possibility of a relation between the second law (in the form of the impossibility of a Maxwellian demon) and irreducible dispersions associated with pure states (represented by Heisenberg s uncertainty principle) was suggested by Slater (10). His suggestion was not adopted, however, because Demers (11) proved that dispersions associated with pure states are insufficient to account for the implications of the second law, especially with regard to heavy atoms at low pressures. In the present work, we can relate the second law to quantal dispersions of mixed states because we have accepted the existence of dispersions of mixed states that are irreducible. 

As the number of analyte molecules used in an analysis approaches the single molecule level, the relative uncertainty in the number of molecules present becomes huge. Molecular shot noise is the term that we have given to this fundamental uncertainty in a chemical measurement. It is an irreducible and fundamental limitation of chemical analysis at the single molecule limit only by analyzing a large sample can we obtain precise information that is characteristic of the composition of the sample. 

We have shown that quantum dynamical systems store information in their dynamics. The information is accessed via measurement. Closer inspection would suggest even that information is created through measurement. The key conclusion is that, since both processes are represented by a 3-state QFG constructed from the same internal quantum dynamics, it is the means of observation alone that affects the amount of memory. This was illustrated with the particular examples of the spin-1 particle in a magnetic field. Depending on the choice of observable the spin-1 particle generates different process languages. We showed that these could be analyzed in terms of the block entropy—a measure of uncertainty, the entropy rate—a measure of irreducible randomness, and the excess entropy—a measure of structure. Knowing the (deterministic) QFG representation, these quantities can be calculated in closed form. 

Some researchers classify the different types of uncertainty into aleatory and epistemic (Der Kiureghian and Ditlevsen 2009). Aleatory uncertainty refers to those types of uncertainty that are inherent in nature and, therefore, irreducible by definition. For example, the variability in material properties, variability in earthquake loading, etc. are all irreducible in nature. Epistemic uncertainty refers to those types of uncertainty that may be reduced when more information is available. For example, when an improved model can be used to predict the response of a structure, then the uncertainty regarding the model prediction would decrease, thereby decreasing the overall uncertainty. 

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