Logical Inference (English)

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Logical Inference (English)

Logical Inference (English)

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INFER (Logical Inference(s) in English) is a software program capable of making logical inferences based on simple sentences. For example if we type in ” John shot a partridge”, the program will print out “John is a hunter”. If we type in “John married Ann”, the program will print out “John is husband of Ann”, “Ann is wife of John”. Works only on selected simple sentences and their equivalents in meaning.

The act or process of deriving logical conclusions from premises known or assumed to be true. b. The act of reasoning from factual knowledge or evidence.

Optimizing an experimental design is a complex task when a model is required for indirect reconstruction of physical parameters from the sensor readings. In this work, a formulation is proposed to unify the probabilistic reconstruction of mechanical parameters and an optimization problem. An information-theoretic framework combined with a new metric of information density is formulated providing several comparative advantages: (i) a straightforward way to extend the formulation to incorporate additional concurrent models, as well as new unknowns such as experimental design parameters in a probabilistic way; (ii) the model causality required by Bayes’ theorem is overridden, allowing generalization of contingent models; and (iii) a simpler formulation that avoids the characteristic complex denominator of Bayes’ theorem when reconstructing model parameters. The first step allows the solving of multiple-model reconstructions. Further extensions could be easily extracted, such as robust model reconstruction, or adding alternative dimensions to the problem to accommodate future needs.

Inverse problems are used in various fields, including medical imaging, nondestructive testing, mathematical finance, astronomy, geophysics, or sub-surface prospecting, whenever interrogating phenomena or properties of a system that cannot be readily quantified. The inverse problem can be defined in opposition to the forward problem. Given a physical system, the forward problem consists of using an idealized model of that system to predict the outcome of possible experiments. In contrast, the inverse problem is posed to interrogate or reconstruct an unknown part of the system given an observed set of output data.
This reconstruction problem was historically first solved in a deterministic way, providing a unique answer to the unknown parameters [1,2,3]. However, if the degree of certainty and reliability of the parameters is relevant, a probabilistic approach is required. This was introduced using the framework of Bayesian statistics by Cox and Jaynes [4] based on Cox’s postulates [5], and still being developed [6,7,8,9,10,11,12,13]. Its central idea is that the unknown is defined as a probability density function over the model parameters to be reconstructed, and this probability is updated with the experimental information and linked with a model through Bayes’ theorem. An alternative theoretical framework was posed by Tarantola [14] based on the idea of conjunction of states of information (theoretical, experimental, and prior information, generally on model parameters). The axioms of probability theory apply to different situations: the Bayesian perspective is the traditional statistical analysis of random phenomena, whereas the information states criterion is the description of (more or less) subjective states of information on a system. However, the collection of applications successfully solved by Ensemble Kalman Filter-type algorithms (EnKF) are not directly solvable by the proposed formulation, at least without profound adaptations.
Furthermore, its delicate formulation poses difficulties when modifying and extending it to solve real-world needs. To overcome this, we propose an information-theoretic reconstruction framework, which is built on a new metric of information density that drops Cox’s normalization in favor of simplifications. This metric is used with the concept of combining information density functions from two independent sources: (i) experimental measurements and (ii) mathematical models, over the same data (observations and model parameters) with the aim of finding which ones are all plausible at the same time. This new framework ultimately allows the straightforward solving of problems combining multiple concurrent models, or conveniently solve as experimental design, sensor design and placement problems in specific cases as in the ultrasonics testing in a probabilistic way, which only recently has been computed from the Bayesian perspective [15,16,17,18]. Beyond this, new dimensions can be defined into the problem to accommodate future needs. Moreover, models are not required to be causal, paving the way to contingent models such as stochastic associations, for instance, whose scope extends to applications such as image reconstruction, face recognition or complex physics-based model parameters reconstruction.
The two mentioned starting ingredients at the top are an experiment performed to capture some measurements, in box 1, and, in box 2, an idealization of the experimental system made throughout a model, which allows simulation of the measurements, but depends on the model parameters, which are the unknowns of the problem. In box 3, to treat the observations from box 1 in an uncertain way, they are described by means of the concept of information density over the theoretically possible space of observations, formally defined in Section 2.1 and Section 2.2. In box 4, the pairs of values of sought model parameters and simulated observations are analogously defined by means of their joint information density. In box 5, both sources of information, experiment and model, are combined as described in Section 2.2. In box 6, the probabilistic reconstruction answer is yielded as described in Section 2.4.
This scheme solves the basic form of the reconstruction inverse problem, assuming a single model and a predetermined way of measuring. However, the formulation proposed below has the strength of being easily extended to solve practical problems explained in Section 2.3, where the former assumptions need not be made.
In this work we propose a new technique to optimize the experimental design of a testing system or sensor and illustrate it for the particular case of characterizing a viscoelastic material, step by step. First, the information-theoretic inverse problem framework is formulated, then, the practical method is detailed describing the process of parametrization, the operation with discrete observation data or signals, and two key extensions: to hypothesis testing and to experimental design optimization. The proposal is illustrated with a practical example to reconstruct a mechanical model from a tensile testing.
Assuming that the two sources of information are the experimental observations and an idealized model that simulates observations for given model parameters, two basic variables stem from this premise: observations and model parameters.
The observations O are, in the most general case, vectors compounding a set of signals oi(t), but may also be a single signal, analog or digitally sampled, sets of values, or even a single measurement value. Although a unique space of observations O can be defined to contain all possible observations, depending on their origin they can be either observed Oo={ooi(t)} or modeled Om={omi(t)}. Examples of observations may be ultrasonic or seismic recorded signals, optical, X-ray or thermographic images, or any measurement based on any physical magnitude used to interrogate the system under study.
The model parameters M are analogously a set of diverse physical parameters, which define a manifold H. They are the input of the mathematical model that simulates the experimental behavior and its measurable by an output. They may stand for damage parameters, pathology or sought mechanical properties, for instance, that feed models that simulate the observations described above. In the numerical example in Section 4, combining three sources of information is tested: model-based forecast, observation, and experimental design parameters for its optimization.

2.2. Definition of Information Density and Its Operations

To treat this data O and M in an uncertain way, we do not define univocal values, but information densities over them. The information density f(x) over either of them (x for generality) is defined from the conception of Cox [5] as a degree of belief or certainty that the values of x are plausible. Therefore, the probabilities that are established as a consequence of this logical framework are objective and the logical relations in that axiomatization [19,20]. They can be understood as states of knowledge, in contrast to the physical propensity of a phenomenon. A more detailed discussion is provided in [21]. The present definition of information density is compatible with either the evidential, logical and even subjective theoretical frameworks described in [21].
In particular, we formally define the information density f(x) of an event or value x as a nonnegative real f(x)R+ that is zero (f(x)=0) when its value is impossible, and the larger the more plausible. Two logical inference operations introduce a structure to the space of all probability distributions. Starting from the and and or operator definition for Boolean logic (which can adopt the values of true or false, without intermediate degrees of certainty), over two probability distributions Pa and Pb that may represent two different sources of information a and b about the same events.
Note that the normalization requirement of either Kolmogorov axioms or Cox’s postulates (Kolmogorov axioms state that the probability P of any events AB satisfy [22],
  • Non-negativity: P(A)0.
  • Finite additivity: P(AB)=P(A)+P(B)A,B|AB=.
  • Normalization: P(Ω)=1.)
is not imposed here, which will strongly simplify the formulation, as shown later. In particular, dropping the normalization axiom in the definition of the information density f simplifies the final formulation in comparison with the Bayesian inverse problem as well as theory of Tarantola.
A main cornerstone of this formulation is that the relationship between the model parameters and the observations provided by a model need not to be an implication due to a cause-effect, which requires to define the conditional probability of Bayesian statistics. Instead, just the conjunction of information densities needs to be defined, in which the causality between model and observations may be inverted or even not exist, as further discussed in [21]. These two characteristics define the relationship between model and observation. One uses probability as logic, and alternative one interprets it as information content. They will be shown below to allow the solving of reconstruction problems with multiple concurrent models, also paving the way to contingent models such as stochastic associations, as well as experimental design and placement problems, in a simple and straightforward way, both conceptually and computationally.
Therefore, we define the information contents that come from the observations as fo(O), and that provided by the model as fm(O,M), in the sense that the model couples values of model parameters M with observations O, yielding degrees of certainty f when the fed values in the model is fulfilled or not with a range of degrees of plausibility.
The origin of the uncertainties is, therefore, incorporated into the interpretation of probability as a measure of relative plausibility of the various possibilities obtained from the available information. This interpretation is not well known in the engineering community where there is a wide-spread belief that probability only applies to aleatory uncertainty (inherent randomness in nature) and not to epistemic uncertainty (missing information). Jaynes [4] noted that the assumption of inherent randomness is an example of what he called the Mind-Projection Fallacy: our uncertainty is ascribed to an inherent property of nature, or, more generally, our models of reality are confused with reality.
The interpretation of the final inferred model probability can be used either to identify a set of plausible values, or to find the most probable one (expected, i.e., that with maximal information density, argmax f(M)), or, following Tarantola [14], just to falsify inconsistent models (those with low f), since according to Poppers falsationism [23], that is the only thing we can assert.
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