On the a priori and a posteriori assessment of probabilities

We argue that in spite of their apparent dissimilarity, the methodologies employed in the a priori and a posteriori assessment of probabilities can both be justified by appeal to a single principle of inductive reasoning, viz., the principle of symmetry. The difference between these two methodologies consists in the way in which information about the single-trial probabilities in a repeatable chance process is extracted from the constraints imposed by this principle. In the case of a posteriori reasoning, these constraints inform the analysis by fixing an a posteriori determinant of the probabilities, whereas, in the case of a priori reasoning, they imply certain claims which then serve as the basis for subsequent probabilistic deductions. In a given context of inquiry, the particular form which a priori or a posteriori reason may take depends, in large part, on the strength of the underlying symmetry assumed: the stronger the symmetry, the more information can be acquired a priori and the less information about the long-run behavior of the process is needed for an a posteriori assessment of the probabilities. In the context of this framework, frequency-based reasoning emerges as a limiting case of a posteriori reasoning, and reasoning about simple games of chance, as a limiting case of a priori reasoning. Between these two extremes, both a priori and a posteriori reasoning can take a variety of intermediate forms.     

Identification of a new tetragonal phase in the Nb-Ti-Al system

During the course of investigation of the phase equilibria in the Nb-Ti-Al system, a new phase, designated theta, was observed in three Nb-Ti-40at.%Al alloys following ageing at low temperatures (less than 1000 C). Using X-ray diffraction and transmission electron microscopy techniques, this phase was determined to have a bct structure, with lattice parameters a₀ = 5.106 A and c₀ = 28.168A, a space group of I 4₁/amd (number 141) and composition near 45at.%Nb25at.%Ti-30at.%Al. An orientation relationship between the new tetragonal phase and the gamma-TiAl phase of [101]_gamma//[120]_theta and (111)_gamma//(001)_theta was observed within the microstructure.     

Deceptive updating and minimal information methods

The technique of minimizing information (infomin) has been commonly employed as a general method for both choosing and updating a subjective probability function. We argue that, in a wide class of cases, the use of infomin methods fails to cohere with our standard conception of rational degrees of belief. We introduce the notion of a deceptive updating method and argue that non-deceptiveness is a necessary condition for rational coherence. Infomin has been criticized on the grounds that there are no higher order probabilities that ??support?? it, but the appeal to higher order probabilities is a substantial assumption that some might reject. Our elementary arguments from deceptiveness do not rely on this assumption. While deceptiveness implies lack of higher order support, the converse does not, in general, hold, which indicates that deceptiveness is a more objectionable property. We offer a new proof of the claim that infomin updating of any strictly-positive prior with respect to conditional-probability constraints is deceptive. In the case of expected-value constraints, infomin updating of the uniform prior is deceptive for some random variables but not for others. We establish both a necessary condition and a sufficient condition (which extends the scope of the phenomenon beyond cases previously considered) for deceptiveness in this setting. Along the way, we clarify the relation which obtains between the strong notion of higher order support, in which the higher order probability is defined over the full space of first order probabilities, and the apparently weaker notion, in which it is defined over some smaller parameter space. We show that under certain natural assumptions, the two are equivalent. Finally, we offer an interpretation of Jaynes, according to which his own appeal to infomin methods avoids the incoherencies discussed in this paper.     

Epitaxial Ni-Al thin films on NaCl using a Ag buffer layer

Epitaxial nanoscale [001] films of Ni _x Al_100-x (x = 62.5) have been prepared by physical vapour deposition on to a thin film of Ag [001] on NaCl (001) faces with occasional hillocks. The Ag film contains numerous dislocations and stacking faults and has a rms surface roughness of 2 nm. The Ni-Al film is ordered in the B2 structure and reveals many dislocations as well as antiphase boundaries between ordered domains. The formation of subgrains in the Ni-Al film results in severe height variations up to 30 nm across the surface. A cross-sectional model for the growth of both films is presented.     

Biased information and the exchange paradox

This paper presents a new solution to the well-known exchange paradox, or what is sometimes referred to as the two-envelope paradox. Many recent commentators have analyzed the paradox in terms of the agent’s biased concern for the contents of his own arbitrarily chosen envelope, claiming that such bias violates the manifest symmetry of the situation. Such analyses, however, fail to make clear exactly how the symmetry of the situation is violated by the agent’s hypothetical conclusion that he ought to switch envelopes on the assumption that his own envelope contains some specific amount of money. In this paper, I offer an explanation of this fact based on the idea that the agent’s deliberations are not only constrained by the epistemic symmetry reflected in the agent’s uniform ignorance as to the contents of the two envelopes, but also by a deeper methodological symmetry that manifests itself in the intuition that the two envelopes constitute equally legitimate sources of potential information. I interpret this intuition as implying that the agent’s final decision should reflect a point of rational equilibrium arrived at through an iterated process of counterfactual self-reflection, whereby the agent takes account of what he would have thought had he instead chosen the other envelope. I provide a formal model of this method of counterfactual self-reflection and show, in particular, that it cannot issue in the paradoxical conclusion that the agent ought to switch regardless of how much money is in his envelope. In this way, by correcting for the bias in the agent’s reasoning, the paradox is resolved.