Branch-counting in the Everett interpretation of quantum mechanics

A defence is offered of a version of the branch-counting rule in the Everett interpretation (otherwise known as many worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule and deliver a notion of objective probability similar to naive frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc : it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose–Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.

Reconciling Meticulous Divine Providence with Objective Chance

Randomness can be defined in terms of objective probability: an event is random just in case its objective probability (in the circumstances) is other than zero or one. There is a tension between objective probability and divine providence: if God has arranged for E to occur, then its objective probability would seem to be one. I will first show that this tension creates problems for six models of how to combine worldly chance with divine providence: determinism, Molinism, Thomism, divine luck, the multiverse, and van Inwagen’s theory of chance. I will then develop two new solutions to this problem.

Betting on the Open Future

A.N. Prior considered an objection to open future views, viz. that they are inconsistent with our ordinary practices of betting. Prior worried that, on open future views, if we bet on rain, and then it does rain, I could refuse to grant the payout on grounds that the proposition you bet was true was not true at the time of the bet. The author argues that this objection fails, by developing a picture of betting on which we are not betting on anything like current truth. He then considers the objection that his view is inconsistent with the idea that there are non-zero probabilities of future events; he argues that though our credence in a given future contingent proposition may be zero, the objective probability of the relevant event may nevertheless be high. The author develops a comparison between this view and parallel views about the probability of conditionals and probabilities in fictions.

Standard of proof in common law: Mathematical explication and probative value of statistical data

As a result of recent amendments to the procedural legislation of Ukraine, one may observe a tendency in judicial practice to differentiate the standards of proof depending on the type of litigation. Thus, in commercial litigation the so-called standard of “probability of evidence” applies, while in criminal proceedings – “beyond a reasonable doubt” standard applies. The purpose of this study was to find the rational justification for the differentiation of the standards of proof applied in civil (commercial) and criminal cases and to explain how the same fact is considered proven for the purposes of civil lawsuit and not proven for the purposes of criminal charge. The study is based on the methodology of Bayesian decision theory. The paper demonstrated how the principles of Bayesian decision theory can be applied to judicial fact-finding. According to Bayesian theory, the standard of proof applied depends on the ratio of the false positive error disutility to false negative error disutility. Since both types of error have the same disutility in a civil litigation, the threshold value of conviction is 50+ percent. In a criminal case, on the other hand, the disutility of false positive error considerably exceeds the disutility of the false negative one, and therefore the threshold value of conviction shall be much higher, amounting to 90 percent. Bayesian decision theory is premised on probabilistic assessments. And since the concept of probability has many meanings, the results of the application of Bayesian theory to judicial fact-finding can be interpreted in a variety of ways. When dealing with statistical evidence, it is crucial to distinguish between subjective and objective probability. Statistics indicate objective probability, while the standard of proof refers to subjective probability. Yet, in some cases, especially when statistical data is the only available evidence, the subjective probability may be roughly equivalent to the objective probability. In such cases, statistics cannot be ignored

Travelers’ Bi-Attribute Decision Making on the Risky Mode Choice with Flow-Dependent Salience Theory

The purpose of this paper is to provide new insights into travelers’ bi-attribute (travel time and travel cost) risky mode choice behavior with one risky option (i.e., the highway) and one non-risky option (i.e., the transit) from the long-term planning perspective. In the classical Wardropian User Equilibrium principle, travelers make their choice decisions only based on the mean travel times, which might be an unrealistic behavioral assumption. In this paper, an alternative approach is proposed to partially remedy this unrealistic behavioral assumption with flow-dependent salience theory, based on which we study travelers’ context-dependent bi-attribute mode choice behavior, focusing on the effect of travelers’ salience characteristic. Travelers’ attention is drawn to the bi-attribute salient travel utility, and then the objective probability of each state for the risky world is distorted in favor of this bi-attribute salient travel utility. A long-term bi-attribute salient user equilibrium will be achieved when no traveler can improve their bi-attribute salient travel utility by unilaterally changing the choice decisions. Conditions for the existence and uniqueness of the bi-attribute salient user equilibrium are presented, and based on the equilibrium results, we analyze travelers’ risk attitudes in this bi-attribute risky choice problem. Finally, numerical examples are conducted to examine the sensitivity of equilibrium solutions to the input parameters, which are cost difference and salience bias.

Two Non-senses of “Probability”

This chapter engages in some ground-clearing. Two concepts have been proposed to play the role of objective probability. One is associated with the idea that probability involves mere counting of possibilities (often wrongly attributed to Laplace). The other is frequentism, the idea that probability can be defined as long-run relative frequency in some actual or hypothetical sequence of events. Associated with the idea that probability is merely a matter of counting of possibilities is a temptation to believe that there is a principle, called the Principle of Indifference, which can generate probabilities out of ignorance. In this chapter the reasons that neither of these approaches can achieve its goal are rehearsed, with reference to historical discussions in the eighteenth and nineteenth centuries. It includes some of the prehistory of discussions of what has come to be known, misleadingly, as Bertrand’s paradox.

Counterfactuals and Probability

Dorothy Edgington has been a resolute defender of an NTV account of conditionals, according to which a conditional does not express a proposition that makes a categorical claim about the world, but instead make a qualified claim, or express a conditional belief, qualified by or conditional on the proposition expressed by the antecedent. Unlike some philosophers who defend an NTV view for indicative conditionals, but not for subjunctive or counterfactual conditionals, Edgington argues for the more radical thesis that both kinds of conditionals should be given a non-propositional analysis. This chapter considers Edgington’s NTV account of subjunctive conditionals, the role of objective probability in the account, and its relation to the possible-worlds propositional analysis of subjunctive conditionals.

Disagreement and Deep Agnosticism

One defense of the “steadfast” position in cases of peer disagreement appeals to the idea that it's rational for you to remain deeply agnostic about relevant propositions concerning your peer's judgment, that is, to assign no credence value at all to such propositions. Thus, according to this view, since you need not assign any value to the proposition that your peer's judgment is likely to be correct, you need not conciliate, since you can remain deeply agnostic on the question of how the likelihood of your peer's judgment bears on the likelihood of your own. This paper argues that the case for deep agnosticism as a response to peer disagreement fails. Deep agnosticism (as a general thesis) implies that it is sometimes permissible to withhold judgment about whether there is a non-zero chance of a proposition's being true. However, in cases of disagreement where deep agnosticism is supposed to support the steadfast position, such withholding isn't rational. This is because of constraints placed on rational credence by objective probability or chance, which ensure that rational credence adequately reflects strength of evidence.

Priority Threats to Financial Security in the conditions of the Modern Economy Transformation

Introduction. The shadow economy is one of the main threats to the financial security of the state. Strengthening Ukraine’s economy is impossible without reducing the level of shadowing. Official data show that at the beginning of 2020 more than a quarter of GDP is in the shadows, but the real level of shadowing, in our opinion, is much higher. Purpose. The aim of the article is to deepen the theoretical and methodological foundations of substantiation of priority threats to financial security in the conditions of transformation processes in the modern economy, clarification of the optimal method of calculating the shadow economy in Ukraine and monitoring the impact of the shadow sector on financial security of Ukraine. Results. It was found that the system of threats to the financial security of Ukraine includes trends, phenomena and factors that actually or potentially complicate or make it impossible to preserve national values ​​in the financial sphere and the realization of national interests of Ukraine. Threats to security conditions are considered as certain factors or factors that create a danger, namely the existing and objective probability of negative impact on a complex system. Our own view on the classification of factors influencing the level of financial security is presented. It is noted that most often the factors influencing the level of financial security are interrelated, causing cyclical poverty and a permanently low level of financial security. It is analyzed that today the level of the shadow economy is 25-50%. To create the necessary conditions for reducing the level of the shadow sector of Ukraine’s economy, it is necessary to carry out a number of tax reforms, transformations in the banking system (to increase non-cash transactions, create an open database of NBU and MFIs), anti-corruption measures and more. Systematizing the methodological approaches to the assessment of the shadow sector, it was found that the main methods of assessing the level of the shadow sector are: public expenditure – retail trade; electric; unprofitable enterprises; monetary; method of minimal shadow economy. Conclusions. It is proved that the level of financial security of the state decreases with increasing level of shadow economy. This is due to the reduction of the tax base and, consequently, to the reduction of tax revenues to the budgets of all levels. Keywords: financial security; financial security of the state; financial security of households; transformational economy; shadow economy.

Stochastic representation decision theory: How probabilities and values are entangled dual characteristics in cognitive processes

Humans are notoriously bad at understanding probabilities, exhibiting a host of biases and distortions that are context dependent. This has serious consequences on how we assess risks and make decisions. Several theories have been developed to replace the normative rational expectation theory at the foundation of economics. These approaches essentially assume that (subjective) probabilities weight multiplicatively the utilities of the alternatives offered to the decision maker, although evidence suggest that probability weights and utilities are often not separable in the mind of the decision maker. In this context, we introduce a simple and efficient framework on how to describe the inherently probabilistic human decision-making process, based on a representation of the deliberation activity leading to a choice through stochastic processes, the simplest of which is a random walk. Our model leads naturally to the hypothesis that probabilities and utilities are entangled dual characteristics of the real human decision making process. It predicts the famous fourfold pattern of risk preferences. Through the analysis of choice probabilities, it is possible to identify two previously postulated features of prospect theory: the inverse S-shaped subjective probability as a function of the objective probability and risk-seeking behavior in the loss domain. It also predicts observed violations of stochastic dominance, while it does not when the dominance is “evident”. Extending the model to account for human finite deliberation time and the effect of time pressure on choice, it provides other sound predictions: inverse relation between choice probability and response time, preference reversal with time pressure, and an inverse double-S-shaped probability weighting function. Our theory, which offers many more predictions for future tests, has strong implications for psychology, economics and artificial intelligence.