Reasoning About Measurements

This chapter explains the approach of ‘operationalism’, which in a physical theory admits only concepts associated with concrete experimental procedures, and lays out its consequences for propositions about measurements, their logical structure, and states. It illustrates these with toy examples where the ability to perform measurements is limited by design. For systems composed of several constituents this chapter introduces the notions of composite and reduced states, statistical independence, and correlations. It examines what it means for multiple systems to be prepared identically, and how this is represented mathematically. The operational requirement that there must be procedures to measure and prepare a state is examined, and the ensuing constraints derived. It is argued that these constraint leave only one alternative to classical probability theory that is consistent, universal, and fully operational, namely, quantum theory.

Running it up the flagpole to see if anyone salutes: A response to Woodward on causal and explanatory asymmetries

Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and explanatory asymmetries” argues that various statistical independence relations not only help us to uncover the directions of causal and explanatory relations in our world, but also are the worldly basis of causal and explanatory directions. We raise questions about Woodward’s envisioned epistemology, but our primary focus is on his metaphysics. We argue that any alleged connection between statistical (in)dependence and causal/explanatory direction is contingent, at best. The directions of causal/explanatory relations in our world seem not to depend on the statistical (in)dependence relations in our world (conceived of either as frequency patterns or as relations among chances). Thus, we doubt that statistical (in)dependence relations are the worldly basis of causal and explanatory directions.

Statistical independence in mathematics–the key to a Gaussian law

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

Random Permutations, Non-Decreasing Subsequences and Statistical Independence

In this paper, we show how the longest non-decreasing subsequence, identified in the graph of the paired marginal ranks of the observations, allows the construction of a statistic for the development of an independence test in bivariate vectors. The test works in the case of discrete and continuous data. Since the present procedure does not require the continuity of the variables, it expands the proposal introduced in Independence tests for continuous random variables based on the longest increasing subsequence (2014). We show the efficiency of the procedure in detecting dependence in real cases and through simulations.

Recovering Secrets From Prefix-Dependent Leakage

We discuss how to recover a secret bitstring given partial information obtained during a computation over that string, assuming the computation is a deterministic algorithm processing the secret bits sequentially. That abstract situation models certain types of side-channel attacks against discrete logarithm and RSA-based cryptosystems, where the adversary obtains information not on the secret exponent directly, but instead on the group or ring element that varies at each step of the exponentiation algorithm.Our main result shows that for a leakage of a single bit per iteration, under suitable statistical independence assumptions, one can recover the whole secret bitstring in polynomial time. We also discuss how to cope with imperfect leakage, extend the model to k-bit leaks, and show how our algorithm yields attacks on popular cryptosystems such as (EC)DSA.