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The article `Bell Nonlocality in Classical Systems Coexisting with Other System Types' by Chiribella et al. defines `classical' in a quantum context in a way that ignores noncommuting quantum projectors, and is hence inconsistent with Hilbert-space quantum theory., Comment: 3 pages

A Bayesian nonparametric method of James, Lijoi \& Prunster (2009) used to predict future values of observations from normalized random measures with independent increments is modified to a class of models based on negative binomial processes for which the increments are not independent, but are independent conditional on an underlying gamma variable. Like in James et al., the new algorithm is formulated in terms of two variables, one a function of the past observations, and the other an updating by means of a new observation. We outline an application of the procedure to population genetics, for the construction of realisations of genealogical trees and coalescents from samples of alleles.

This is a Reply to the Comment by Vaidman in arXiv:2306.16756 on the paper: R. B. Griffiths, Phys. Rev. A 107, 062219 (2023), Comment: Added a reference to previous version so that it agrees with published version

Developing a quantum analog of the modern classical theory of causation, as formulated by Pearl and others using directed acyclic graphs, requires a theory of random or stochastic time development at the microscopic level, where the noncommutation of Hilbert-space projectors cannot be ignored. The Consistent Histories approach provides such a theory. How it works is shown by applying it to simple examples involving beam splitters and a Mach-Zehnder interferometer. It justifies the usual laboratory intuition that properly tested apparatus can reveal the earlier microscopic cause (e.g., radioactive decay) of a later macroscopic outcome. The general approach is further illustrated by how it resolves the Bell inequalities paradox. The difficulties encountered in an approach known as Quantum Causal Models can be traced to its lack of a satisfactory theory of quantum random processes., Comment: 13 pages, 1 figure. V2: The text has been completely rewritten, but the basic conclusions remain unchanged

A protocol for transmission of information between two parties introduced by Salih et al., Phys. Rev. Lett. 110 (2013) 170502 (hereafter SLAZ), involves sending quantum amplitude back and forth through a quantum channel in a series of steps, rather than simply sending a signal in one direction. The authors claimed that their protocol was ``counterfactual'' in the sense that while a quantum channel is needed to connect the parties, its actual usage becomes vanishingly small in the asymptotic limit as the number of steps tends to infinity. Here we show that this claim is incorrect because it uses probabilistic reasoning that is not valid at intermediate times in the presence of quantum interference. When ill-defined probabilities are replaced with a well-defined measure of channel usage here called ``Cost'', equal to the absolute square of the amplitude sent through the channel, the total Cost does not go to zero in the asymptotic limit of a large number of steps, but is bounded below by a rigorous inequality. A detailed analysis shows that this bound is satisfied in the SLAZ protocol. The analysis leading to the bound uses the fact that the Gram matrix formed by inner products of a collection of pure quantum states is additive over Hilbert subspaces and invariant under unitary time transformations. Its off-diagonal elements, which in general are not positive, play a significant role in the formal argument as well as providing a somewhat strange way of visualizing the transfer of information., Comment: Ver. 3: 19 pages. Error corrected in Ref. [9]. Minor changes (abstract, etc.) to make it resemble the published version

Recombination is a fundamental evolutionary force, but it is difficult to quantify because the effect of a recombination event on patterns of variation in a sample of genetic data can be hard to discern. Estimators for the recombination rate, which are usually based on the idea of integrating over the unobserved possible evolutionary histories of a sample, can therefore be noisy. Here we consider a related question: how would an estimator behave if the evolutionary history actually was observed? This would offer an upper bound on the performance of estimators used in practice. In this paper we derive an expression for the maximum likelihood estimator for the recombination rate based on a continuously observed, multi-locus, Wright--Fisher diffusion of haplotype frequencies, complementing existing work for an estimator of selection. We show that, contrary to selection, the estimator has unusual properties because the observed information matrix can explode in finite time whereupon the recombination parameter is learned without error. We also show that the recombination estimator is robust to the presence of selection in the sense that incorporating selection into the model leaves the estimator unchanged. We study the properties of the estimator by simulation and show that its distribution can be quite sensitive to the underlying mutation rates., Comment: 28 pages, 3 figures

Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - \alpha \{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < \alpha < \infty$, with an initial condition $u(0, x) = \delta(x - x_0)$. This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller's solution. For any $\alpha$ and $x_0 > 0$ we calculate the distribution of the random variable $A_n(s; t)$, defined as the finite number of ancestors at a time $s$ in the past of a sample of size $n$ taken from the infinite population of a Feller diffusion at a time $t$ since since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time $t$ back, conditional on non-extinction as $t \to \infty$. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times., Comment: 32 pages, 6 figures. New diagrams have been added and some rewording to sections 1 to 6. Section 7 of the original manuscript contained an error which has necessitated rewriting the original sections 7 to 9 as a new and more straightforward Section 7 which contains new results as a theorem and a corollary

We study a large class of long-range random walks which take values on the vertices of an N dimensional hypercube. These processes are connected with multivariate Bernoulli autoregression., Comment: 25 pages, 1 figure

The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Pedagogical derivations are given for known results that the limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with discrete multi-type branching processes. For the sub-critical process the previously unsolved quasi-stationary distribution is obtained to first order in the overall mutation rate, which is assumed to be small. The sampling distribution over allele types for a sample of given finite size is found to agree to first order in mutation rates with the analogous sampling distribution for a Wright-Fisher diffusion with constant population size., Comment: 37 pages, 2 figures. Propositions 1 and 2 and Lemma 1 reworked, and clarification of various points

In Phys. Rev. A 101, 022117 (2020), it was argued that Bell inequalities are based on classical, not quantum, physics, and hence their violation in experiments provides no support for the claimed existence of peculiar nonlocal and superluminal influences in the real (quantum) world. This Reply to Lambare's Comment, Phys. Rev. A 104, 066201 (2021), on that paper seeks to clarify some issues related to the correct use of Hilbert space quantum mechanics for identifying the microscopic causes of later macroscopic measurement outcomes, a matter not properly addressed by Bell, who used classical hidden variables in place of the Hilbert subspaces (equivalently, their projectors) employed by von Neumann in his \it{Mathematical Foundations of Quantum Mechanics}., Comment: This Reply to Lambare arXiv:2102.07524 is close to the published version

The two-parameter Poisson-Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson-Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman's coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson-Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman's Polya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available., Comment: Final version, to appear on Stochastic Processes and their Applications

We consider various forms of a process, which we call {\em gluing}, for combining two or more complementary quantum channel pairs $(\mathcal{B},\mathcal{C})$ to form a composite. One type of gluing combines a perfect channel with a second channel to produce a \emph{generalized erasure channel} pair $(\mathcal{B}_g,\mathcal{C}_g)$. We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the \emph{dephrasure channel} of Leditzky et al. For both (i) and (ii), $(\mathcal{B}_g,\mathcal{C}_g)$ depends on the damping parameter $0\leq p\leq 1$ and a parameter $0 \leq \lambda \leq 1$ that characterizes the gluing process. In both cases we study $Q^{(1)}(\mathcal{B}_g)$ and $Q^{(1)}(\mathcal{C}_g)$, where $Q^{(1)}$ is the channel coherent information, and determine the regions in the $(p,\lambda)$ plane where each is zero or positive, confirming previous results for (ii). A somewhat surprising result for which we lack any intuitive explanation is that $Q^{(1)}(\mathcal{C}_g)$ is zero for $\lambda \leq 1/2$ when $p=0$, but is strictly positive (though perhaps extremely small) for all values of $\lambda> 0$ when $p$ is positive by even the smallest amount. In addition we study the nonadditivity of $Q^{(1)}(\mathcal{B}_g)$ for two identical channels in parallel. It occurs in a well-defined region of the $(p,\lambda)$ plane in case (i). In case (ii) we have extended previous results for the dephrasure channel without, however, identifying the full range of $(p,\lambda)$ values where nonadditivity occurs. Again, an intuitive explanation is lacking., Comment: 13 pages (two-column), 6 figures, v2 matches published version

We introduce a general class of random walks on the $N$-hypercube, study cut-off for the mixing time, and provide several types of representation for the transition probabilities. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an "almost-perfect" mixing in at most three steps. In other words, the total variation distance between the three steps transition and the stationary distribution decreases geometrically in $N$, which is the dimension of the hypercube. In some cases, the walk mixes almost-perfectly in exactly two steps. Notice that a well-known result (Theorem 1 in Diaconis and Shahshahani (1986)) shows that there exist no random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps., Comment: 27 pages, 1 figure

For a given set of random variables $X_1,\ldots,X_d$ we seek as large a family as possible of random variables $Y_1,\ldots,Y_d$ such that the marginal laws and the laws of the sums match: $Y_i\,{\buildrel d \over =}\,X_i$ and $\sum_iY_i\,{\buildrel d \over =}\,\sum_iX_i$. Under the assumption that $X_1,\ldots,X_d$ are independent and belong to any of the Meixner classes, we give a full characterisation of the random variables $Y_1,\ldots,Y_d$ and propose a practical construction by means of a finite mean square expansion. When $X_1,\ldots,X_d$ are identically distributed but not necessarily independent, using a symmetry-balancing approach we provide a universal construction with sufficient symmetry to satisfy the more stringent requirement that, for any symmetric function $g$, $g(Y)\,{\buildrel d \over =}\,g(X)$., Comment: 2 figures

In quantum physics the term `contextual' can be used in more than one way. One usage, here called `Bell contextual' since the idea goes back to Bell, is that if $A$, $B$ and $C$ are three quantum observables, with $A$ compatible (i.e., commuting) with $B$ and also with $C$, whereas $B$ and $C$ are incompatible, a measurement of $A$ might yield a different result (indicating that quantum mechanics is contextual) depending upon whether $A$ is measured along with $B$ (the $\{A,B\}$ context) or with $C$ (the $\{A,C\}$ context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell noncontextual: the outcome of a particular $A$ measurement when $A$ is measured along with $B$ would have been exactly the same if $A$ had, instead, been measured along with $C$. A different definition, here called `globally (non)contextual' refers to whether or not there is ('noncontextual') or is not ('contextual') a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well-defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus has a tensor product structure, a fact sometimes overlooked., Comment: 14 pages, 1 figure. v2: Significant modifications of Sec. 5, but basic conclusions are unchanged

It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. Claims to the contrary based on quantum violations of Bell inequalities are shown to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wavefunction collapse and Schr\"odinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students., Comment: v3: Minor changes and corrections. This version agrees pretty well with published paper

The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of the sample up to the most recent common ancestor with additional mutations occurring on the lineage from the most recent common ancestor to the time origin if complete coalescence occurs before the origin. The sampling distribution leads to an approximation for the transition density in the diffusion with small mutation rates. This new solution has interest because the transition density in a Wright-Fisher diffusion with general mutation rates is not known., Comment: 26 pages, 3 figures, Sections 2 and 7 expanded

The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright-Fisher diffusion. The approach is different from Burden and Tang (2016,2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate lambda., Comment: 14 pages, 0 figures

The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright-Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the forward Kolmogorov equation, the stationary distribution is obtained to leading order as a set of line densities on the edges of the sample space, corresponding to states for which one island is bi-allelic and the other island is non-segregating, and a set of point masses at the corners of the sample space, corresponding to states for which both islands are simultaneously non-segregating. Analytic results for the corner probabilities and line densities are verified independently using the backward generator and for the corner probabilities using the coalescent., Comment: 30 pages, 5 figures, added new section on Alternative Approaches, more extensive numerical simulations

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q_n(x,y};N,p) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n=0,1,.., N. sum_{n=0}^N rho^nQ_n(x,y);N,p) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The \chi^2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.