Equally Probable Research Articles

Random walks on a line and algebraic curves

This work is devoted to the studying the generating function of the first hitting time of the positive semi-axis under the homogeneous discrete integer random walk on a line. In the first part of the work the increments are supposed to be independent. Recurrent relations for the probabilities allow us to write the system of equations for the sought generating function. Applying the resultants technique, we succeed to reduce this system to a single equation. Then we can study it by calculating the genus of the correposponding plane algrebraic curve via analyzing its singularities. In the work we write the sought equations for some random walks and we show that if the increments take all integer values from $-2$ to $2$, or from $-1$ to $3$ with equal probabilities or they take equally probable values $-1$ and $4$, then the curve is rational, while this is not true in the first case. In the second part of the work we consider a symmetric process, the increments take the values $-1$, $0$, $1$, but then we suppose a non-zero correlation of each next increment with the previous one. For such process the equation for the generating function defines an elliptic curve depending on the square of the correlation coefficient for neighbouring increments if all increments are non-zero and it defines a hyperelliptic curve of genus $2$. The degeneration criterion of the latter is the presence of multiple roots of a sixth order polynomial under general symmetrically distributed conditional probabilities.

Evolution of memes

The paper considers the evolution of a population of individuals, where each one initially possesses a certain number of strategies the memory of which does not exceed a depth of 2. All individuals randomly enter into competition in pairs at each stage of evolution. A random pair of individuals conducts a competition between pairs of all their randomly selected strategies when they are interacting. These strategies compete in pairs according to the iterated prisoner's dilemma. In such struggle, strategies earn evolutionary advantage points according to a given payout matrix. The strategy with the most points wins. Two strategies come into this game twice to negate an impact of the first move. The first game starts by one strategy, the second game starts by another one. The winnings are determined by the outcome of both these games. After this competition the winning strategy of one individual replaces the corresponding losing strategy of another individual. Thus, there is an exchange of more "successful" strategies between individuals with the loss of lost strategies. The evolution of the population of such individuals was carried out until the stage of stationary state. There were established patterns of changes in basic properties of strategies of average individual during evolution. It is shown that in the process of evolution the aggression of an individual increases, tenting to the maximum value. The stationary set of strategies of an individual consists of strategies of maximum memory depth and complexity with a certain number of primitive strategies. The complexity and memory depth of an individual's strategies turns out to be evolutionary beneficial. In the stationary state the number of primitive strategies in an individual depends on their initial distribution to individuals. The paper considers two initial distributions, where the first corresponds to the equal probability of any strategy in the distribution by individuals, and the seconds corresponds to equally probable choice in terms of memory depth. The variety of strategies in the process of evolution decreases significantly, making up only a small part of the initial strategies present in the population.

Damped electrostatic ion acoustic solitary wave structures in quantum plasmas with Bohm potential and spin effects

Nonlinear properties of ion acoustic solitary waves are studied in the case of dense magnetized plasmas. The degenerate electrons with relative density effects from their spin states in the same direction and from equally probable up and down spinning states are taken up separately. Quantum statistical as well as quantum tunneling effects for both types of electrons are taken. The ions have large inertia and are considered classically, whereas the electrons are degenerate. The collisions of ions and electrons with neutral atoms are considered. We derive the deformed Korteweg de–Vries (DKdV) equation for small amplitude electrostatic potential disturbances by employing the reductive perturbation technique. The Runge–Kutta method is applied to solve numerically the DKdV equation. The analytical solution of DKdV is also presented with time dependence. We discuss the profiles for velocity, amplitude, and time variations in solitons for the cases when all the electrons are spinning in the same direction and for the case when there is equal probability of electrons having spin up and spin down. We have found that the wave is unstable because of the collisions between neutral gas molecules and the charged plasmas particles in the presence of degenerate electrons.

Monty Hall Problem and the Principle of Equal Probability in Measurement Theory

In this paper, we study the principle of equal probability (i.e., unless we have sufficient reason to regard one possible case as more probable than another, we treat them as equally probable) in measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics with the Copenhagen interpretation. This turn from physics to language does not only realize theremarkable extensionof quantum mechanicsbut alsoestablish the method of science. Our study will be executed in the easy example of the Monty Hall problem. Although our argument is simple, we believe that it is worth pointing out the fact that the principle of equal probability can be, for the first time, clarified in measurement theory (based on the dualism) and not the conventional statistics (based on Kolmogorov’s probability theory).

Bayesian covariance matrix estimation using a mixture of decomposable graphical models

We present a Bayesian approach to estimating a covariance matrix by using a prior that is a mixture over all decomposable graphs, with the probability of each graph size specified by the user and graphs of equal size assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs in more efficiently estimating the covariance matrix. The prior requires knowing the number of decomposable graphs for each graph size and we give a simulation method for estimating these counts. We also present a Markov chain Monte Carlo method for estimating the posterior distribution of the covariance matrix that is much more efficient than current methods. Both the prior and the simulation method to evaluate the prior apply generally to any decomposable graphical model.

IMPROVING THE PERFORMANCE OF PROTEIN THREADING USING INSERTION/DELETION FREQUENCY ARRAYS

As a protein evolves, not every part of the amino acid sequence has an equal probability of being deleted or for allowing insertions, because not every amino acid plays an equally important role in maintaining the protein structure. However, the most prevalent models in fold recognition methods treat every amino acid deletion and insertion as equally probable events. We have analyzed the alignment patterns for homologous and analogous sequences to determine patterns of insertion and deletion, and used that information to determine the statistics of insertions and deletions for different amino acids of a target sequence. We define these patterns as insertion/deletion (indel) frequency arrays (IFAs). By applying IFAs to the protein threading problem, we have been able to improve the alignment accuracy, especially for proteins with low sequence identity. We have also demonstrated that the application of this information can lead to an improvement in fold recognition.

Cooperativity in Ordinary Ice and Breaking of Hydrogen Bonds

The total interaction energy between two H-bonded water molecules in a condensed phase is composed of a binding energy between them and an energy due to a cooperative effect. An approximate simple expression is suggested for the dependence of the interaction energy between two H-bonded water molecules on the number of neighboring water molecules with which they are H-bonded. Using this expression, the probabilities of breaking a H bond with various numbers of H-bonded neighbors are estimated. These probabilities are used in computer simulations of the breaking of specified fractions of H bonds in an ordinary (hexagonal) ice. A large "piece" of hexagonal ice (up to 8 millions molecules) is built up, and various percentages of H bonds are considered broken. It is shown that 62-63% of H bonds must be broken in order to disintegrate the "piece" of ice into disconnected clusters. This value is only a little larger than the percolation threshold (61%) predicted both by the percolation theory for tetrahedral ice and by simulations in which all H bonds were considered equally probable to be broken. When the percentage of broken bonds is smaller than 62-63%, there is a network of H-bonded molecules which contains the overwhelming majority of water molecules. This result contradicts some models of water which consider that water consists of a mixture of water clusters of various sizes. The distribution of water molecules with unequal probabilities for breaking is compared with the simulation involving equal probabilities for breaking. It was found that in the former case, there is an enhanced number of water monomers without H bonds, that the numbers of 2- and 3-bonded molecules are smaller, and the number of 4-bonded molecules is larger than in the latter case.

The Not-Preference-Based Hoede-Bakker Index

The paper concerns a certain modification of the generalized Hoede-Bakker index - a notion defined for a social network of players. In the original Hoede-Bakker set up, preferences of players are involved. It is assumed that a player has an inclination either to accept or to reject a proposal, but due to the influence of others, his final decision may be different from his original inclination. In this paper, we propose the not-preference-based (NPB) generalized Hoede-Bakker index, where feasible strategies instead of players' inclinations are considered. We show that if all feasible strategy profiles are equally probable, then the NPB generalized Hoede-Bakker index is a ‘net' Success, i.e., ‘Success - Failure', where Success and Failure of a player is defined as the probability that the player is successful and fails, respectively. Moreover, under the assumption of equal probabilities of all feasible strategy profiles, we show that the probability that a player is lucky (Luck) equals the probability that he fails (Failure). Since Success - Luck = Decisiveness, it follows that, under the same assumption, the NPB generalized Hoede-Bakker index is equal to the probability that a player is decisive.

Bayesian Covariance Matrix Estimation Using a Mixture of Decomposable Graphical Models

A Bayesian approach is used to estimate the covariance matrix of Gaussian data. Ideas from Gaussian graphical models and model selection are used to construct a prior for the covariance matrix that is a mixture over all decomposable graphs. For this prior the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both in identifying the correctdecomposable graph and in more efficiently estimating the covariance matrix.

Microvessels of hamster buccal pouch under conditions of reduced systemic blood pressure

Using implanted camera we studied hamster buccal pouch microvessels under conditions of reduced blood pressure. The correlation (R=0.30-0.85) between the diameter and pressure was observed in 42% vessels. During progressive decrease in blood pressure, R for the majority of left venules was below zero, while left arterioles had equal probability of having R<0 and R>0. For right venules the positive and negative correlations were equally probable, while right arterioles were predominantly characterized by R>0. Heterogeneous reactions in microvessels of different diameter were revealed.

Probability and Symmetry

The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing a generalized approach to conditionalization. The technique is illustrated by application to simple examples.

On the space-group frequency in organic structures

Donohue (1985) revises the space-group frequency distribution table arrived at by Mighell, Himes & Rodgers (1983), based on about 30 000 organic compounds, on the grounds that space-group determination, in certain cases where symmetry of the second kind is not present, automatically determines an additional structure (the enantiomeric structure). He cites the example of space group P2~ for an I. amino acid determining the enantiomeric crystal for the D form. Ill cases where enantiomeric space groups are part of the list of 230 space groups, he adds the observed data under each category and gives equal weight based on the total for each space group. This results in promoting space group P2~2~21 to second place instead of P]. This method of treating the data seems, to the author, questionable. Firstly, from the standpoint of statistical analysis, enlarging the observed data to include additional data which are unobserved, however strong the arguments may be for them, is itself questionable. Secondly, the arguments advanced for such enlargements, although possibly valid theoretically, cannot always make such enantiomorphs readily realizable in practice. The procedure of doubling the observed value for an enantiomorphic space group (complementing them in the case of both space groups being part of the 230) implies equal probability of occurrence in practice and should not be based on theoretical feasibility alone. The common and trivial cases when enantiomeric distinguishability of crystal structures with corresponding identity of enantiomeric molecular species such as L and D amino acids need not always be true such as in cases where the enantiomers result from readily interconvertible conformers (rotational isomers), not uncommon in 'disymmetric' organic structures. Simple examples are the cases of hydrogen peroxide (Srinivasan, 1970) and ammonium oxalate (Robertson, 1965) where the distinguishability is non-existent in solution but structurally they take enantiomorphic space groups, presumably with equal probability of occurrence. In other organic structures with asymmetric centres, different conformational species are possible as in the case of D-alloisoleucine hydrochloride which has two forms both having the same space group, a result one would not have anticipated (Srinivasan, Varughese & Swaminathan, 1974). Although in simple disymmetric cases such as hydrogen peroxide or other symmetrically substituted peroxides, disulfides (Foss, 1954) etc. the enantiomeric forms may be expected, from energy considerations, to be equally probable, this, again, will lead to complications in terms of observability in complex systems of organic structures. As an extreme example, a small protein crystallizing in an enantiomorphic space group cannot be expected to have its enantiomeric structure in practice. It would thus seem that structural and conformational variations vitiate the assumptions of equal probability of occurrence of enantiomeric structures, which assumption is implicit in the procedure adopted for enlargement of observed data. The proper procedure would appear to be to confine oneself to the 'observed data' only under each space group, irrespective of whether enantiomeric space groups are or are not part of the 230 space-group listings. However, if enantiomeric structures are experimentally determined [which, in some cases might hold surprising and unexpected results as for NaCIO3 and NaBrO3 (Beurskens Kerssen, Kroon, Endeman, Vanlaar & Bijvoet, 1963)] they should be included since they are established as structures with different physical properties. Thus, the sole criterion should be that experimentally observed and established structures enter the statistics. It would thus seem appropriate that the revision as proposed by Donohue is not warranted and the earlier result of Mighell et al. (1983) should be accepted and space group P] restored to second place and P21212~ to third place.

Intramolecular energy distribution in the thermal decomposition of hydrazoic acid

A major assumption in the Rice--Ramsberger--Cassel--Marcus theory (RRCM theory) is that the states of an activated molecule are filled with equal probability [i, 2]. In a real molecule, the transitions in general are not equally probable, so it is of interest to examine deviations from the RRCM model. This is necessary particularly in order to set up any theory that incorporates explicitly the detailed energy redistribution in an active molecule.

A test of Premack’s “indifference principle”

For two female college Ss, different procedures were used to produce comparable probabilities of lever responding for points and music. Both of these equally probable responses then were made contingent upon responding on a third lever on CRF, FR-5, and FR-10 reinforcement schedules. For all three schedules, the two equally probable responses produced comparable reinforced response increments. In a second experiment, a sequence of procedures was used which produced equal probabilities of responding for music and points, then a higher probability of responding for music relative to responding for points, then the converse probabilities. Responding on a CRF schedule varied directly with the probability of the reinforcing response. The results of both experiments were interpreted as support for Premack’s indifference principle.

Optimal variable length codes (arbitrary symbol cost and equal code word probability)

The problem of constructing minimum-redundancy prefix codes for the general discrete noiseless channel without constraints is solved for unequal code letter costs, provided that the symbols encoded are assumed to be equally probable. A graphical technique is developed for solving the problem for which the code words are equally probable and are constructed from r symbols where r is greater than or equal to two. A method is given for constructing an optimal exhaustive prefix code. This method is then generalized to the extent that the exhaustive constraint is deleted, thereby resulting in an algorithm, designated ACE for arbitrary symbol cost and equal code word probability, which solves the stated problem.

MODIFIED POISSON PROBABILITY LAW FOR POINT PATTERN MORE REGULAR THAN RANDOM1

MAPPING, and using map evidence, for incisive, descriptive statements about spatial distributions is a fundamental operation in geographic research. In terms of their abstract properties, map distributions exist in one of the three lowest geometric dimensions -the use of points, lines, and areas to represent a distribution. Map analysis is the description of the density and pattern of this geometry. The map itself may serve as the descriptive statement, or the many subtleties of map distributions may be summarized into symbols, words, or numbers. Highly regular or obviously systematic patterns are found infrequently in geographic investigations. Usually, the patterns display no obvious order or system. In interpreting such patterns, there is a temptation to dismiss analysis with the statement that the distribution is irregular or, possibly, to say that it is a distribution. To say that a distribution is irregular neither effectively describes it nor suggests cause. To say that a distribution is random, in a nontechnical sense, is to say that the pattern has no discernible order and that cause is undeterminable. In the terminology of mathematical statistics, the term random has a precise meaning which refers to the process generating a pattern and the pattern is the realization of a theoretical process. A process is synonymous with pure chance because each event has an equal probability of occurrence. In terms of a map pattern, pure chance means that each map location has an equal probability of receiving a symbol. Since it is highly unlikely that geographic distributions, particularly locational patterns involving human decisions, are the result of equally probable events, it is expected that most map patterns reflect some system or order. It is for this reason that map patterns are examined for evidence of a spatial process. The search for

Mixed Model Variance Analysis with Normal Error and Possibly Non-Normal Other Random Effects: Part I: The Univariate Case

Mixed Model Variance Analysis with Normal Error and Possibly Non-Normal Other Random Effects: Part I: The Univariate Case