Joint probability distribution

Mutual information, effectively measures the degree to which two probability distributions or, in the context of CA, two sites or blocks - are correlated. Given probability distributions pi and pj and the joint probability distribution py, 1 is defined by ... 

To apply the above method, we must decide the distribution of parameter values to explore. One immediate answer would be to impose on the parameters an appropriate joint probability distribution, but this would require us to know it, or at least to have a reasonable idea of what it might be. 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

Bricogne, G. (1988) A Bayesian statistical theory ofthe phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors, Acta Cryst., A44, 517-545. 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

When it comes to the covariance structure, however, problems become acute. Total inversion requires that a joint probability distribution is known for observations and parameters. This is usually not a problem for observations. The covariance structure among the parameters of the model becomes more obscure how do we estimate the a priori correlation coefficient between age and initial Sr ratio in our isochron example without infringing seriously the objectivity of error assessment When the a priori covariance structure between the observations and the model parameters is estimated, the chances that we actually resort to unsupported and unjustified speculation become immense. Total inversion must be well-understood in order for it not to end up as a formal exercise of consistency between a priori and a posteriori estimates. 

Giacovazzo, C. and SUiqi, D. (2001). The method of joint probability distribution functions apphed to MAD techniques. The two-wavelength case for acentric crystals. Acta Crystallogr. A 57, 700-707. 

The wave function P contains all information of the joint probability distribution of the electrons. For example, the two-electron density is obtained from the wave function by integration over the spin and space coordinates of all but two electrons. It describes the joint probability of finding electron 1 at r, and electron 2 at r2. The two-electron density cannot be obtained from elastic Bragg scattering. 

We first treat a stiff system as a generic unconstrained system. We consider a joint probability distribution T (g) for all 3N coordinates of a stiff system, soft and hard, given by... 

The normalized joint probability distribution for an ensemble of such generic variables will be denoted by (X), which is normalized such that dX dX 0 X) = 1. The evolution of such a probability distribution is... 

On the other hand, one may attribute fixed values to Xs + l9...9Xr and consider the joint probability distribution of the remaining variables Xl9...9Xs. This is called the conditional probability o/X, ...,XS, conditional onIs+b...,Ir having the prescribed values xs + l9. ..9xr.It will be denoted by >... 

The expression (2.1) is not the precise number of reactive collisions, but the average. The actual number fluctuates around it and we want to find the resulting fluctuations in the rij around the macroscopic values determined by (2.2). In order to describe them one needs the joint probability distribution P(n, t). Although it is written as a function of all rij, it is defined on the accessible sublattice alone. Alternatively one may regard it as a distribution over the whole physical octant, which is zero on all points that are not accessible. 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

We are interested in the fluctuations around these macroscopic values and therefore introduce the joint probability distribution P(nx, nY, t). It obeys the M-equation... 

Subdivide the total volume Q into cells A and call nk the number of particles in cell X. The cells must be so small that inside each of them the above mentioned condition of homogeneity prevails. Let P( nk, t) be the joint probability distribution of all nk. At t + dt it will have changed because of two kinds of possible processes. Firstly, the nk inside each separate cell X may change by an event that creates or annihilates a particle. In the master equation for P( nk, t) this gives a corresponding term for each separate cell. 

We demonstrate the method on the following concrete - if somewhat trivial - example. A swarm of particles is moving freely in space, but each particle has a probability a per unit time to disappear, through spontaneous decay or through a reactive collision. To cover the latter possibility we allow a to depend on v. The (r, u)-space is decomposed in cells A and nx is the number of particles in cell X. The joint probability distribution P( nx, t) varies through decay and through the motion of the particles. The decay is described by... 

Exercise. The neutrons in a nuclear reactor behave as free particles until they are absorbed, scatter, or cause fission and thereby produce more neutrons. The master equation for the joint probability distribution of the occupation numbers nk of the phase space cells X is... 

A physical system S that evolves probabilistically in time can be mathematically described by a time-dependent random variable X(t). It is assumed that (1) one can measure values xlt x2, x3,.. . , xn of X(t) at instants ti,t2,ti,...,tn representing the possible states of the system S and (2) one can define a set of joint probability distribution functions... 

Prompted by these considerations, Gillespie [388] introduced the reaction probability density function p (x, l), which is a joint probability distribution on the space of the continuous variable x (0 < x < oc) and the discrete variable l (1 = 1,..., to0). This function is used as p (x, l) Ax to define the probability that given the state n(t) at time t, the next event will occur in the infinitesimal time interval (t + x,t + x + Ax), AND will be an Ri event. Our first step toward finding a legitimate method for assigning numerical values to x and l is to derive, from the elementary conditional probability hi At, an analytical expression for p (x, l). To this end, we now calculate the probability p (x, l) Ax as the product po (x), the probability at time t that no event will occur in the time interval (t, t + x) TIMES a/ Ax, the subsequent probability that an R.i... 

The previous example involved a two-dimensional system (involving two independent dynamic species). Thus the CME followed from the two-dimensional reaction diagram. For systems with more species, the dimension of the problem grows accordingly. For a system with three species, say A, B, and C, the CME tracks the three-dimensional probability of A molecules, m B molecules, and n C molecules present at time t. In general, the mathematical description of an A-dimensional system is the joint probability distribution... 

Imparato and Peliti " show how the work distribution for a given protocol can be obtained using a joint probability distribution for the work and the initial state, and apply it to system represented by a coordinate in a mean field. They show that it leads to the JE, and demonstrate some numerical issues when applied to large systems where it becomes difficult to observe fluctuations. They also provide an analytical treatment of a driven Brownian particle, obtaining the probability distribution for the work and FR for this system. ... 

Fig. 24.8. Computational simulation analysis of conformational dynamics in T4 lysozyme enzymatic reaction, (a) Histograms of fopen calculated from a simulated single-molecule conformational change trajectory, assuming a multiple consecutive Poisson rate processes representing multiple ramdom walk steps, (b) Two-dimensional joint probability distributions <5 (tj, Tj+i) of adjacent pair fopen times. The distribution <5(ri, Ti+i) shows clearly a characteristic diagonal feature of memory effect in the topen, reflecting that a long topen time tends to be followed by a long one and a short fopen time tends to be followed by a short one...

Joint probability distribution function of the local velocity and acceleration, Eq. (44)... 

FIGURE 8.7 The top graph shows the separate probabilities over the entire range 0 to 1 for two isomorphous heavy atom derivatives I and II. Both are bimodal and each separately predicts the two most likely phase angles for the native structure factor. At the bottom the joint probability distribution strongly predicts a single most probable phase. 

Figure 8.7 shows the phase probability distributions for two independent heavy atom derivatives, and how they combine to yield a joint probability distribution, and a relatively unambiguous choice of the correct phase angle. Because this determination is entirely a matter of computing and combining phase probabilities, it can readily be carried out for each reflection hkl in a digital computer. 

The techniques (e.g., weighted histogram analysis, equal weight rule, etc.) utilized for one-component systems or incompressible binary mixtures can be readily carried over to compressible systems. Since the system is described by two order parameters one monitors the joint probability distribution of... 

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