Probability theory conditional expectation

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

One may say, perhaps, that some factor must still be introduced in the theoretical expression to obtain the correct magnitude of 6 and the experimental observations offer a means of evaluating this. But, unfortunately, the theoretical probabilities do not have even the right relative values. They decrease with quantum number while for the experimental values Tolman and Badger found a decided increase. The absolute values which they calculated may be in error for the reasons given above, but more perfect resolution would be expected to increase the trend they observed rather than to eliminate it. It would seem, therefore, that the predictions of the new quantum theory, while they may apply to some ideal system, do not describe the conditions we have experimentally observed in the case of hydrogen chloride. 

Nearly every inference we make with respect to any future event is more or less doubtful. If the circumstances are favourable, a forecast may be made with a greater degree of confidence than if the conditions are not so disposed. A prediction made in ignorance of the determining conditions is obviously less trustworthy than one based upon a more extensive knowledge. If a sportsman missed his bird more frequently than he hit, we could safely infer that in any future shot he would be more likely to miss than to hit. In the absence of any conventional standard of comparison, we could convey no idea of the degree of the correctness of our judgment. The theory of probability seeks to determine the amount of reason which we may have to expect any event when we have not sufficient data to determine with certainty whether it will occur or not and when the data will admit of the application of mathematical methods. 

The Reverend Thomas Bayes [1702-1761] was a British mathematician and Presbyterian minister. He is well known for his paper An essay towards solving a problem in the doctrine of chances [14], which was submitted by Richard Price two years after Bayes death. In this work, he interpreted probability of any event as the chance of the event expected upon its happening. There were ten propositions in his essay and Proposition 3,5 and 9 are particularly important. Proposition 3 stated that the probability of an event X conditional on another event Y is simply the ratio of the probability of both events to the probability of the event Y. This is the definition of conditional probability. In Proposition 5, he introduced the concept of conditional probability and showed that it can be expressed regardless of the order in which the events occur. Therefore, the concern in conditional probability and Bayes theorem is on correlation but not causality. The consequence of Proposition 3 and 5 is the Bayes theorem even though this was not what Bayes emphasized in his article. In Proposition 9, he used a billiard example to demonstrate his theory. The work was republished in modern notation by G. A. Barnard [13]. In 1774, Pierre-Simon Laplace extended the results by Bayes in his article Memoire sur la probabilite des causes par les evenements (in French). He treated probability as a tool for filling up the gap of knowledge. The Bayes theorem is one of the most frequently encountered eponyms in the literature of statistics. 

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