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It will be clear from the previous chapters that once we have built the state space, and decided which selection function β and which acceptance function α is to be used, the equilibrium density δ is determined. If we make sure that the chain is reversible and β is symmetrical in its arguments, then δ only depends on α. So, if we want to enforce some desirable properties on this density we have to do that by selecting a proper acceptance function, and, when reversibility and symmetry are not implied, a proper move set and selection function as well. The realization of what seem to be desirable properties of a density function should be carefully considered. Some important aspects, such as the computational consequences for the implementation of the chain, are not always apparent from the density function. Also, what seems to be an advantage may not be under our control, or may already be inherent to chains. For example, a small score variance once we have obtained a chain that produces low average score, would be advantageous, because that would mean that the chain is almost all the time in a low score state, and thus selecting almost exclusively high quality configurations. Minimizing the score variance for all chains is not desirable, as will become clear in the later chapters, and if we are able to obtain chains with very low mean score, the variance has to be low as well, because the scores have a lower bound. What we certainly do not want is that the density function exhibits a bias towards certain states, that is not based on the score assigned to that state.
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