Candidate density

To proceed, first, assume we have some well-behaved candidate density that integrates to the proper number of electrons, N. In that case, the first theorem indicates that this density determines a candidate wave function and Hamiltonian. That being the case, we can evaluate the energy expectation value... 

In Chapter 7 we develop a method for finding a Markov chain that has good mixing properties. We will use the Metropolis-Hastings algorithm with heavy-tailed independent candidate density. We then discuss the problem of statistical inference on the sample from the Markov chain. We want to base our inferences on an approximately random sample from the posterior. This requires that we determine... 

Draw a random value of 9 from the candidate density go( ). 

Calculate the weight for that value as the ratio of the target to the scaled up candidate density... 

Note W recognize this is the shape of a normal 0,1) distribution. However, we will only use the formula giving the shape, and not our knowledge of what the posterior having that shape actually is.) Let us use the normal(0,2 ) candidate density. Figure 2.1 shows the unsealed target and the candidate density. We want the smallest value ofM such that for all 0,... 

Figure 2.3 Scaled up candidate density and unsealed target.

The preliminary steps are to first calculate the unsealed posterior and the candidate density over a fine mesh of 6 values over the main range of the posterior and graph them. Take logarithms of the unsealed posterior and candidate density and graph them to make sure the candidate density dominates the unsealed posterior in the tails. Use the computer to find the maximum value of unsealed posterior divided by the candidate density... 

Finally, take all the 0 values that have indicator value Indi = 1 into our accepted sample. These are the values i where Ui < Wi. In Minitab we use the Unstack command. These accepted values will be a random sample of size n from the posterior distribution g 9 yii- ,yn)- Generally the final sample size n will be less than the initial sample size N, except in the case where the initial candidate distribution is proportional to the unsealed target and all candidates will be accepted. To get as many candidates accepted as possible, the candidate density should have a shape as similar as possible to the target, yet still dominate it. 

The candidate density go 9) must dominate the scaled up target g 0) f y 0). This means we can find a value M such that M x go( ) > fivW) for all values of 0. It is especially important that the candidate density have heavier tails than the unsealed target. 

Take logarithms of the densities and graph them. If the log(candidate density) lies above the log(target density) in the tails, then the candidate dominates the target. 

We want to generate a random sample from the posterior given in Exercise 3.1. First we draw a random sample of size 100000 from a Laplace 0,l) candidate density. Use acceptance-rejection-sampling to reshape it to be a random sample from the posterior. 

In Section 6.1 we show how the Metropolis-Hastings algorithm can be used to find a Markov chain that has the posterior as its long-run distribution in the case of a single parameter. There are two kinds of candidate densities we can use, random-walk candidate densities, or independent candidate densities. We see how the chain... 

We should note that having the candidate density q 0, O ) close to the target g 6 y) leads to more candidates being accepted. In fact, when the candidate density is exactly the same shape as the target... 

Single Parameter with a Random-Walk Candidate Density... 

Metropolis et al. (1953) considered Markov chains with a random-walk candidate distribution. Suppose we look at the case where there is a single parameter 0. For a random-walk candidate generating distribution the candidate is drawn from a symmetric distribution centered at the current value. Thus the candidate density is given by... 

Figure 6.1 Six consecutive draws from a Metropolis-Hastings chain with a random-walk candidate density. Note the candidate density is centered around the current value.

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