Unstable property estimates from PTDW

We know from Section 2.3 that the infinite-mean nature of the weight variable is certainly not a desirable property. To find out in a real example whether this has a severe impact on the estimation results, we study the basic model described in Sec- tion 1.2. To be consistent, the same set of temperatures was used in both PTDW and PT simulations, this along with observed swap rates of the PT simulation are listed in Table 3.1. While it is entropically favourable for the polymers to move inde- pendently, there is an energetic tendency for the hydrophilic tails of both polymers to interact with each other within the membrane and form a dimer. The equilibrium of the system is a balance between these two driving forces. At high temperatures, the entropy dominates and monomer state is predominant, and at low temperatures, the energy dominates and dimer state is predominant. There is an energy barrier between these two states and this is reflected by the low swap rate betweenT2 and

T3 in Table 3.1.

We first examine the distribution of the (log) weights of PTDW to check that it is indeed a stable distribution, as mentioned in Section 2.3. One way to do this is to partition the log weights into contiguous blocks and compare their distribution functions. In Figure 3.1, we show the histogram of log weight and a

Temperature T1 T2 T3 T4 T5

0.3 0.48 0.85 1.3 2.0

Swap rates 0.38 0.05 0.56 0.52

Table 3.1: Temperatures and observed swap rates for the PT simulation. The

PTDW simulation used the same set of temperatures.

0 1000 2000 3000 0 10 20 30 log(weights) count (a) ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●● ●● ● ●●●●● ● ● ● ●●●● ● ● ● 0 5 10 15 20 25 0 5 10 15 20 25 log(weights)[seq(10001, 15000, 10)] log(w eights)[seq(15001, 20001, 10)] (b)

Figure 3.1: Histogram of logw from the PTDW simulation (Figure 3.1a) and q-q plot of log weights corresponding to MC steps 10001—15000 (horizontal axis) and to steps 15001—20001 (vertical axis) (Figure 3.1b). The liney =xis plotted in red in the q-q plot.

Quantile-Quantile (q-q) plot for two contiguous sets of log weights. The q-q plot is an effective graphical technique to inspect if two sets of data share a common distribution. The axes of a q-q plot are quantiles of the datasets. For continuous distributions theα-quantile (α ∈[0,1]) is simply given by F−1(α), where F is the distribution function, so, for example, the 0.5-quantile is just the median. As we can see in Figure 3.1b, the quantile points fall along the liney=x, suggesting that the two sets of log weights have similar distributions.

The mean potential energy of the system was used as a property for com- parison, since it can be obtained easily from the simulations without additional calculations. To monitor convergence, we plot cumulative average of the estimates at each temperature. The results are shown side-by-side in Figure 3.2. In the case of PTDW, the cumulative average is defined as the weighted average:

PT PTDW PTDW−ST −220 −210 −200 0 5000 10000 15000 20000 0 5000 10000 15000 20000 0 5000 10000 15000 20000 steps ×103 cum ulativ e a ver age of U temperature T=0.3 T=0.48 T=0.85 T=1.3 T=2

Figure 3.2: Cumulative average of potential energy computed by the method of PT (left), PTDW with raw weights (middle), and PTDW with stratified truncation (right). It is worth noting that because stratified truncation was applied based on the energy values of each temperature trajectory, the stratified weights became dependent on temperature; whereas when it is not applied, the same set of weights, namely the raw weights, are used in property estimates at all temperatures.

Un(x) = Pn i=1w(i)U(x(i)) Pn i=1w(i) .

From the PT plot we see a much quicker convergence at higher temperatures than at lower temperatures; whereas in the case of PTDW (no stratified truncation), the behaviour is irregular, especially at higher temperatures (T ≥ 0.85). We note that changes in energy as simulation goes along are negligible compare to changes in DW weights, which can be as large as 1014, and so these “abrupt jumps” must be due to the large variability in the weights, where adding one observation with its associated weight significantly changes the values of Un(x) accumulated from previous observations. So we see that, for the system we are interested in, raw weights from PTDW cannot be used to estimate expectations.

Post-processing the raw weights with the stratified truncation procedure, denoted as PTDW-ST, was conducted and the results are shown in the rightmost of Figure 3.2. Although one can play with different settings of the strata sizes, here we used 50 strata with equal number of samples in each stratum. It is observed that instability of the cumulative average is reduced considerably, however, the energy estimates do not agree with those from the PT simulation—most notably

they disagree entirely at T = 0.48, with PT estimate being typical of the dimer state and PTDW-ST estimate being typical of the monomer state. The fact that there is a large discrepancy between the PT and the PTDW-ST estimates at this temperature let us presume that at least one of the two methods is converging to the wrong value. In the next section, we present evidence against the PTDW-ST estimate.3

Despite not being able to estimate averages reliably, the PTDW method did show its capability to quickly locate the low energy state, which can be seen by the rapid decrease ofUn(x) atT = 0.3, irrespective of whether stratified truncation was applied. In contrast, more than 5×106 MC steps were needed for equilibration in the PT method.

3.2.2 A model-dependent strategy for verification and efficient sam-