sljeff/dots.ocr 🖼️🔢📝 → 📝

[{"bbox": [1355, 199, 2103, 262], "category": "Page-header", "text": "A PREPRINT - SEPTEMBER 5, 2022"}, {"bbox": [698, 455, 1846, 1010], "category": "Picture"}, {"bbox": [441, 1051, 2103, 1153], "category": "Caption", "text": "Figure 7: Algorithmic performance measured in ES/s (effective samples per second), for the eight highest energy KL coefficients $\ heta_k$, $k = 1, \dots, 8$, for both RWMH (blue) and MLDA (red)."}, {"bbox": [450, 1208, 1251, 1782], "category": "Picture"}, {"bbox": [1292, 1208, 2088, 1782], "category": "Picture"}, {"bbox": [526, 1821, 2021, 1878], "category": "Caption", "text": "Figure 8: The true (blue) and measured (red) densities of prey (left) and predators (right)."}, {"bbox": [441, 1970, 2103, 2073], "category": "Text", "text": "and perturbing the calculated values $N(t^)$ and $P(t^)$ with independent Gaussian noise $\epsilon \sim N(0, 1)$ (Fig. 8). Our aim is to predict the mean density of predators $\mathbb{E}(P)$ over the same period."}, {"bbox": [441, 2086, 2103, 2364], "category": "Text", "text": "The solutions of the ODE system in Eq. Eq. (36) can be approximated by a suitable numerical integration scheme. We use an explicit, adaptive Runge-Kutta method of order 5(4) [46]. For the finest level $l = 2$, we integrate over the entire time domain $T_2 = [0, 12]$ and use the entire dataset to compute the likelihood function, while for the coarse levels, we stop integration early, so that $T_1 = [0, 8]$ and $T_0 = [0, 4]$, and use only the corresponding subsets of the data to compute the likelihood functions."}, {"bbox": [441, 2381, 2103, 2528], "category": "Text", "text": "We assume that we possess some prior knowledge about the parameters, and use informed pri-
ors $N_0 \sim N(10.8, 1)$, $P_0 \sim N(5.3, 1)$, $a \sim N(2.5, 0.5)$, $b \sim \ ext{Inv-Gamma}(1.0, 0.5)$, $c \sim \ ext{Inv-Gamma}(1.0, 0.5)$ and $d \sim N(1.2, 0.3)$."}, {"bbox": [441, 2542, 2103, 2871], "category": "Text", "text": "To demonstrate the multilevel variance reduction feature, we ran the MLDA sampler with randomi-
sation of the subchain length as described in Section 2.3 and then compared the (multilevel) MLDA
estimator in Eq. Eq. (21), which uses both the coarse and fine samples, with a standard MCMC es-
timator based only on the samples produced by MLDA on the fine level. In both cases, we used the
three-level model hierarchy as described above and employed the Differential Evolution Markov
Chain (DE-MC_Z) proposal [48] on the coarsest level. The coarsest level proposal kernel was au-
tomatically tuned during burn-in to achieve an acceptance rate between 0.2 and 0.5. The subchain"}]