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Estimating Distributions of Parameters in Nonlinear State Space Models with Replica Exchange Particle Marginal Metropolis–Hastings Method
Extracting latent nonlinear dynamics from observed time-series data is important for understanding a dynamic system against the background of the observed data. A state space model is a probabilistic graphical model for time-series data, which describes the probabilistic dependence between latent variables at subsequent times and between latent variables and observations. Since, in many situations, the values of the parameters in the state space model are unknown, estimating the parameters from observations is an important task. The particle marginal Metropolis–Hastings (PMMH) method is a method for estimating the marginal posterior distribution of parameters obtained by marginalization over the distribution of latent variables in the state space model. Although, in principle, we can estimate the marginal posterior distribution of parameters by iterating this method infinitely, the estimated result depends on the initial values for a finite number of times in practice. In this paper, we propose a replica exchange particle marginal Metropolis–Hastings (REPMMH) method as a method to improve this problem by combining the PMMH method with the replica exchange method. By using the proposed method, we simultaneously realize a global search at a high temperature and a local fine search at a low temperature. We evaluate the proposed method using simulated data obtained from the Izhikevich neuron model and Lévy-driven stochastic volatility model, and we show that the proposed REPMMH method improves the problem of the initial value dependence in the PMMH method, and realizes efficient sampling of parameters in the state space models compared with existing methods.
Journal Article
Superman family adventures. Monkey Metropolis!
The City of Metropolis runs into monkey mayhem when Beppo's friend goes crazy, and Titano, the giant angry ape, appears.
Book
High-dimensional Bayesian inference via the unadjusted Langevin algorithm
We consider in this paper the problem of sampling a high-dimensional probability distribution π having a density w.r.t the Lebesgue measure on ℝd, known up to a normalization constant
x
↦
π
x
=
e
−
U
x
/
∫
ℝ
d
e
−
U
y
d
y
. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that U is continuously differentiable, ▽U is globally Lipschitz and U is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order 2 and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.
Journal Article
NONASYMPTOTIC CONVERGENCE ANALYSIS FOR THE UNADJUSTED LANGEVIN ALGORITHM
In this paper, we study a method to sample from a target distribution π over ℝd having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with π. For both constant and decreasing step sizes in the Euler discretization, we obtain nonasymptotic bounds for the convergence to the target distribution π in total variation distance. A particular attention is paid to the dependency on the dimension d, to demonstrate the applicability of this method in the high-dimensional setting. These bounds improve and extend the results of Dalalyan.
Journal Article
Productivity, Sustainability, and Economic Growth in Metropolises: Estimates of Long-Time Commuting Effects in Developing Countries
Purpose: The study aimed to investigate the long-time commuting patterns of workers in the developing world, observing metropolises’ samples. The main question to be answered was whether there is scope for labor policy changes towards reducing long-time commuting to obtain productivity gains based on life quality improvements. Theoretical background: Although previous research indicates the presence of symmetric patterns in the developing world, specifically in long-time commuting, there is still a gap in checking the possibilities for labor productivity growth as a consequence of worker´s life quality improvement. Methodology: In this study, we observed six of the biggest metropolises located in very different geographic regions – New Delhi (India), Mexico City (Mexico), São Paulo (Brazil), Manilla (Philippines), Nairobi (Kenya), and Accra (Ghana). Simple random samples of workers in the Metropolitan Areas were surveyed electronically, by ‘Google-Forms e-survey’ during the second half of 2019. Considering error margins below 5 percentage points and with a 95 percent point confidence level, the authors used proportion (p) sample distributions to draw inferences about the population of workers. Results and conclusion: The results showed that long-time commuters are between 12 and 26 percent of the workers. More than 65 percent of workers in all the cities were interested in reducing commuting time. More than half of the workers agreed that reducing commuting time could improve labor productivity, and the same share is aware of the negative effects on quality of life and health. Labor policy changes in these six metropolises have the potential to affect more than 6.5 million workers. Research implications: The study advances in verifying the long-time commuting patterns and consequences, opening doors for labor policy changes towards gaining productivity by reducing time spent commuting, as previous literature has partially done. Originality/value: The approach of the work and its results contribute to expanding empirical research related to the assumptions of labor productivity growth based on life quality improvements.
Journal Article
COUNTEREXAMPLES FOR OPTIMAL SCALING OF METROPOLIS–HASTINGS CHAINS WITH ROUGH TARGET DENSITIES
For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n
−1 and as
n
−
1
3
with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions of the order of 𝒞¹(ℝ) for RWM and 𝒞³(ℝ) for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent H. For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as
n
−
1
H
and as
n
−
1
2
+
H
and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis–Hastings algorithms (possibly depending on a random environment).
Journal Article
Examples of Adaptive MCMC
We investigate the use of adaptive MCMC algorithms to automatically tune the Markov chain parameters during a run. Examples include the Adaptive Metropolis (AM) multivariate algorithm of Haario, Saksman, and Tamminen (2001), Metropolis-within-Gibbs algorithms for nonconjugate hierarchical models, regionally adjusted Metropolis algorithms, and logarithmic scalings. Computer simulations indicate that the algorithms perform very well compared to nonadaptive algorithms, even in high dimension.
Journal Article
Procedural Reconstruction of 3D Indoor Models from Lidar Data Using Reversible Jump Markov Chain Monte Carlo
Automated reconstruction of Building Information Models (BIMs) from point clouds has been an intensive and challenging research topic for decades. Traditionally, 3D models of indoor environments are reconstructed purely by data-driven methods, which are susceptible to erroneous and incomplete data. Procedural-based methods such as the shape grammar are more robust to uncertainty and incompleteness of the data as they exploit the regularity and repetition of structural elements and architectural design principles in the reconstruction. Nevertheless, these methods are often limited to simple architectural styles: the so-called Manhattan design. In this paper, we propose a new method based on a combination of a shape grammar and a data-driven process for procedural modelling of indoor environments from a point cloud. The core idea behind the integration is to apply a stochastic process based on reversible jump Markov Chain Monte Carlo (rjMCMC) to guide the automated application of grammar rules in the derivation of a 3D indoor model. Experiments on synthetic and real data sets show the applicability of the method to efficiently generate 3D indoor models of both Manhattan and non-Manhattan environments with high accuracy, completeness, and correctness.
Journal Article
COUPLING AND CONVERGENCE FOR HAMILTONIAN MONTE CARLO
Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L¹ Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal target distributions are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given error ϵ are a direct consequence of contractivity. These bounds show that HMC can overcome diffusive behavior if the duration of the Hamiltonian dynamics is adjusted appropriately.
Journal Article