Optimal Bayesian Estimator Research Articles

Deriving cloud droplet number concentration from surface-based remote sensors with an emphasis on lidar measurements

Abstract. Given the importance of constraining cloud droplet number concentrations (Nd) in low-level clouds, we explore two methods for retrieving Nd from surface-based remote sensing that emphasize the information content in lidar measurements. Because Nd is the zeroth moment of the droplet size distribution (DSD), and all remote sensing approaches respond to DSD moments that are at least 2 orders of magnitude greater than the zeroth moment, deriving Nd from remote sensing measurements has significant uncertainty. At minimum, such algorithms require the extrapolation of information from two other measurements that respond to different moments of the DSD. Lidar, for instance, is sensitive to the second moment (cross-sectional area) of the DSD, while other measures from microwave sensors respond to higher-order moments. We develop methods using a simple lidar forward model that demonstrates that the depth to the maximum in lidar-attenuated backscatter (Rmax⁡) is strongly sensitive to Nd when some measure of the liquid water content vertical profile is given or assumed. Knowledge of Rmax⁡ to within 5 m can constrain Nd to within several tens of percent. However, operational lidar networks provide vertical resolutions of > 15 m, making a direct calculation of Nd from Rmax⁡ very uncertain. Therefore, we develop a Bayesian optimal estimation algorithm that brings additional information to the inversion such as lidar-derived extinction and radar reflectivity near the cloud top. This statistical approach provides reasonable characterizations of Nd and effective radius (re) to within approximately a factor of 2 and 30 %, respectively. By comparing surface-derived cloud properties with MODIS satellite and aircraft data collected during the MARCUS and CAPRICORN II campaigns, we demonstrate the utility of the methodology.

Single trial Bayesian inference by population vector readout in the barn owl's sound localization system.

Bayesian models have proven effective in characterizing perception, behavior, and neural encoding across diverse species and systems. The neural implementation of Bayesian inference in the barn owl's sound localization system and behavior has been previously explained by a non-uniform population code model. This model specifies the neural population activity pattern required for a population vector readout to match the optimal Bayesian estimate. While prior analyses focused on trial-averaged comparisons of model predictions with behavior and single-neuron responses, it remains unknown whether this model can accurately approximate Bayesian inference on single trials under varying sensory reliability, a fundamental condition for natural perception and behavior. In this study, we utilized mathematical analysis and simulations to demonstrate that decoding a non-uniform population code via a population vector readout approximates the Bayesian estimate on single trials for varying sensory reliabilities. Our findings provide additional support for the non-uniform population code model as a viable explanation for the barn owl's sound localization pathway and behavior.

Lithospheric strength of the Anatolian plateau and implications for strong earthquake activity in Turkey

On February 6, 2023, the doublet earthquake including two main shocks with magnitudes MW7.8 and MW7.5, occurred near the western side of the East Anatolian Fault at the southeast boundary of the Anatolian Plateau in Turkey. Based on the WGM2012 Bouguer gravity anomaly data and the Etopo1 topography data, this study first introduced a joint inversion of admittance and coherence functions and used the Bayesian optimal parameter estimation method to obtain the effective elastic thickness Te and loading ratio F of the lithosphere for various tectonic units in the Anatolian Plateau. Secondly, we discussed the characteristics and influencing factors of the lithospheric mechanical strength and analyzed its relationship with seismic activity. The lithospheric mechanical strength of the Anatolian Plateau showed clear lateral heterogeneity and a "weak-strong-weak" spatial pattern from east to west, reflecting various tectonic processes. At last, the strong seismic activity was found where the lithospheric strength was low in the Anatolian Plate. We also incorporated GPS strain rate and other results to investigate the tectonic background and primary causes of the MW7.8 and MW7.5 doublet earthquakes in Turkey. The results have a good insight into urban safety design in the Turkish region, including post-disaster rehabilitation, earthquake hazard assessment, and loss reduction.

Emergence of belief-like representations through reinforcement learning.

To behave adaptively, animals must learn to predict future reward, or value. To do this, animals are thought to learn reward predictions using reinforcement learning. However, in contrast to classical models, animals must learn to estimate value using only incomplete state information. Previous work suggests that animals estimate value in partially observable tasks by first forming "beliefs"-optimal Bayesian estimates of the hidden states in the task. Although this is one way to solve the problem of partial observability, it is not the only way, nor is it the most computationally scalable solution in complex, real-world environments. Here we show that a recurrent neural network (RNN) can learn to estimate value directly from observations, generating reward prediction errors that resemble those observed experimentally, without any explicit objective of estimating beliefs. We integrate statistical, functional, and dynamical systems perspectives on beliefs to show that the RNN's learned representation encodes belief information, but only when the RNN's capacity is sufficiently large. These results illustrate how animals can estimate value in tasks without explicitly estimating beliefs, yielding a representation useful for systems with limited capacity.

The self and the Bayesian brain: Testing probabilistic models of body ownership through a self-localization task

Simple multisensory manipulations can induce the illusory misattribution of external objects to one's own body, allowing to experimentally investigate body ownership. In this context, body ownership has been conceptualized as the result of the online Bayesian optimal estimation of the probability that one object belongs to the body from the congruence of multisensory inputs. This idea has been highly influential, as it provided a quantitative basis to bottom-up accounts of self-consciousness. However, empirical evidence fully supporting this view is scarce, as the optimality of the putative inference process has not been assessed rigorously. This pre-registered study aimed at filling this gap by testing a Bayesian model of hand ownership based on spatial and temporal visuo-proprioceptive congruences. Model predictions were compared to data from a virtual-reality reaching task, whereby reaching errors induced by a spatio-temporally mismatching virtual hand have been used as an implicit proxy of hand ownership. To rigorously test optimality, we compared the Bayesian model versus alternative non-Bayesian models of multisensory integration, and independently assess unisensory components and compare them to model estimates. We found that individually measured values of proprioceptive precision correlated with those fitted from our reaching task, providing compelling evidence that the underlying visuo-proprioceptive integration process approximates Bayesian optimality. Furthermore, reaching errors correlated with explicit ownership ratings at the single individual and trial level. Taken together, these results provide novel evidence that body ownership, a key component of self-consciousness, can be truly described as the bottom-up, behaviourally optimal processing of multisensory inputs.

Optimal Bayesian estimation of Gaussian mixtures with growing number of components

We study Bayesian estimation of finite mixture models in a general setup where the number of components is unknown and allowed to grow with the sample size. An assumption on growing number of components is a natural one as the degree of heterogeneity present in the sample can grow and new components can arise as sample size increases, allowing full flexibility in modeling the complexity of data. This however will lead to a high-dimensional model which poses great challenges for estimation. We novelly employ the idea of a sample size dependent prior in a Bayesian model and establish a number of important theoretical results. We first show that under mild conditions on the prior, the posterior distribution concentrates around the true mixing distribution at a near optimal rate with respect to the Wasserstein distance. Under a separation condition on the true mixing distribution, we further show that a better and adaptive convergence rate can be achieved, and the number of components can be consistently estimated. Furthermore, we derive optimal convergence rates for the higher-order mixture models where the number of components diverges arbitrarily fast. In addition, we suggest a simple recipe for using Dirichlet process (DP) mixture prior for estimating the finite mixture models and provide theoretical guarantees. In particular, we provide a novel solution for adopting the number of clusters in a DP mixture model as an estimate of the number of components in a finite mixture model. Simulation study and real data applications are carried out demonstrating the utilities of our method.

Tuning-Free Bayesian Estimation Algorithms for Faulty Sensor Signals in State-Space

Sensors provide insights into the industrial processes, while misleading sensor outputs may result in inappropriate decisions or even catastrophic accidents. In this article, the Bayesian estimation algorithms are developed to estimate unforeseen signals in sensor outputs without tuning. The optimal Bayesian estimation method is first derived by incorporating a Gaussian distribution specifying potential unmodeled dynamics into the measurement equation. Since its performance depends on tuning parameters, an iterative Bayesian estimation algorithm is developed using the variational inference technique. Specifically, an inverse Wishart distribution is introduced to describe the predicted covariance of abnormal signals. We then estimate it together with the other independent Gaussian distributions to conditionally approximate the joint posterior distribution, by which the effects of tuning parameters can be replaced adaptively. Testing the proposed algorithms through a simulated electromechanical brake model and a real experimental system shows that the proposed algorithm can satisfactorily estimate additive sensor faults online and services as a sensor monitor that simultaneously provides the locations and magnitudes of faulty signals without tuning.

Erratum: Quantum state smoothing as an optimal Bayesian estimation problem with three different cost functions [Phys. Rev. A 104 , 032213 (2021)

Erratum: Quantum state smoothing as an optimal Bayesian estimation problem with three different cost functions [Phys. Rev. A <b>104</b> , 032213 (2021)

Stochastic resonance in Bayesian estimation and CRLB for nonlinear system

In this work, noise enhanced parameter estimation problems are investigated for a general nonlinear system, where an additive noise is added to the nonlinear system input and a Bayesian estimator is developed based on the noise modified output. The optimal probability distribution of the additive noises is formulated successively for minimizing the mean square error (MSE) of the optimal Bayesian estimation and the Cramer–Rao lower bound (CRLB). Then the optimal additive noises for the two different noise enhanced optimization problems are explicitly derived as constant vectors, which implies the randomization of constant vectors is not beneficial to these optimizations. Finally, numerical results are presented to illustrate the theoretical results.

Radiative Transfer Speed-Up Combining Optimal Spectral Sampling With a Machine Learning Approach

The Orbiting Carbon Observatories-2 and -3 make space-based measurements in the oxygen A-band and the weak and strong carbon dioxide (CO2) bands using the Atmospheric Carbon Observations from Space (ACOS) retrieval. Within ACOS, a Bayesian optimal estimation approach is employed to retrieve the column-averaged CO2 dry air mole fraction from these measurements. This retrieval requires a large number of polarized, multiple-scattering radiative transfer calculations for each iteration. These calculations take up the majority of the processing time for each retrieval and slow down the algorithm to the point that reprocessing data from the mission over multiple years becomes especially time consuming. To accelerate the radiative transfer model and, thereby, ease this bottleneck, we have developed a novel approach that enables modeling of the full spectra for the three OCO-2/3 instrument bands from radiances calculated at a small subset of monochromatic wavelengths. This allows for a reduction of the number of monochromatic calculations by a factor of 10, which can be achieved with radiance errors of less than 0.01% with respect to the existing algorithm and is easily tunable to a desired accuracy-speed trade-off. For the ACOS retrieval, this speeds up the over-retrievals by about a factor of two. The technique may be applicable to similar retrieval algorithms for other greenhouse gas sensors with large data volumes, such as GeoCarb, GOSAT-3, and CO2M.

Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution

In this paper, several related estimation problems are addressed from a Bayesian point of view, and optimal estimators are obtained for each of them when some natural loss functions are considered. The problems considered are the estimation of a regression curve, a conditional distribution function, a conditional density, and even the conditional distribution itself. These problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, and parametric and nonparametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function commonly used in estimating a real function of the unknown parameter. The cornerstone of these Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate the results.

Sea Surface Skin Temperature Retrieval from FY-3C/VIRR

The visible and infrared scanning radiometer (VIRR) onboard the Fengyun-3C (FY-3C) meteorological satellite has 11 μm and 12 μm channels, which are capable of sea surface temperature (SST) observations. This study is based on atmospheric radiative transfer modeling (RTM) by applying Bayesian cloud detection theory and optimal estimation (OE) to obtain sea surface skin temperature (SSTskin) from VIRR in the Northwest Pacific. The inter-calibration of FY-3C/VIRR 11 μm and 12 μm brightness temperature (BT) is carried out using the Moderate Resolution Imaging Spectroradiometer (MODIS) as the reference sensor. Bayesian cloud detection and OE SST retrieval with the calibration BT data is performed to obtain SSTskin. The SSTskin retrievals are compared with the buoy SST with a temporal window of 1 h and a spatial window of 0.01°. The bias is −0.12 °C, and the standard deviation is 0.52 °C. Comparisons of the retrieved SSTskin with the AVHRR (Advanced Very High Resolution Radiometer) SSTskin from European Space Agency Sea Surface Temperature Climate Change Initiative (ESA SST CCI) project show the bias of 0.08 °C and the standard deviation of 0.55 °C. The results indicate that the VIRR SSTskin are consistent with AVHRR SSTskin and buoy SST.

Active recursive Bayesian inference using Rényi information measures

• An active recursive inference method based on Rényi information measures. • Posterior probability changes can enhance both query optimization and stopping criterion in the RBI process. • Using Rényi entropy and α -divergence unifies the multi-objective RBI problem. • The proposed unified framework enhances both speed and accuracy, in the presence of an adversarial prior. Recursive Bayesian inference (RBI) framework provides optimal Bayesian latent variable estimates in real-time settings with streaming noisy observations. Active RBI attempts to effectively select queries that lead to more informative observations to reduce uncertainty until the process is stopped at a certain confidence level. However, the mismatch between querying objective and stopping criterion creates the conundrum of improving the performance of one objective at the expense of deteriorating the other. Moreover, conventional active querying methods stagger in the presence of misleading prior information. Inspired by information theoretic approaches, we propose an active RBI framework where query and stopping criterion are jointly selected through a unified objective based on Rényi information measures. The proposed unified formulation enables us to jointly enhance speed and accuracy in the RBI process. We theoretically demonstrate that the proposed objective encourages exploration in the presence of misleading prior. Furthermore, we motivate our framework by proving a geometrical representation for active querying and decision making on a probability simplex. We provide empirical and experimental studies on two applications including restaurant recommendation and brain-computer interface (BCI) typing systems and demonstrate that our method outperforms comparable approaches by improving both speed and accuracy.

Retrieval of Sea Surface Temperature From HY-1B COCTS

The Chinese Ocean Color and Temperature Scanner (COCTS) on board HY-1 series satellites has two thermal infrared channels with the spectrum range of 10.30-11.40 &#x03BC;m and 11.40-12.50 &#x03BC;m for sea surface temperature (SST) observations. To reprocess the Haiyang-1B (HY-1B) COCTS SST, the Bayesian cloud detection and optimal estimation (OE) SST retrieval were applied to COCTS data in this study. The Bayesian cloud detection algorithm that has been developed is based on the Bayes&#x2019; theorem and uses simulation of COCTS observations. The MODerate resolution atmospheric TRANsmission (MODTRAN) model was used for simulation of COCTS brightness temperatures. SSTs were retrieved from COCTS by OE from 2009 to 2011 in the northwest Pacific. Comparison of COCTS OE SST with in situ SST showed that the COCTS SSTs are cooler than buoy measurements by &#x2013;0.23 &#x00B0;C on average, and the standard deviation (SD) of differences was 0.51 &#x00B0;C. A large component of the mean difference is attributable to the cool skin effect at the ocean surface (typically &#x2013;0.15 to &#x2013;0.2 &#x00B0;C), the remainder being attributable to simulation and calibration biases. The mean difference of COCTS OE SST with matched skin temperatures from the Advanced Along Track Scanning Radiometer (AATSR) is closer to zero, being &#x2013;0.09 &#x00B0;C, with a SD of 0.49 &#x00B0;C. These validation results of COCTS OE SST demonstrate that Bayesian cloud detection and OE SST retrieval algorithm work well for improving COCTS SST accuracy, and show the potential of these methods to help develop SST products for operational HY-1 satellites, HY-1C and HY-1D.

Machine-Learning-Based Parameter Estimation of Gaussian Quantum States

We propose a machine learning framework for parameter estimation of single mode Gaussian quantum states. Under a Bayesian framework, our approach estimates parameters of suitable prior distributions from measured data. For phase-space displacement and squeezing parameter estimation, this is achieved by introducing Expectation-Maximization (EM) based algorithms, while for phase parameter estimation an empirical Bayes method is applied. The estimated prior distribution parameters along with the observed data are used for finding the optimal Bayesian estimate of the unknown displacement, squeezing, and phase parameters. Our simulation results show that the proposed algorithms have estimation performance that is very close to that of Genie Aided Bayesian estimators, that assume perfect knowledge of the prior parameters. In practical scenarios, when numerical values of the prior distribution parameters are not known beforehand, our proposed methods can be used to find optimal Bayesian estimates from the observed measurement data.

LOS/NLOS Estimators for mmWave Cellular Systems With Blockages

Designers of millimeter wave (mmWave) cellular systems need to evaluate line-of-sight (LOS) maps to provide good service to users in urban scenarios. In this letter, we derive estimators to obtain LOS maps in scenarios with potential blocking elements. Applying previous stochastic geometry results, we formulate the optimal Bayesian estimator of the LOS map using a limited number of actual measurements at different locations. The computational cost of the optimal estimator is derived and is proven to be exponential in the number of available data points. An approximation is discussed, which brings the computational complexity from exponential to quasilinear and allows the implementation of a practical estimator. Finally, we compare numerically the optimal estimator and the approximation with other estimators from the literature and also with an original heuristic estimator with good performance and low computational cost. For the comparison, both synthetic layouts and a real layout of Chicago have been used.

A Bayesian framework for deriving sector-based methane emissions from top-down fluxes

Atmospheric methane observations are used to test methane emission inventories as the sum of emissions should correspond to observed methane concentrations. Typically, concentrations are inversely projected to a net flux through an atmospheric chemistry-transport model. Current methods to partition net fluxes to underlying sector-based emissions often scale fluxes based on the relative weight of sectors in a prior inventory. However, this approach imposes correlation between emission sectors which may not exist. Here we present a Bayesian optimal estimation method that projects inverse methane fluxes directly to emission sectors while accounting uncertainty structure and spatial resolution of prior fluxes and emissions. We apply this method to satellite-derived fluxes over the U.S. and at higher resolution over the Permian Basin to demonstrate that we can characterize a sector-based emission budget. This approach provides more robust comparisons between different top-down estimates, critical for assessing the efficacy of policies intended to reduce emissions.

Quantum state smoothing as an optimal Bayesian estimation problem with three different cost functions

Quantum state smoothing is a technique to estimate an unknown true state of an open quantum system based on partial measurement information both prior and posterior to the time of interest. In this paper, we show that the smoothed quantum state is an optimal state estimator; that is, it minimizes a risk (expected cost) function. Specifically, we show that the smoothed quantum state is optimal with respect to two cost functions: the trace-square deviation from and the relative entropy to the unknown true state. However, when we consider a related risk function, the linear infidelity, we find, contrary to what one might expect, that the smoothed state is not optimal. For this case, we derive the optimal state estimator, which we call the lustrated smoothed state. It is a pure state, the eigenstate of the smoothed quantum state with the largest eigenvalue.

Sports Event Level Measurement Indicator System Using Multisensor Information Fusion

In view of the imperfection of the measurement index of the level of sports events, the coverage rate of the measurement index is low and the stability is poor. Therefore, this paper puts forward a sports event level measurement index system based on multisensor information fusion. First, the simulated annealing algorithm is used to cluster the grouped sensors and fuse the optimal Bayesian estimation of compatible sensors. Second, the relative entropy measure method is used to expand the compatibility measure of the sensor information in the group, and the optimal Bayesian estimation value of the consistent measure test is obtained. The outliers are eliminated, the optimal fusion value is obtained by the overall weighted statistical fusion, and the alternative measure index system is constructed. Finally, analytic hierarchy process (AHP) is used to calculate the weight of each alternative index, so as to achieve the final measurement index. The results show that the standard deviation of clustering average energy consumption is low, and the energy loss is small. The system can effectively construct the measurement index, and the index coverage rate is as high as 95%.

Optimal Bayesian estimation for random dot product graphs

Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data.