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We propose a new class of financial volatility models, called the REcurrent Conditional Heteroskedastic (RECH) models, to improve both in-sample analysis and out-ofsample forecasting of the traditional conditional heteroskedastic models. In particular, we incorporate auxiliary deterministic processes, governed by recurrent neural networks, into the conditional variance of the traditional conditional heteroskedastic models, e.g. GARCH-type models, to flexibly capture the dynamics of the underlying volatility. RECH models can detect interesting effects in financial volatility overlooked by the existing conditional heteroskedastic models such as the GARCH, GJR and EGARCH. The new models often have good out-of-sample forecasts while still explaining well the stylized facts of financial volatility by retaining the well-established features of econometric GARCH-type models. These properties are illustrated through simulation studies and applications to thirty-one stock indices and exchange rate data. . An user-friendly software package together with the examples reported in the paper are available at https://github.com/vbayeslab., Comment: 47 pages, 17 figures, 22 tables

The Stochastic Volatility (SV) model and its variants are widely used in the financial sector while recurrent neural network (RNN) models are successfully used in many large-scale industrial applications of Deep Learning. Our article combines these two methods in a non-trivial way and proposes a model, which we call the Statistical Recurrent Stochastic Volatility (SR-SV) model, to capture the dynamics of stochastic volatility. The proposed model is able to capture complex volatility effects (e.g., non-linearity and long-memory auto-dependence) overlooked by the conventional SV models, is statistically interpretable and has an impressive out-of-sample forecast performance. These properties are carefully discussed and illustrated through extensive simulation studies and applications to five international stock index datasets: The German stock index DAX30, the Hong Kong stock index HSI50, the France market index CAC40, the US stock market index SP500 and the Canada market index TSX250. An user-friendly software package together with the examples reported in the paper are available at \url{https://github.com/vbayeslab}., Comment: 51 pages, 16 figure, 18 tables

Harmonic generation from solid surfaces is a promising tool for producing high energy attosecond pulses. We report shaping of the harmonic spectrum to achieve the bandwidth necessary for attosecond pulse generation. The shaping is demonstrated for lower as well as for higher harmonics using single and two-pulse pumping. The measured harmonic field autocorrelation function exhibits attosecond spikes in good agreement with the harmonic spectrum. Double slit experiments reveal a high spatial coherence of the harmonic beam., Comment: 9 pages, 8 figures

This article addresses the problem of efficient Bayesian inference in dynamic systems using particle methods and makes a number of contributions. First, we develop a correlated pseudo-marginal (CPM) approach for Bayesian inference in state space (SS) models that is based on filtering the disturbances, rather than the states. This approach is useful when the state transition density is intractable or inefficient to compute, and also when the dimension of the disturbance is lower than the dimension of the state. Second, we propose a block pseudo-marginal (BPM) method that uses as the estimate of the likelihood the average of G independent unbiased estimates of the likelihood. We associate a set of underlying uniform of standard normal random numbers used to construct each of the individual unbiased likelihood estimates and then use component-wise Markov Chain Monte Carlo to update the parameter vector jointly with one set of these random numbers at a time. This induces a correlation of approximately 1-1/G between the logs of the estimated likelihood at the proposed and current values of the model parameters. Third, we show for some non-stationary state space models that the BPM approach is much more efficient than the CPM approach, because it is difficult to translate the high correlation in the underlying random numbers to high correlation between the logs of the likelihood estimates. Although our focus has been on applying the BPM method to state space models, our results and approach can be used in a wide range of applications of the PM method, such as panel data models, subsampling problems and approximate Bayesian computation.

Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating $2^{J}$ terms, with $J$ the number of discrete variables. Currently, data augmentation methods are used to carry out inference for discrete copula and, in practice, the computation becomes infeasible when $J$ is large. Our article proposes two new fast Bayesian approaches for estimating high dimensional copulas with discrete margins, or a combination of discrete and continuous margins. Both methods are based on recent advances in Bayesian methodology that work with an unbiased estimate of the likelihood rather than the likelihood itself, and our key observation is that we can estimate the likelihood of a discrete copula unbiasedly with much less computation than evaluating the likelihood exactly or with current simulation methods that are based on augmenting the model with latent variables. The first approach builds on the pseudo marginal method that allows Markov chain Monte Carlo simulation from the posterior distribution using only an unbiased estimate of the likelihood. The second approach is based on a Variational Bayes approximation to the posterior and also uses an unbiased estimate of the likelihood. We show that Monte Carlo and randomised quasi Monte Carlo methods can be used with both approaches to reduce the variability of the estimate of the likelihood, and hence enable us to carry out Bayesian inference for high values of $J$ for some classes of copulas where the computation was previously too expensive. Our article also introduces {\em a correlated quasi random number pseudo marginal} approach into the literature. The methodology is illustrated through several real and simulated data examples., Comment: 63 pages and 25 figures

The pseudo-marginal (PM) approach is increasingly used for Bayesian inference in statistical models, where the likelihood is intractable but can be estimated unbiasedly. %Examples include random effect models, state-space models and data subsampling in big-data settings. Deligiannidis et al. (2016) show how the PM approach can be made much more efficient by correlating the underlying Monte Carlo (MC) random numbers used to form the estimate of the likelihood at the current and proposed values of the unknown parameters. Their approach greatly speeds up the standard PM algorithm, as it requires a much smaller number of samples or particles to form the optimal likelihood estimate. Our paper presents an alternative implementation of the correlated PM approach, called the block PM, which divides the underlying random numbers into blocks so that the likelihood estimates for the proposed and current values of the parameters only differ by the random numbers in one block. We show that this implementation of the correlated PM can be much more efficient for some specific problems than the implementation in Deligiannidis et al. (2016); for example when the likelihood is estimated by subsampling or the likelihood is a product of terms each of which is given by an integral which can be estimated unbiasedly by randomised quasi-Monte Carlo. Our article provides methodology and guidelines for efficiently implementing the block PM. A second advantage of the the block PM is that it provides a direct way to control the correlation between the logarithms of the estimates of the likelihood at the current and proposed values of the parameters than the implementation in Deligiannidis et al. (2016). We obtain methods and guidelines for selecting the optimal number of samples based on idealized but realistic assumptions., Comment: 41 pages, 6 tables , 4 figures

This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This perspective sheds a new light on the foundations and the mathematical analysis of this class of methods. A key consequence is the equivalence between the backward and ancestral particle Markov chain Monte Carlo methods, and Gibbs sampling of a many-body Feynman-Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques to derive explicit non-asymptotic Taylor type series of the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler. These results provide sharp quan- titative estimates of the convergence properties of conditional particle Markov chain models with respect to the time horizon and the size of the systems. We illustrate the implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of conditional particle samplers, and explicit and non-asymptotic Lp-mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called SMC2) type particle methodologies.

This paper is concerned with Bayesian inference when the likelihood is analytically intractable but can be unbiasedly estimated. We propose an annealed importance sampling procedure for estimating expectations with respect to the posterior. The proposed algorithm is useful in cases where finding a good proposal density is challenging, and when estimates of the marginal likelihood are required. The effect of likelihood estimation is investigated, and the results provide guidelines on how to set up the precision of the likelihood estimation in order to optimally implement the procedure. The methodological results are empirically demonstrated in several simulated and real data examples., Comment: 27 pages, 2 figures

We construct an adaptive independent Metropolis-Hastings sampler that uses a mixture of normals as a proposal distribution. To take full advantage of the potential of adaptive sampling our algorithm updates the mixture of normals frequently, starting early in the chain. The algorithm is built for speed and reliability and its sampling performance is evaluated with real and simulated examples. Our article outlines conditions for adaptive sampling to hold and gives a readily accessible proof that under these conditions the sampling scheme generates iterates that converge to the target distribution., Comment: 35 pages and 6 figures