Performing exact posterior inference in complex generative models is often difficult or impossible due to an expensive to evaluate or intractable likelihood function. Approximate Bayesian computation (ABC) is an inference framework that constructs an approximation to the true likelihood based on the similarity between the observed and simulated data as measured by a predefined set of summary statistics. Although the choice of appropriate problem-specific summary statistics crucially influences the quality of the likelihood approximation and hence also the quality of the posterior sample in ABC, there are only few principled general-purpose approaches to the selection or construction of such summary statistics. In this paper, we develop a novel framework for this task using kernel-based distribution regression. We model the functional relationship between data distributions and the optimal choice (with respect to a loss function) of summary statistics using kernel-based distribution regression. We show that our approach can be implemented in a computationally and statistically efficient way using the random Fourier features framework for large-scale kernel learning. In addition to that, our framework shows superior performance when compared to related methods on toy and real-world problems.
A second category consists of methods that construct summary statistics from auxiliary models. An example of this approach is indirect score ABC @cite_1 . Here, a score vector that is calculated from the partial derivatives of the auxiliary likelihood plays the role of the summary statistics. Motivated by the fact that the score of the observed data is zero when the auxiliary model parameters are set by maximum-likelihood estimation (MLE), the method searches the parameter space for values whose simulated data produce a score close to zero under the same auxiliary model parameters. Thus, the discrepancy measure between the observed and simulated data is defined in terms of scores of the simulated data at the parameter values estimated with MLE from the observed data. A detailed review of this class of approaches can be found in .
@cite_1
2021468256 2106596127 2128306087 2032616735
We consider estimation of network cardinality by distributed anonymous strategies relying on statistical inference methods. In particular, we focus on the relative Mean Square Error (MSE) of Maximum Likelihood (ML) estimators based on either the maximum or the average of M-dimensional vectors randomly generated at each node. In the case of continuous probability distributions, we show that the relative MSE achieved by the max-based strategy decreases as 1 M, independently of the used distribution, while that of the average-based estimator scales approximately as 2 M. We then introduce a novel strategy based on the average of M-dimensional vectors locally generated from Bernoulli random variables. In this case, the ML estimator, which is the Least Common Multiple (LCM) of the denominators of the irreducible fractions corresponding to the M elements of the average vector, leads to an MSE which goes exponentially to zero as M increases. We then discuss the effects of finite-precision arithmetics in practical dynamic implementations. Numerical experiments reveal that the MSE of the strategy based on Bernoulli trials is two order of magnitude smaller than that based on continuous random variables, at a price of one order of magnitude slower convergence time. of the number of bits required to write down the observed data, has been reformulated to extend the classical maximum likelihood principle. The principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models; i.e., of the model structures. The principle rests on a new way to interpret and construct a universal prior distribution for the integers, which makes sense even when the parameter is an individual object. Truncated realvalued parameters are converted to integers by dividing them by their precision, and their prior is determined from the universal prior for the integers by optimizing the precision. 1. Introduction. In this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data, when each observation is given with some precision. Instead of attempting at an absolutely shortest description, which would be futile, we look for the optimum relative to a class of parametrically given distributions. This Minimum Description Length (MDL) principle, which we introduced in a less comprehensive form in [25], turns out to degenerate to the more familiar Maximum Likelihood (ML) principle in case the number of parameters in the models is fixed, so that the description length of the parameters themselves can be ignored. In another extreme case, where the parameters determine the data, it similarly degenerates to Jaynes's principle of maximum entropy, [14]. But the main power of the new criterion is that it permits estimates of the entire model, its parameters, their number, and even the way the parameters appear in the model; i.e., the model structure. Hence, there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or, perhaps, over parameterized. Approximate Bayesian computation (ABC) has become a popular technique to facilitate Bayesian inference from complex models. In this article we present an ABC approximation designed to perform biased filtering for a Hidden Markov Model when the likelihood function is intractable. We use a sequential Monte Carlo (SMC) algorithm to both fit and sample from our ABC approximation of the target probability density. This approach is shown to, empirically, be more accurate w.r.t. the original filter than competing methods. The theoretical bias of our method is investigated; it is shown that the bias goes to zero at the expense of increased computational effort. Our approach is illustrated on a constrained sequential lasso for portfolio allocation to 15 constituents of the FTSE 100 share index. Approximate Bayesian computation (ABC) constitutes a class of computational methods rooted in Bayesian statistics. In all model-based statistical inference, the likelihood function is of central importance, since it expresses the probability of the observed data under a particular statistical model, and thus quantifies the support data lend to particular values of parameters and to choices among different models. For simple models, an analytical formula for the likelihood function can typically be derived. However, for more complex models, an analytical formula might be elusive or the likelihood function might be computationally very costly to evaluate. ABC methods bypass the evaluation of the likelihood function. In this way, ABC methods widen the realm of models for which statistical inference can be considered. ABC methods are mathematically well-founded, but they inevitably make assumptions and approximations whose impact needs to be carefully assessed. Furthermore, the wider application domain of ABC exacerbates the challenges of parameter estimation and model selection. ABC has rapidly gained popularity over the last years and in particular for the analysis of complex problems arising in biological sciences (e.g., in population genetics, ecology, epidemiology, and systems biology).