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Register a new userOn the Consistency of Maximum Likelihood Estimators for Causal Network Identification
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Added by
Xiaotian Xie
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider the problem of identifying parameters of a particular class of Markov chains, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph. Incoming edges to a node in the graph indicate that the state of the node at a particular time instant is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular nodes state variable is a convex combination of the parental node states in the previous time instant and an additional Bernoulli noise random variable. This paper focuses on the problem of edge weight identification using Maximum Likelihood (ML) estimation and proves that the ML estimator is strongly consistent for two variants of the BAR model. We additionally derive closed-form estimators for the aforementioned two variants and prove their strong consistency.
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Network analysis needs tools to infer distributions over graphs of arbitrary size from a single graph. Assuming the distribution is generated by a continuous latent space model which obeys certain natural symmetry and smoothness properties, we establish three levels of consistency for non-parametric maximum likelihood inference as the number of nodes grows: (i) the estimated locations of all nodes converge in probability on their true locations; (ii) the distribution over locations in the latent space converges on the true distribution; and (iii) the distribution over graphs of arbitrary size converges.
We consider a dynamic version of the stochastic block model, in which the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution is modeled through a hidden Markov chain on the nodes memberships. We prove the consistency (as the number of nodes and time steps increase) of the maximum likelihood and variational estimators of the model parameters, and obtain upper bounds on the rates of convergence of these estimators. We also explore the particular case where the number of time steps is fixed and connectivity parameters are allowed to vary.
Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the `sparsity property, the intervals based on these estimators are larger than the standard interval by an order of magnitude. Furthermore, a simple asymptotic confidence interval construction in the `sparse case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense.
In this paper, we study the asymptotic normality of the conditional maximum likelihood (ML) estimators for the truncated regression model and the Tobit model. We show that under the general setting assumed in his book, the conjectures made by Hayashi (2000) footnote{see page 516, and page 520 of Hayashi (2000).} about the asymptotic normality of the conditional ML estimators for both models are true, namely, a sufficient condition is the nonsingularity of $mathbf{x_tx_t}$.
Consider a setting with $N$ independent individuals, each with an unknown parameter, $p_i in [0, 1]$ drawn from some unknown distribution $P^star$. After observing the outcomes of $t$ independent Bernoulli trials, i.e., $X_i sim text{Binomial}(t, p_i)$ per individual, our objective is to accurately estimate $P^star$. This problem arises in numerous domains, including the social sciences, psychology, health-care, and biology, where the size of the population under study is usually large while the number of observations per individual is often limited. Our main result shows that, in the regime where $t ll N$, the maximum likelihood estimator (MLE) is both statistically minimax optimal and efficiently computable. Precisely, for sufficiently large $N$, the MLE achieves the information theoretic optimal error bound of $mathcal{O}(frac{1}{t})$ for $t < clog{N}$, with regards to the earth movers distance (between the estimated and true distributions). More generally, in an exponentially large interval of $t$ beyond $c log{N}$, the MLE achieves the minimax error bound of $mathcal{O}(frac{1}{sqrt{tlog N}})$. In contrast, regardless of how large $N$ is, the naive plug-in estimator for this problem only achieves the sub-optimal error of $Theta(frac{1}{sqrt{t}})$.
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