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### Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models. Consider a hidden Markov model (HMM) with ...
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### Exploiting conditional independence for inference in Bayesian networks

How is conditional independence used for making probabilistic inference in Bayes networks easier or more efficient? For example, given the following Bayes network: Let's say I want to compute <...
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### Bayesian estimation with lower dimensional prior

Let a statistical model of a random variable $X$ with parameter $\theta \in R^m$ be represented by a density function $p(X=x|\theta)$. Assume that the prior, $q(\cdot)$, is on a lower dimensional ...
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A lot of recent literature in Bayesian approach to inverse problems involves Adaptive priors, i.e - priors that depend on noise level. A lot of articles deal with optimization of contraction rates ...
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### Bayes statistics precisely formulated

I am trying to learn something about Bayesian statistics, however, I am struggling already with the simplest equations and, moreover, with the very basic questions: What are we given? What is our goal?...
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Suppose we have iid boolean variables $X_1,...,X_T = X_{1:T}$ and the associated deterministic parameters $k_1,...,k_T=k_{1:T}$ and $c_1,...,c_T=c_{1:T}$, where for each $t \in \mathbb{N}$, $k_{t} \in ... 1answer 254 views ### Base schemes and Bayesian priors One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme. In Bayesian ... 0answers 42 views ### A canonical example of the non-existence of predictive probability distribution Section 3 of Fortini et al. (2000) states that Given$(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of$x_n$given$(x_1, \dots, x_{n-1})$with respect to$P$need not ... 1answer 141 views ### Orthogonal decomposition of conditional expectations Suppose I have a random variable$x$and a set of conditional distributions on$x$. Here is an example where the conditionals are nested: $$q_1 := E(x|y_1), \quad q_2 := E(x|y_1,y_2),\quad q_3 := E(x|... 1answer 258 views ### Rate of convergence of Bayesian posterior Suppose a data generating process (DGP) is parameterized by some unknown parameter \theta_0, say P_{\theta_0}, and we want to estimate the value of \theta_0 using Bayesian method. Let \pi(\... 1answer 201 views ### In what sense is the Bayesian posterior mean a “convex combination”? I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate x \in \mathbb{R}^n from two signals with zero mean, normally ... 2answers 443 views ### Probability spaces involved in using Bayesian Inference I am currently reading "Statistical and Inductive Inference by Minimum Message Length" by C.S. Wallace. In this, Wallace gives a fairly informal account of Bayesian Inference which, in the case ... 0answers 78 views ### Simultaneous multiple perturbations in Markov chain Monte Carlo I'm coding a McMC algorithm for geophysical applications. Using the Metropolis-Hastings scheme to accept/reject the proposed models is smth that i thought i completely understood, but i don't. To be ... 1answer 75 views ### What is the problem with this model parameter estimation algorithm? In a statistical model with parameters \theta and unobserved laten variables Z, the model likelihood is$$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$The standard way to estimate$\theta$... 1answer 138 views ### Parameter estimation using bayesian update on moduli space? Scientists take a set of data points, say in${\mathbb R}^2$, and, assuming that this data should fit a polynomial of degree$d$(or an exponential, etc.), they estimate parameters. I would think ... 0answers 103 views ### Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods? Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ... 2answers 88 views ### What is the likehood function in the noise free observation case In the nonlinear Bayesian Tracking problem, if we consider the noise exists only in the state equation : x[k] = f(x[k-1],v[k-1]) where vk-1 here is an iid process noise sequence And we suppose that ... 1answer 372 views ### bayes theorem on histogram [closed] How can we apply Bayes theorem on histogram ? 2answers 378 views ### How to deal with this Chicken-And-Egg problem ? Let's imagine designing an odds pattern for a game, in which players bet for win or lose. Suppose the probablity of winning is$p$, thus the probablity of losing is$1-p$. Now imagine$n_1$people ... 1answer 593 views ### Exploiting conditional independence working with covariance matrices I have a Bayesian network where the number of nodes is potentially large. I've conditioned on some of the nodes (observed data) and I'm trying to draw samples from the distribution remaining nodes (... 1answer 100 views ### Continuous-time Markov chain to sample Bayesian posterior distribution Given a Bayesian network and evidence for the values of a subset of the variables, a standard question is to compute the posterior distribution on the remaining variables. The Gibbs sampling technique ... 2answers 5k views ### A “simple” explanation of the concept of D-separation in a Bayesian Network? Hello everyone. I'm looking for a "simple" explanation of the concept of D-separation in a Bayesian Network. As far as I know the definition is "two variables (nodes) in the network are D-Separated ... 1answer 298 views ### Conditional probability and independence Suppose that we have vectors of events$\{H_1,...,H_n\}$and$\{D_1,...,D_m\}$. Consider the following two sets of conditions: Condition set 1 (1)$P(H_i H_j)=0$for any$i\neq j$and$\sum_iP(H_i)=...
Consider a Wiener process with zero drift, infintesimal variance $\sigma^2$, and an unknown starting value $\nu$. That is, \begin{align} Y_t \sim \mathcal{N}(\nu, t\sigma^2). \end{align} Now, ...