The bayesian tag has no usage guidance.

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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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### Adaptive priors

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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244 views

### 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 ...

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### Bayesian model to estimate the parameter of a Bernoulli law

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 ...

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### 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 ...

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### 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 ...

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### 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 := ...

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229 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 ...

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### 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 ...

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### 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 ...

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76 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 ...

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73 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 $\theta$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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577 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 ...

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### 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 ...

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### 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 ...

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297 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 ...

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### Estimating Wiener process parameters

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, ...

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### Derivatives of conditional expectations

Let $X$, $Y$ and $Z$ be independent, real-valued random variables, probably with continuous density functions. Define $A = X + Y$ and $B = X + Z$. Consider the regular conditional expectation ...

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### What can be said about an infinite linear chain of conjugate prior distributions?

We can sample a discrete value from the multinomial distribution.
We can also sample the parameters of the multinomial distribution from its conjugate prior the dirichlet distribution.
Since the ...

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### In Bayesian statistics, must I use a marginalized prior in conjunction with a marginalized distribution?// [closed]

Suppose I have some sampling distribution g(x,y,z) which has been marginalized over some variables (say y and z) giving us the marginal distribution which we'll call gx(x).
Suppose I now wish to use ...

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### Are all probabilities conditional probabilities? [closed]

We know that $P(A\mid B) = \frac{P(A \cap B)}{P(B)}$. So $P(B) = P(A\mid B)P(A \cap B)$. Thus are all probabilities conditional probabilities? Can one make a probability more accurate by introducing a ...

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### Probability estimates for pairwise majority votes

This is related to the rank aggregation question I asked previously.
I have items $I_1, \ldots, I_N$ and the observations of a number of pairwise trials which pit pairs $I_i$ and $I_j$ against ...