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Questions tagged [bayesian-probability]

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3
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2answers
208 views

Multivariate normal concentration

If $X\sim N(0,\Sigma)$ for some $d$-dimensional normal distribution, then $X = \Sigma^{1/2} Z$ where $Z\sim (0,I)$. How to compute the following quantity? $$ \operatorname{var} (X^T X) = \...
1
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0answers
30 views

Bayesian methods in online setting

Imagine the following (very concrete) model: We have a series of random variables $x_k$ with values in $\lbrace 0, 1\rbrace$. We assume $x_k \mid p_k \sim \operatorname{Alt}(p_k),$ where $p_0 \sim R(0,...
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0answers
21 views

Bayesian parameter estimation

I am generally not that knowledgeable for math, so if my question is too broad or inaccurate, please let me know. I am currently reading a paragraph of one paper (https://www.fil.ion.ucl.ac.uk/spm/...
2
votes
2answers
142 views

Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors

Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
-1
votes
2answers
120 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 ...
0
votes
1answer
121 views

Shannon problem

Since a few days, I try in my research to model / formalize a source of Shannon a little weird, and I can't do it at all. First of all, I explain to you its operating principle and then I describe it ...
3
votes
1answer
227 views

Updating Geman and Geman (1984) on image restoration

I am reading the seminal paper Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine ...
0
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0answers
53 views

Probability distribution of the Hadamard ratio of two degenerate multivariate Gaussian distributions?

This question pertains to the theory of Hadamard/elementwise functions of multivariate r.v.s/random vectors, which is unfortunately not a very popular topic: References for the theory of Hadamard ...
0
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1answer
76 views

How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the marginal distribution of each eigenvalue, using whatever you can. Background In my ...
2
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3answers
20k 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 ...
0
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1answer
62 views

Learning a Gaussian from noisy observations

Is it possible to learn a distribution over the parameters ($K=\Sigma^{-1}$ and $\mu$) of a Gaussian from noisy measurements of $X$? (Starting with some appropriate prior over the parameters) I know ...
6
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0answers
118 views

Existence of stick breaking representations for random measures

The Dirichlet process has a roughly size ordered representation in terms of beta random variables, called a stick-breaking representation (Sethuraman, 1994). Similar results hold for the beta process, ...
1
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0answers
35 views

RMHMC sampling in non-parametric setup

The aim is to sample distributions using Fisher information (as mass matrix in Hamiltonian MCMC sampling). Details can be found in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.580&...
1
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1answer
82 views

Accounting for unobserved events in baysian learning

I wanted to use Bayes theorem to help me automate the task of deciding if I should ignore events, but I am not sure how to update the posterior if I do The simple story goes like this: An event $y_i$...
1
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1answer
54 views

A problem with elementary inequality involving probabilities and Brier scoring rule

I am trying to prove certain relations between certain values of the so called Brier inaccuracy measure (Brier scoring rule). Given a vector $p = (p_1, \ldots p_n)$, where $p_1 + \ldots p_n = 1$ and $...
1
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0answers
34 views

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 ...
1
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1answer
123 views

convergence of Bayesian posterior with non iid data

Let $(\epsilon_t)_t$ be a sequence of iid random variables, distributed according to the density $f:\mathbb{R}\to (0,\infty)$ and $$ x_t = q( \theta^\star, x_1,x_2, \ldots, x_{t-1}) + \epsilon_t \,. ...
9
votes
1answer
266 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 ...
0
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0answers
84 views

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 ...
1
vote
1answer
40 views

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 ...
2
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1answer
105 views

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 ...
2
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2answers
304 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 goal?...
8
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1answer
288 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 ...
1
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2answers
615 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 ...
2
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0answers
50 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 ...
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1answer
279 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|...
7
votes
1answer
605 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(\...
1
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0answers
87 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 ...
2
votes
1answer
79 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 $\theta$ ...
3
votes
1answer
154 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 ...
5
votes
1answer
1k views

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 ...
2
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0answers
181 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 ...
4
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6answers
1k views

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 ...
1
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1answer
867 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 (...
-1
votes
2answers
435 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 ...
1
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1answer
121 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 ...
1
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1answer
325 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)=...
3
votes
1answer
3k views

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 $E_Y(a,...
2
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0answers
921 views

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, ...
5
votes
3answers
1k views

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 ...
-4
votes
2answers
935 views

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 ...