Questions tagged [bayesian-probability]
The bayesian-probability tag has no usage guidance.
83
questions
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What does the KL being symmetric tell us about the distributions?
Suppose two probability density functions, $p$ and $q$, such that $\text{KL}(q||p) = \text{KL}(p||q) \neq 0$. Intuitively, does that tell us anything interesting about the nature of these densities?
9
votes
1
answer
293
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Who introduced the term hyperparameter?
I am trying to find the earliest use of the term hyperparameter. Currently, it is used in machine learning but it must have had earlier uses in statistics or optimization theory. Even the multivolume ...
9
votes
1
answer
457
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 ...
8
votes
1
answer
318
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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 ...
7
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1
answer
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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(\...
6
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1
answer
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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 ...
6
votes
0
answers
193
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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, ...
5
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6
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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 ...
5
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3
answers
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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 ...
5
votes
1
answer
325
views
Bounding the sensitivity of a posterior mean to changes in a single data point
There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently ...
4
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2
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215
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Do these distributions have a name already?
In playing with some math finance stuff I ran into the following distribution and I was curious if someone had a name for it or has studied it or worked with it already.
To start, let $\Delta^n$ be ...
4
votes
1
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450
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Gaussian process kernel parameter tuning
I am reading on gaussian processes and there are multiple resources that say how the parameters of the prior (kernel, mean) can be fitted based on data,specifically by choosing those that maximize the ...
4
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0
answers
227
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Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed
Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with
$$X_n \sim \mathtt{Binomial}(n,1-q),$$
and
$$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$
where $q \in (...
4
votes
0
answers
636
views
Bayesian Networks and Polytree
I am a bit puzzled by the use of polytree to infer a posterior in a Bayesian Network (BN).
BN are defined as directed acyclic graphs. A polytree is DAG whose underlying undirected graph is a tree. ...
3
votes
2
answers
607
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Parametrising a sparse orthogonal matrix
I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $AA^𝑇...
3
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2
answers
408
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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?
$$
\operatorname{var} (X^T X)
=
\...
3
votes
1
answer
267
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A quantity associated to a probability measure space
Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows:
The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)...
3
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1
answer
408
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Updating Geman and Geman (1984) on image restoration
I am reading the seminal paper
Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine ...
3
votes
1
answer
169
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Parameter estimation using bayesian update on moduli space?
Scientists take a set of data points, say in ${\mathbb R}^2$, and, assuming that this data should fit a polynomial of degree $d$ (or an exponential, etc.), they estimate parameters.
I would think ...
3
votes
1
answer
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Derivatives of conditional expectations
Let $X$, $Y$ and $Z$ be independent, real-valued random variables, probably with continuous density functions. Define $A = X + Y$ and $B = X + Z$. Consider the regular conditional expectation $E_Y(a,...
3
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1
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Bayesian Inference with Student-t likelihood
Suppose I've observed $x$ from a Student-t distribution with unknown $\mu$, and I'd now like to infer $\mu$. Since the t-distribution isn't exponential family, there's no conjugate prior available, ...
3
votes
0
answers
69
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Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
3
votes
0
answers
152
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Minimizing an f-divergence and Jeffrey's Rule
My question is about f-divergences and Richard Jeffrey's (1965) rule for updating probabilities in the light of partial information.
The set-up:
Let $p: \mathcal{F} \rightarrow [0,1]$ be a ...
3
votes
0
answers
82
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Have stick-breaking priors with non-iid atoms been considered, and if not, why not?
Roughly speaking, a stick-breaking prior is a random discrete probability measure $P$ on a measurable space $\mathcal X$ of the form
$$P=\sum_{j\ge1}w_j\delta_{\theta_j}$$
where $(w_j)_{j\ge1}$ is a ...
2
votes
3
answers
34k
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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 ...
2
votes
2
answers
461
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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?...
2
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1
answer
175
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Derive equation for regularized logistic regression with batch updates
I am trying to understand this paper by Chapelle and Li "An Empirical Evaluation of Thompson Sampling" (2011). In particular, I am failing to derive the equations in algorithm 3 (page 6). ...
2
votes
1
answer
505
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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 ...
2
votes
1
answer
87
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 $\theta$ ...
2
votes
0
answers
134
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Sum of arrival times of Chinese Restaurant Process (CRP)
Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
2
votes
0
answers
59
views
Concentration of posterior probability around a tiny fraction of the prior volume
In the context of approximating the evidence $Z$ in a Bayesian inference setting
$$
Z = \int d\theta \mathcal L (\theta)\pi (\theta)
$$
with $\mathcal L$ the likelihood, $\pi$ the prior, John Skilling'...
2
votes
0
answers
95
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Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?
Nowadays there are many papers on the number theory using heuristics.
I have read some of them.
But I have no clear understanding of the Bayesian Probability(subjective probability).
The concept of ...
2
votes
0
answers
102
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Convergence of Bayesian posterior
Let $\Delta [0,1]$ denote the set of all probability distributions on the unit interval.
Let $\mu \in \Delta [0,1]$ denote an arbitrary prior. Importantly, $\mu$ does not necessarily admit a density ...
2
votes
1
answer
717
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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,...
2
votes
2
answers
500
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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 ...
2
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0
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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 ...
2
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0
answers
256
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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 ...
2
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0
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1k
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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, ...
1
vote
1
answer
109
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Bayesian inverse problems on non-separable Banach spaces
I am now studying Bayesian inverse problems. In the note of Dashti and Stuart https://arxiv.org/abs/1302.6989, they mentioned that "... when considering a non-separable Banach space $B$, it is ...
1
vote
1
answer
305
views
Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
1
vote
1
answer
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Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
1
vote
1
answer
74
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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
vote
1
answer
117
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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
vote
1
answer
392
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)=...
1
vote
1
answer
125
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Conditional Gaussians in infinite dimensions
I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...
1
vote
1
answer
123
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Conditional density for random effects prediction in GLMM
I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of ...
1
vote
1
answer
329
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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 \,.
...
1
vote
1
answer
730
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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 ...
1
vote
1
answer
45
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 ...
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2
answers
1k
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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 ...