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Peakedness of conditioned distributions

I'm struggling to prove the following: Let $X,Y,Z$ be iid random variables (with pdf $f$) that are unimodal and symmetric around 0. Then $X \mid (X = Z)$ is more peaked than $X \mid \left(\tfrac12 X +...
Pavel Kocourek's user avatar
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18 views

Equality Issue in Deriving Covariance Update for Kalman Filter

I am currently working on deriving the Kalman Gain from the covariance of the updated state and have encountered an equality issue that I am unable to resolve. Below are the derivation steps and the ...
user528329's user avatar
1 vote
1 answer
76 views

Bayes classifiers with cost of misclassification

A minimum ECM classifier disciminate the features $\underline{x}$ to belong to class $t$ ($\delta(\underline{x}) = t$) if $\forall j \ne t$: $$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne ...
BiasedBayes's user avatar
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0 answers
13 views

Understanding relation of 2 dependent, integral equations which are nested in a Bayesian Expectation

I'm trying hard to try understand the recursive nature between two equations in a recent macroeconomics paper, but my question mainly relates to how mathematically such recursive equations can depend ...
Justin Lee's user avatar
1 vote
0 answers
67 views

Gibbs Priors form a Martingale

I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...
BayesRayes's user avatar
2 votes
1 answer
209 views

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 ...
Grandes Jorasses's user avatar
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1 answer
89 views

Existence and uniqueness of a posterior distribution

I am wondering about the existence and uniqueness of a posterior distribution. While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...
CoilyUlver's user avatar
9 votes
1 answer
313 views

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 ...
ACR's user avatar
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30 views

Canonical information geometry for probability distributions on different parameter spaces

I am interested in a canonical information geometry on spaces of probability distributions containing distributions with different parameter spaces. Let me give some context and practical motivation ...
Lance's user avatar
  • 203
3 votes
0 answers
69 views

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(...
Jarwin's user avatar
  • 81
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0 answers
72 views

Probability distribution for a Bayesian Update

I am struggling with a process like this: $$X_t=\begin{cases} \frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\ \frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
DreDev's user avatar
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1 answer
102 views

How does this Bayesian updating work $z_i=f+a_i+\epsilon_i$

$z_i=f+a_i+\epsilon_i$ ,where $f\sim N(\bar{f},\sigma_{f}^2)$ ; $a_i\sim N(\bar{a_{i}},\sigma_{a}^2)$; $\epsilon_i\sim N(0,\sigma_{\epsilon}^2)$. We can see the signals $\{z_i\}$ where $i\subseteq {1,...
yunfan Yang's user avatar
1 vote
0 answers
103 views

Curvature of randomly generated B-spline curve

I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...
MatEZ's user avatar
  • 31
2 votes
1 answer
196 views

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). ...
denvercoder9's user avatar
2 votes
0 answers
60 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'...
long_john's user avatar
1 vote
1 answer
120 views

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 ...
T. Huynh's user avatar
1 vote
1 answer
136 views

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 ...
user2379888's user avatar
4 votes
2 answers
216 views

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 ...
Jess Boling's user avatar
3 votes
1 answer
270 views

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)...
Ali Taghavi's user avatar
4 votes
1 answer
531 views

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 ...
john's user avatar
  • 141
1 vote
0 answers
35 views

Estimation of probability matrix from samples at different time intervals

I am given discrete-time Markov chain that evolves on a finite subset $\{1,\dots,n\}$. This Markov chain is time-homogeneous and has a transition matrix $P$ that I want to estimate. Let $X_t$ be the ...
N. Gast's user avatar
  • 552
0 votes
1 answer
170 views

CLT for random variables with positive support (e.g. exponential)

I have a bunch of iid $\{X_i\}$ with $X_i \sim \exp(\lambda)$ - let's say $\lambda = 1$. Now, classic version of CLT tells me: \begin{equation} \sqrt{n}\left(1-\bar{X}_n\right) \rightarrow \mathcal{N}\...
qwert's user avatar
  • 89
1 vote
0 answers
62 views

2d interpolation minimizing the integral of the norm of the Hessian

It is well known that cubic interpolation is the solution of the interpolation problem that minimizes the integral of the square of the second derivative: $$ min_{f \text{ s.t. } f(x_i)=y_i} \int (f''(...
Bernard 's user avatar
0 votes
1 answer
162 views

Lower bound for reduced variance after conditioning

Let $X$ be a random variable with variance $\tau^2$ and $Y$ be another random variable such that $Y-X$ is independent of $X$ and has mean zero and variance $\sigma^2$. (One can think of $Y$ as a noisy ...
Nima's user avatar
  • 3
-1 votes
1 answer
55 views

Linear operator over a simplex space in a multinomial distribution parameter estimation problem

This is actually a variant of a well-known problem of how the parameters of a multinomial distribution can be estimated by maximum likelihood, and this arises from a final year project I undertook ...
Hephaes's user avatar
1 vote
1 answer
324 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 ...
RandomMatrices's user avatar
1 vote
0 answers
78 views

Bayesian inference of stochastically evolving model parameters

I have a question related to self-calibration in radio interferometry, but I will try to phrase it as generic as possible. I have a set of data points, $D = \{ d_{0, t_0}, d_{1, t_0}, ..., d_{M, t_0}, ...
Sketos's user avatar
  • 29
1 vote
1 answer
2k views

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)=\...
wuhanichina's user avatar
3 votes
0 answers
164 views

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 ...
jw7642's user avatar
  • 91
1 vote
0 answers
56 views

Quantitative bounds on convergence of Bayesian posterior

Let $Y$ be a random variable in $[0,1]$, and let $X_1, X_2, \ldots$ be a sequence of random variables in $[0,1]$. Suppose that the $X_i$'s are conditionally i.i.d given $Y$ ; in other words, I'd like ...
sd234's user avatar
  • 173
2 votes
0 answers
96 views

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 ...
gualterio's user avatar
  • 1,073
4 votes
0 answers
228 views

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 (...
as1's user avatar
  • 91
-1 votes
1 answer
339 views

Proving the existence of a symmetric Bayesian Nash equilibrium

I am currently faced with the following question: Consider the public goods game. Suppose that there are $I > 2$ players and that the public goods is supplied (with benefit of 1 for all players) ...
user157299's user avatar
0 votes
0 answers
45 views

restriction of a formula with matrix inverse multiplied by a vector

I'm trying to reproduce a proof from this paper but I'm stuck in one point (Lemma 6). The general subject is bayesian model for multi-armed bandit problem solved with Thompson sampling. I think I ...
Martyna's user avatar
1 vote
0 answers
60 views

Bayesian posterior consistency when prior distribution is induced by a diffusion

Let $\Pi_{b,\sigma}$ be a prior distribution on $\{z_t\}_{t<T}\in C_0[0,T]$ induced by the following diffusion: \begin{align} d\tilde z_t&=b(\tilde z_t,t)dt+\sigma(\tilde z_t,t) dW_t, ~...
user467491's user avatar
2 votes
0 answers
111 views

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 ...
user_newbie10's user avatar
1 vote
1 answer
130 views

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 ...
e4e5ke2's user avatar
  • 13
0 votes
1 answer
474 views

Optimal solution to cross entropy loss in the continuous case

This could be a simple question but I don't have a satisfying answer. Setup. Suppose that we have $K$ different classes, and consider cross entropy loss which maps a probability vector in the ...
Xi Wu's user avatar
  • 143
9 votes
3 answers
467 views

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?
HesterJ's user avatar
  • 123
3 votes
0 answers
82 views

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 ...
mathducky's user avatar
3 votes
1 answer
2k views

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, ...
Luke's user avatar
  • 41
5 votes
1 answer
330 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 ...
Ben Golub's user avatar
  • 1,058
4 votes
0 answers
651 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. ...
Bremen's user avatar
  • 41
3 votes
2 answers
631 views

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...
HesterJ's user avatar
  • 123
1 vote
0 answers
437 views

Gaussian Integrals over Spheres

I'm after a reference for an integral. In particular, I am looking a way to approximate or calculate the following: $$ \int \limits_{\| \theta \|_2 = 1} e^{(-(\theta - \mu)^T \Sigma (\theta - \mu))} ...
user550008's user avatar
1 vote
1 answer
115 views

The expectation of binary logistics regression with respect to Gaussian distribution

I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If ...
Dalek's user avatar
  • 37
2 votes
1 answer
732 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,...
Joe's user avatar
  • 151
1 vote
0 answers
64 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/...
Seok Jun Hong's user avatar
0 votes
1 answer
160 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 ...
lulu2612's user avatar
3 votes
1 answer
418 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 ...
rjm's user avatar
  • 75