Questions tagged [bayesian-probability]
The bayesian-probability tag has no usage guidance.
80
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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 ...
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18
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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 ...
- 203
3
votes
0
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64
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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 $$Y = \Phi(...
- 81
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0
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62
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Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$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)(...
- 21
0
votes
1
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101
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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,...
- 11
1
vote
0
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58
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 &...
- 21
0
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0
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33
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Likelihood ratio when true distribution is known
Let $X_{i}, i=1,...,N,$ follow a distribution $p(x)$. Now, we calculate the likelihood ratio between $p$ and a different distribution $q$ using samples $X_i$:
$$
\frac{p(X_i)}{q(X_i)},\ \text{or}\ \...
2
votes
1
answer
138
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Derive equation for regularized logistic regression with batch updates
I am trying to understand this paper by Chapelle and Li "An Empirical Evaluation of Thompson Sampling" (2011). In particular, I am failing to derive the equations in algorithm 3 (page 6). ...
- 59
2
votes
0
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54
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Concentration of posterior probability around a tiny fraction of the prior volume
In the context of approximating the evidence $Z$ in a Bayesian inference setting
$$
Z = \int d\theta \mathcal L (\theta)\pi (\theta)
$$
with $\mathcal L$ the likelihood, $\pi$ the prior, John Skilling'...
- 21
1
vote
1
answer
97
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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 ...
- 41
1
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1
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97
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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 ...
- 369
4
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2
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214
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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 ...
- 628
3
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1
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255
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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)...
- 303
3
votes
1
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343
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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 ...
- 131
1
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0
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33
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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 ...
- 552
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1
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143
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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}\...
- 89
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0
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49
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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''(...
- 407
0
votes
1
answer
121
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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 ...
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-1
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1
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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 ...
- 1
1
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1
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266
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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
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
1
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0
answers
77
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Bayesian inference of stochastically evolving model parameters
I have a question related to self-calibration in radio interferometry, but I will try to phrase it as generic as possible. I have a set of data points, $D = \{ d_{0, t_0}, d_{1, t_0}, ..., d_{M, t_0}, ...
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1
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1
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1k
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Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 41
3
votes
0
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151
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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 ...
- 91
1
vote
0
answers
51
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 ...
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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 ...
- 1,043
4
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0
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201
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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 (...
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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) ...
- 29
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0
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41
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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 ...
- 1
1
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0
answers
53
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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, ~...
- 11
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0
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88
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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 ...
- 121
1
vote
1
answer
107
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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 ...
- 13
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1
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391
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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 ...
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9
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3
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410
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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?
- 123
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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 ...
- 31
3
votes
1
answer
2k
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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, ...
- 41
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1
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303
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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 ...
- 1,058
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0
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539
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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. ...
- 41
3
votes
2
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539
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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...
- 123
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0
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362
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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))} ...
- 11
1
vote
1
answer
109
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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 ...
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votes
1
answer
697
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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,...
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0
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58
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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/...
0
votes
1
answer
143
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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 ...
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3
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1
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394
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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 ...
- 75
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1
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88
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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 ...
6
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0
answers
183
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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, ...
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2
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460
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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 ...
- 305
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0
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46
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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&...
- 157
1
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1
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72
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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 $...
- 135
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1
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133
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Learning a Gaussian from noisy observations
Is it possible to learn a distribution over the parameters ($K=\Sigma^{-1}$ and $\mu$) of a Gaussian from noisy measurements of $X$? (Starting with some appropriate prior over the parameters)
I know ...
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