Questions tagged [bayesian-probability]
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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 +...
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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 ...
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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 ...
- 11
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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(...
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Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$X_t=\begin{cases}
\frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\
\frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
- 21
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How does this Bayesian updating work $z_i=f+a_i+\epsilon_i$
$z_i=f+a_i+\epsilon_i$ ，where $f\sim N(\bar{f},\sigma_{f}^2)$ ; $a_i\sim N(\bar{a_{i}},\sigma_{a}^2)$; $\epsilon_i\sim N(0,\sigma_{\epsilon}^2)$. We can see the signals $\{z_i\}$ where $i\subseteq {1,...
- 11
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Curvature of randomly generated B-spline curve
I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...
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Derive equation for regularized logistic regression with batch updates
I am trying to understand this paper by Chapelle and Li "An Empirical Evaluation of Thompson Sampling" (2011). In particular, I am failing to derive the equations in algorithm 3 (page 6). ...
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Concentration of posterior probability around a tiny fraction of the prior volume
In the context of approximating the evidence $Z$ in a Bayesian inference setting
$$
Z = \int d\theta \mathcal L (\theta)\pi (\theta)
$$
with $\mathcal L$ the likelihood, $\pi$ the prior, John Skilling'...
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