Questions tagged [bayesian]
The approach and interpretation of probability associated with Bayes' theorem; usually used as opposed to the frequentist approach. It can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain. A Bayesian probabilist starts with some prior probability, and evaluates the evidence in favour of a hypothesis by combining the prior with the likelihood function of the observed data.
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Optimal strategy for clients calling a rate limited API?
SO the problem definition is:
Have X clients talking to a single API and that API has some upper bound, N, on the number of requests it will handle before it shapes i.e. the traffic and takes ten ...
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Can probability theory be made fully computable?
I’ve been reading about objective Bayesian theories lately and came upon the concept of universal priors and specifically, the Solomonoff prior. This seemed to answer my initial query about whether a ...
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Is the variance of a posterior distribution always less than the variance of the prior?
A prior distribution is given by distribution $f_\theta(\theta)$, with variance $\sigma^2$.
A posterior distribution is $g_\theta(\theta)=h(x,\theta)\cdot f_\theta(\theta)$, where $x$ is our sample.
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Reference for the expected MSE of the optimal estimator in a Gaussian mixture channel
Let $X$ be a $d$-dimensional random vector drawn from a Gaussian mixture
$$
X \sim \sum_{k=1}^K \pi_k \, \mathcal{N}_d(\mu_k, \Sigma_k),
$$
and let
$$
Y = X + N, \quad N \sim \mathcal{N}_d(0, \Sigma_N)...
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What is the meaning of $(I - A^T)^{-1}$ in the linear SEM formulation of the DAG-GNN paper?
In the DAG-GNN paper (Yu et al., NeurIPS 2019, paper link), the authors describe the linear Structural Equation Model (SEM) as follows:
$$
X = A^TX + Z,
$$
where A ${\in R^{m \times m}}$ is the ...
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Mathematically rigorous Bayesian sampling
Broadly: Given a parameter space $\Psi$ (say a Polish space) and a random variable $X$ whose distribution depends on $\psi \in \Psi$, how do we rigorously define sampling $X$, given the uncertainty in ...
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Any methods to find if a binary time series has seasonality OR NOT?
I have a time-series of events that go through a black box over a period of time. I need to determine if there is seasonality within the black box using these events, that are either True or False. ...
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Computing $BF_{01}$ using a Normal and a double exponential
I am currently studying bayesian theory and I was trying to tackle the following problem:
Let $X_1, \ldots, X_n \mid \theta, \sigma^2$ be i.i.d. $N(\theta, \sigma^2)$,
where both $\theta$ (real) and $\...
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Showing $\frac{\mathbb{P}\{\text{data}_t|\text{wrong pmf }\} }{\mathbb{P}\{\text{data}_t|\text{correct pmf }\}}\overset{\text{a.s}}{\to}0$.
Setup: Let $n\in\{2,3,...\}$. For each $t\in\{1,2,...\}$, suppose $X_t$ is distributed i.i.d., taking value $v\in
\{1,...,n\}$ with probability $p_v\in(0,1)$. Let $S_{vt}:=\sum_{\tau=1}^t 𝟙\{X_t=v\}$...
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Bayesian updating question
Say a test of 11 questions, and a person's score is from 0-11, and is categorized into level 1 (score 0-3), level 2 (score 4-7), and level 3 (score 8-11).
The priors of the three levels are $p_1, p_2, ...
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Hessians of Gaussian Random Fields at Minima and Stationary Points
I'm interested in carefully quantifying the number of local minima and stationary points of a Gaussian random field. While working through it, I came across a puzzling conclusion that doesn't seem ...
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Bayes theorem for non-binary events
How does one apply Bayes theorem on non-binary (i.e. not true/false or 1/0) Events?
If we use the example given in the Veritasium video on Bayes, but instead of each disease-test result (i.e. the ...
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Why is the Kalman filter so popular?
This is something I have been trying to understand.
Consider the following local level linear state space model:
Observation equation: $y_t = \mu_t + \epsilon_t$ where $\epsilon_t \sim N(0, \sigma_\...
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Metropolis-Hastings MCMC with a Chinese Restaurant Process proposal
I am using an implementation of Metropolis-Hastings algorithm to figure out the distribution of neuron clusterings, given the precalculated probability of synchronizations between different neurons.
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Asymptotic conditional likelihood ratio with continuous random variables
Setup. Let $p\in[0,1]$ be a continuous random variable with density $f(\cdot)$. Assume that $f$ is bounded, continuously differentiable, and has full support $[0,1]$.
Let $a_1$ and $a_2$ be distinct, ...