Questions tagged [dirac-delta]

Dirac delta is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.

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Can I find the joint probability, knowing that all marginals are Dirac Delta functions?

Given a joint probability density distribution $P(\boldsymbol{\alpha})$ with $\boldsymbol{\alpha}=(\alpha_1,\alpha_2,\alpha_3)$ a variable in $\mathbb{R}^{3}$, such that all marginals $P_i$ are Dirac-...
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MMSE estimator with dirac delta prior pdf

The question is as follows, it's mainly part 3 that I was having problem with. A discrete-valued parameter with the prior pdf $$p(x) = > \sum_{i=1}^2p_i\delta(x-i)$$ is measured with the additive ...
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How to implement a mixture model for Dirac Delta and Normal distributions?

How could I fit data with observations from one Dirac delta component and $n$ normal distributed components? Where $n$ usually is between 1 and 5. My prior knowledge is that one component really is a ...
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Bounding the difference between the PDF of $\hat\beta\sim\mathcal{N}(\beta,\sigma^2)$ and a Dirac delta $\delta_\beta$ in terms of the variance?

Let $\beta \in \mathbb{R}$, let $X$ be a continuous random variable, and let $\hat \beta \sim \mathcal{N}(\beta,\sigma^2)$. Let the pdf of a random variable $A$ be denoted by $p_A$. Consider the ...
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Applying Bayes' rule in a more technical way when densities don't exist

Say $y \mid x \sim \text{Normal}(Ax, B)$ and $x \sim \text{Normal}(c,D)$. Let's assume further that $y \in \mathbb{R}^1$ and $x \in \mathbb{R}^2$. To find $p(x \mid y)$ we can usually do \begin{...
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Metropolis-Hastings with a "Dirac transition"

I'm running the Metropolis-Hastings algorithm on a product space $\tilde E:=I\times E'$, where $I$ is a finite nonempty set and $E'=\bigcup_{i\in I}E'_i$ with $E'_i:=[0,1)^{d_i}$ for $i\in I$. Given ...
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Dirac Delta Function in Bayesian Optimization

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal ...
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joint density of mixed RVs: A random vector and its cardinality [closed]

Suppose $X$ and $Y$ are two continuous random vectors and $ m = |Y|$ is a discrete random variable that denotes the size of the continuous random vector $Y$, i.e. the number of its columns (the ...
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Dirac delta function for multivariate input (in the context of Gaussian processes)

Let's say we have a set of $N$ observations $D = \{\bf X, t\}$ where ${\bf X} = [{\bf x}_1, ..., {\bf x}_N]^T$ are the locations and ${\bf t} = [t_1, ..., t_N]^T$ are the targets. When applying a ...
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Joint conditional density of two iid exponential random variables conditioned on their sum

I have the following question: Suppose $X_1, X_2$ iid $\sim f_X(x)=\theta e^{-\theta x}1\{x\ge 0\}$ and $S=X_1+X_2$. What is $f_{X_1,X_2}(x_1,x_2|S=s)$? The solution from our exercise class to this ...
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Expectation with delta function

In Bishop on page 219, the equality below is given. Unfortunately, I fail to see how one finds the mean $\mu_{a}$ of this distribution. I know the definition of expectation operator, but how does one ...
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Correlation function of Wiener process (brownian)

The following is stated in a text I am using: Consider the Wiener process (Brownian process) $W(t)$. The Wiener process has no derivative $\xi(t) := \frac{d W}{d t}$, reflecting the fact that it ...
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What is the KL divergence of distribution from Dirac delta?

The Kullback–Leibler (KL) divergence of two continuous distributions $P(x)$ and $Q(x)$ is defined as $$D_{KL}(P \mid\mid Q) = \int_{X} P(x) \log{\left[\frac{P(x)}{Q(x)}\right]} dx$$ How can one ...
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how to perform marginalization of posterior to find the predicative distribution for bayesian logistic regression

I want to know how to show that $$p(a) = \int \delta(a - w^T x) q(w) dw$$ is gaussian, where $q$ is gaussian and $x$ is fixed, and $\delta$ is the dirac function. Everything below is just some ...
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How to define general covariance with a dirac delta with correct units

I want to define the covariance of a variable, say $X$, which is the integral over $Y(t)$: $$ {\rm Cov}[X,X'] = \int\int {\rm Cov} [aY(t), a'Y(t')] dt dt'.$$ Just as an example, let the variable $...
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Empirical probability and Dirac distribution

According to Deep Learning p.65(Ian Goodfellow and Yoshua Bengio and Aaron Courville, available online): (...) This can be accomplished by defining PDF using the Dirac delta function $\delta(x)$: ...
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Expectation of Dirac function

I am having some trouble understanding how to derive the expectation of white noise. Assuming we have $W(x)$ which is a continuous white noise process with zero mean and unit variance, Thus $$cov\big(...
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Role of Dirac function in particle filters

Particle approximations to probability densities are often introduced as a weighted sum of Dirac functions $$p(x) \approx \sum_{i=1}^N \omega^i \delta(x-x^i)$$ with the weights $$\omega^i \propto \...
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Dirac Delta function Notation

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is $\pi(x_{1:n}) = \frac{1}{N}\sum^...
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Should Dirac's delta function be regarded as a subclass of the Gaussian distribution?

In Wikidata it is possible to link probability distributions (like everything else) in an ontology, e.g., that the t-distribution is a subclass of the noncentral t-distribution, see, e.g., https://...
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Using Dirac Delta functions for estimating a probability distribution

I'm having some trouble understanding this slide. It's mentioned in the context of gaussian distributions. I sort of understand the Dirac delta "function". The main difficulty I'm having is with the ...
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indicator variable - dirac delta or step function

I am trying to solve the following equation, \begin{equation} = \int_{-\infty}^{\infty} \frac{1}{\sqrt{ (2\pi)^{k_{Y}} | \Sigma |}} \cdot \mathrm{exp} \{ -\frac{1}{2} (Y - Xm)^{T} \Sigma^{-1} (Y ...
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Dirac delta function in likelihood function

I have tried to understand this myself but what I have found on the internet so far has not helped. I have a likelihood function that for part of it has the following statement: d0 is the Dirac ...
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Characteristic function of the Dirac delta?

What is the characteristic function of the Dirac delta function? Is it $e^{i*0}=1$?
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Delta function in monte carlo sampling

I am confused by the dirac delta function in the formulation of monte carlo sampling. http://www.cs.ubc.ca/~arnaud/doucet_johansen_tutorialPF.pdf, for instance, defines in section 3.1 page 8 the ...
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Mutual information with a Dirac delta type pdf

What does the $MI(X,Y)$ convey about $Y$, when one of the probability distributions, $X$ is trivial and has all the probability concentrated at a single point?
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