Questions tagged [normalizing-constant]
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14 questions
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Regarding samples gotten from MCMC
In one article explaining MCMC, I once read the following statement.
The idea of sampling methods is the following. Let’s assume first
that we have a way (MCMC) to draw samples from a probability
...
- 3,089
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1
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Do the following normalizing constants cancel out in Reversible Jump ratio?
We know that a Strauss Point Process has density
$$p(x_{1}, x_{2},..., x_{K})\propto \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$
where ...
- 2,286
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Exponential family admissibility of base measure, sufficient statistic and log partition function
Let
$$ f(y | \eta) = h(y) \exp\left( \eta^\top T(y) + A(\eta) \right)$$
be the exponential family with base density/pmf $h$, sufficient statistic $T$, log partition function $A$ and natural parameter $...
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Monte Carlo Approximation of a Normalizing Constant [duplicate]
I know that one can approximate expectations of a function with respect to a pdf as such
$$
\mathbb{E}_{p(x)}[\phi(x)] = \int \phi(x) p(x) dx \approx \frac{1}{N}\sum_{i=1}^N \phi(x^{(i)}) \qquad\qquad ...
- 667
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Why can't we use Monte carlo approximation for normalising constant in Baye's theorem
The normalising constant $Z$ in baye's theorem is the probability that the model generates the data $D$. $$\begin{align}P(D) &= \int P(D|\theta)P(\theta)d\theta \\
&= E_{\theta \sim p(\theta)}[...
- 1,116
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Why is the normalisation constant in Bayesian not a marginal distribution
The formula for Baye's rule is as follows $$p(\theta |D) = \frac{p(D|\theta)p(\theta)}{\int p(D|\theta)p(\theta)d\theta}$$
where $\int p(D|\theta)p(\theta)d\theta$ is the normalising constant $z$. ...
- 1,116
3
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Simplifying modified Bessel function of the first kind
The modified Bessel function of the first kind shows up in the normalizing constant of a lot of random variables (e.g. the normal product distribution, the noncentral chi-square distribution, the ...
- 21.5k
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How do I compute the closed form normalizing constant for this distribution?
The funnel distribution for random variable $X = (x_1,x_2,..,x_D)$is
$$P(X) = N(x_1|0,9)\prod_{d=2}^D N(x_d | 0,exp(x_1))$$
The closed form normalizing constant for normal distribution $N(x|0,\...
- 701
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551
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weighted sum of posterior Dirichlet distributions
I have the following distribution:
$q(\vec\theta) = \frac{\sum_k \alpha_k}{\sum_k \beta_k \alpha_k} (\sum_k\theta_k\beta_k) \frac{\Gamma(\sum_k \alpha_k)}{\prod_k\Gamma(\alpha_k)} \prod_k \theta_{k}^{...
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What is the relationship between the normalization constants of the normal distribution and the (Inverse-)Wishart distribution?
I was looking at the probability density function of the multivariate normal distribution and that of the inverse-Wishart distribution:
$$
\begin{array}{rccll}
p_{\mathcal{N}}\left(\mathbf{x}; \...
- 398
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1
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141
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Bayesian inference, finding unknown function of parameter
I'm completely new to Baayesian statistics and am having some questioning about the following:
Given a probability distribution defined by the pdf
$\pi(x|\theta) = C(\theta)x^2 e^{-\theta x^{3}}$ ...
- 1
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Deriving the normalizing constant for the multivariate Gaussian
I am trying to derive the normalizing constant for the multivariate Gaussian. The book I'm following suggests diagonalizing the covariance matrix and then using a change of variables.
So, we consider ...
- 2,259
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Dropping the normalization constant in Bayesian inference [duplicate]
Could anyone please explain to me (or provide a situation) where one would drop the normalising constant (or marginal probability term) from Bayes rule when performing Bayesian Inference?
New to ...
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Sampling a random binary matrix with "Gaussian" probability distribution
Let $A_{ij}$ be a $n\times n$ random binary matrix with probability mass function $P(A)$ given by
$$
\log P(A)=-\frac 12 \mathrm{tr}\left[\left(A-M\right)^TV\left(A-M\right)\right] + C,
$$
where $M$ ...
- 1,017