All Questions
Tagged with expected-value bayesian
39
questions
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how can predictive distributions be considered as expectations?
I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
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Bayesian Fubini Tonelli
I am working on a bayesian framework where I place a Gaussian Process on my function $f\sim GP$ and have data $D^n=\{(X_i,Z_i,W_i)\}^n$.
I then have the posterior measure $\mu(f|D^n)$. The posterior ...
- 90
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Expectation over cost-normalized Expected improvements
Are the following two expressions equivalent if we assume the independence of f(x) and C(x)?
$$
E\left[\frac{E\left[\max\left(f(x) - f(x^*), 0\right)\right]} {C(x)}\right]
$$
$$
\frac{E\left[\max\...
17
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Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?
In many areas, we encounter a situation where we compare averages of highly skewed statistics using two unequally sized samples. Typically, this happens when comparing items in an online store. For ...
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Estimating expected value with respect to posterior
I have a neural network and I need to calculate the following:
$$\mathbb{E}_{P(\theta|D)}[f(\theta)]=\frac{\sum_\theta P(D|\theta)P(\theta)f(\theta)}{\sum_\theta P(D|\theta)P(\theta)}$$
Where $f$, ...
- 197
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Why $\mathbb{E}_{(x, y) \sim \mathcal{D}}[f] = \mathbb{E}_{x \sim \mathcal{D}_{X}}[\mathbb{E}_{y \sim \mathcal{D}_{Y|x}}[f|X=x]]$ [duplicate]
I found this equality on p.6 in this document proving that Bayes Predictor is optimal (i.e. it achieves the minimal generalization risk) amongst al hypotheses:
$$
\mathbb{E}_{(x, y) \sim \mathcal{D}}[\...
- 720
2
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1
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Certain approximation in the setting of three expectation values does not make sense to me
I'm currently going through some lecture notes in the field of Bayes optimization and I'm currently looking at a expression looking like this:
$$\mathbb{E}_{x^*} \left[\mathbb{E}_y\left[\left\{\mathbb{...
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Understanding line in the derivation of KL divergence optimising function in Variational Bayes
I am following the derivation of Variational Bayes approach in David Blei's lecture notes, particularly equations (13 - 16).
In particular, the line:
$$
= E_q [\ \log_2 q(Z) ]\ - E_q \left[\ \log_2 \...
- 143
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32
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How to turn an expectation $E[A]$ into a conditional expectation, e.g., $E[A|B=1]$?
How can you turn an expectation $E[A]$ into a conditional expectation, e.g., $E[A|B=1]$, where:
A - continuous random variable, $A \in (-100, 1000)$
B - discrete r.v., $B \in {0, 1}$
A and B are ...
- 13
1
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Expected value of the log of the sum of beta distribution
Can someone help me compute this expression:
$$E[\log(X Y + (1-X)(1-Y))]$$
where $X\sim\operatorname{Beta}(a_1,b_1)$ and $Y\sim\operatorname{Beta}(a_2,b_2)$, and where $X$ and $Y$ are independent.
In ...
- 449
2
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1
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42
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Inconsistency of Bayesian Premium
Given:
The amount of a claim, $X$ is uniformly distributed on the interval $[0,\theta]$
The prior density of $\theta$ is $\pi(\theta) = \frac{500}{\theta^2}, \theta > 500$
Two claims, $x_1=400$ ...
- 31
2
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1
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198
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Bayesian Quadrature of Expectation w.r.t. Kernel Density Estimator Probability Density
I have a model of a physical system, $f(\pmb{x})$, where $f$ is the output of a mathematical model and $\pmb{x}$ are inputs to the model, which are available as observations. My goal is to find the ...
- 207
2
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1
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150
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Bayesian Quadrature to find expectation of unkown function w.r.t. known pdf
I am interested in estimating the integral $\int f(x) P(x) dx$, where $f(x)$ is an expensive function and $P(x)$ is has an analytic form. I would like to evaluate this with as few evaluations of $f(x)$...
- 207
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Why is this integral equal to $1$? (VBIL)
Let $p(y \mid \theta)$ be a likelihood and $\hat{p}_N(y \mid \theta)$ be an unbiased estimator of it.
In VBIL they define $z = \log \hat{p}_N(y \mid \theta) - \log p(y\mid \theta)$ and call its ...
- 2,236
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1
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Explain equation 1.80 in Pattern Recognition and Machine Learning, Bishop
$$E[L] = \sum_k \sum_j \int_{R_j} L_{k,j} p(x, C_k)$$
L is a loss function that returns a real value given a pair (i,j), with i as the index of true class, and j as the index of the predicted class of ...
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