All Questions
Tagged with bayesian-probability pr.probability
45
questions
1
vote
1
answer
76
views
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 $\forall j \ne t$:
$$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne ...
- 11
1
vote
0
answers
67
views
Gibbs Priors form a Martingale
I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...
- 29
2
votes
1
answer
209
views
Sum of arrival times of Chinese Restaurant Process (CRP)
Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
- 131
0
votes
1
answer
89
views
Existence and uniqueness of a posterior distribution
I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...
0
votes
0
answers
72
views
Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$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
0
votes
1
answer
102
views
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
2
votes
0
answers
60
views
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'...
- 21
1
vote
1
answer
136
views
Conditional Gaussians in infinite dimensions
I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...
- 379
4
votes
2
answers
216
views
Do these distributions have a name already?
In playing with some math finance stuff I ran into the following distribution and I was curious if someone had a name for it or has studied it or worked with it already.
To start, let $\Delta^n$ be ...
- 646
3
votes
1
answer
270
views
A quantity associated to a probability measure space
Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows:
The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)...
- 500
0
votes
1
answer
171
views
CLT for random variables with positive support (e.g. exponential)
I have a bunch of iid $\{X_i\}$ with $X_i \sim \exp(\lambda)$ - let's say $\lambda = 1$. Now, classic version of CLT tells me:
\begin{equation}
\sqrt{n}\left(1-\bar{X}_n\right) \rightarrow \mathcal{N}\...
- 89
0
votes
1
answer
162
views
Lower bound for reduced variance after conditioning
Let $X$ be a random variable with variance $\tau^2$ and $Y$ be another random variable such that $Y-X$ is independent of $X$ and has mean zero and variance $\sigma^2$. (One can think of $Y$ as a noisy ...
- 3
1
vote
1
answer
324
views
Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
1
vote
1
answer
2k
views
Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 41
3
votes
0
answers
164
views
Minimizing an f-divergence and Jeffrey's Rule
My question is about f-divergences and Richard Jeffrey's (1965) rule for updating probabilities in the light of partial information.
The set-up:
Let $p: \mathcal{F} \rightarrow [0,1]$ be a ...
- 91