Tagged Questions

0
votes
0answers
16 views

Evidential reasoning in Gaussian Bayesian Networks

I am working on Gaussian Bayesian Networks (GBN) i.e. the Bayesian Networks where all the random variables are continuous in nature. I am seriously trapped in the problem of evidential reasoning in ...
0
votes
1answer
56 views

Given known bivariate normal means and variances, update correlation estimate, $P(\rho)$, with new data?

I'm dealing with two correlated random variables which are modeled via a bivariate normal distribution. I have values for the means ($\mu_x, \mu_y$) and individual variances ($\sigma_x, \sigma_y$) of ...
2
votes
1answer
70 views

Priors for parameters of normal distribution leading to same results as frequentist formula

Given a sample vector $x$ of size $N$ from a normally distributed population. With frequentist methods the population mean is estimated as $\hat{\mu}=\frac{\Sigma{}x_i}{N}$, population sigma is ...
-1
votes
1answer
147 views

Calculating conditional probabilities given a bivariate gaussian

This is a continuation of my previous question. I have two classes, $C_1$ and $C_2$. $C_1$ is a bivariate Gaussian with mean $\mu = (0,0)$ and covariance $\Sigma = I$ $C_2$ is a bivariate ...
1
vote
0answers
59 views

Conjugate prior for a Gaussian model with shifted variance

Consider a set of observations $ \{ y_i \}$ and assume a Gaussian model for these data: $y_i \sim \mathcal{N}(\mu, \sigma^2)$. Suppose the mean parameter $\mu$ is known, but the variance parameter ...
1
vote
1answer
101 views

Bayesian updating - which distribution to use

I'd like to use Bayesian updating to form price expectations to be used in another model. I'm very new to this area, so your help would be highly appreciated. I'm not sure which distribution to use ...
2
votes
2answers
146 views

Identifying a conditional posterior for $\mu$ and $\sigma^2$

My Question Consider the model $X_1, \dots, X_n \overset{\mathrm{iid}}{\sim} \cal N(\mu, \sigma^2), (\mu, \sigma^2) \in (-\infty, \infty) \times (0, \infty)$ with a product prior pdf $\xi(\mu, ...
-1
votes
1answer
136 views

An incremental Gaussian mixture model

Question 1: Suppose that data is modelled by a mixture of K probability distributions which are actually Gaussians. $P(x_i|\theta_j)$ is the probability density of the j'th cluster, for which the ...
0
votes
2answers
171 views

Can anyone tell me why we always use the Gaussian distribution in Machine learning?

For example, we always assumed that the data or signal error is a Gaussian distribution? why? I have asked this question on stackoverflow, the link: ...
1
vote
0answers
88 views

How do I calculate the Bayes error of a multivariate normal Bayesian classifier?

I have a 4 dimensional feature and each of them are independent normal distributions. I want to calculate the bayesian error associated with this classifier. The covariance matrix and the mean have ...
1
vote
1answer
214 views

How do I complete the square with normal likelihood and normal prior?

How do I complete the square from the point I have left off at, and is this correct so far? I have a normal prior for $\beta$ of the form $p(\beta|\sigma^2)\sim \mathcal{N}(0,\sigma^2V)$, to get: ...
2
votes
0answers
532 views

Posterior derivation with prior as Normal-Gamma distribution

I am studying Gary Koop's Bayesian Econometrics , it is bit confusing in the beginning. I will reproduce some results from the textbook here in order to smoothly move to my question. For a simple ...
1
vote
1answer
147 views

How to calculate a posterior for the given model?

Suppose we have a joint distribution on vector $[\mathbf{x}, y]$: $$ p([y, \mathbf{x}] ) = \mathcal{N}\left(\begin{pmatrix} y \\ \mathbf{x}\end{pmatrix}| 0, \begin{pmatrix} k& \mathbf{v} \\ ...
3
votes
0answers
700 views

Multivariate normal posterior

This is a very simple question but I can't find the derivation anywhere on the internet or in a book. I would like to see the derivation of how one Bayesian updates a multivariate normal ...
1
vote
1answer
257 views

Constructing a Bayesian Classifier in R

Does anybody know the way to construct a Bayesian classifier in R for two bi-variate gaussian distributions of which the means and variances are known? The two classes are equi-probable and the ...
1
vote
2answers
228 views

Combination of two Gaussians

Very close to: Finding the Bayesian classifier for a bivariate Gaussian distribution Can somebody tell me the joint Gaussian of the following two Gaussians (in which $C_j = C_1 \text{ or } ...
0
votes
0answers
127 views

Bayesian inference on partitioned multivariate Gaussian

(A cross post after finding more appropriate tags here.) My question is on Bayesian inference of partitioned multivariate Gaussian. To make things easier, suppose there is a 2-dimensional Guassian, ...
1
vote
2answers
412 views

Finding the Bayesian classifier for a bivariate Gaussian distribution

Very close to: Joint Gaussian of two Gaussians I am trying to find the Bayesian classifier for two classes given by the following bivariate Gaussian distributions: $$p(x|c_1) = N(\mu_1, ...
4
votes
2answers
251 views

How to determine the marginal pdf, the posterior?

How to get the marginal pdf of $p(y)$? Do you just integrate out $p({\sigma}^{2})$? Say, the following joint distribution for $y \in {{R}^{d}}$ and ${{\sigma }^{2}}\in {{R}^{d}}$ IG: means inverse ...
3
votes
2answers
179 views

Bayesian updating using $n$ noisy observations of Brownian motion

I am very new to Bayesian inference and can't figure out what may be an elementary problem. Also, please forgive me if I am screwing up the notation -- this is my first foray into Bayesian ...
1
vote
0answers
115 views

Inferring a Gaussian from noisy data

Assume a noise comes from a specific point on a line, noise which I can detect but not completely accurately. My uncertainty we assume to be Gaussian. I want to gather evidence about the real ...
3
votes
3answers
210 views

What is the correct posterior when data are sufficient statistics?

Say you have N observations that are iid. $$ \forall i, \quad p(X_i=x_i|\mu,\sigma,I) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{1}{2\sigma^2}(x_i-\mu)^2\right)$$ then $$ ...
6
votes
1answer
89 views

Correlation between two nodes of a single layer MLP for joint-Gaussian input

Let's say you have a jointly gaussian vector random variable $\mathbf{x}$, with mean $\mathbf{M}$ and covariance $\mathbf{S}$. I now transform each scalar element of $\mathbf{x}$ , say $x_j$, with a ...
1
vote
2answers
417 views

Bayes rule and gaussian PDF

if range of gaussian pdf is not a probability then how come it is used in Bayes rule in the same way as pmf?
2
votes
2answers
416 views

Estimating distribution parameters from few data points

Say I'm doing stats on the height of adults from various countries. I assume the heights of adults from one country are normally distributed, and ignore sex differences (I also ignore the fact that ...