0
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0answers
56 views

Bayes' Theorem Question, with a twist

I have a very old past high school exam question I am trying to solve (for interest only). It's a straightforward application of Bayes' Theorem, with the last part of the question containing a slight ...
1
vote
0answers
21 views

Growing of a score function

The argument that I'm dealing is very specific, I hope to make you understand the problem without going into detail. I have this score function: \begin{align} score = MargL^q + MargL^{\theta} ...
5
votes
0answers
68 views

What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
0
votes
0answers
20 views

Is a prior distribution always a random probability measure?

Let $(\mathcal{X}, \mathcal{B})$ be a measurable space and let its probability measure be $P$. In Bayesian statistics, we may wish to define a prior $\mu$ on the space of all such probability ...
1
vote
1answer
25 views

Conditional probability with bayes rule??

http://cseweb.ucsd.edu/~dasgupta/103/2b.pdf part 2.1.2 implies $P(X|Y \cap Z) = \frac{P(X|Y)}{P(Y|Z)}$ Seems to imply that this is true but if you take bayes, the left hand side is: $P(X|Y \cap Z) = ...
3
votes
2answers
77 views

Conjugate priors make calculations easier but at what cost to the model?

As I understand, when we have a parametric pdf and need to estimate the parameter based on some observed fact, we tend to choose a conjugate prior of the pdf for the parameter. Because conjugate prior ...
0
votes
0answers
13 views

What happens if the recursive bayes is performed without updating the data?

With a relatively good prior, and recursive Bayes' is performed with new data every iteration, the posterior converges to the real value, under ideal circumstances. But what happens if recursive ...
0
votes
0answers
23 views

Matlab Bayesian Newtork toolbox and cotinuous values

I have two doubt, one about theory and one about practical problem. First i have not full understand how to work a bayesian network with continuous values. I have learn that i can approximate P(A) ...
1
vote
1answer
81 views

How to calculate pseudo-determinant for implementing Naive-Bayes

(People who followed Bayesian tag, please read the third paragraph) Problem: I need to calculate pseudo-determinant of a matrix (preferably in MATLAB, but no ...
0
votes
1answer
27 views

Uniform prior distribution multiple results

When I have a simple Bernoulli trial with a certain variable taking, for instance, values 0 and 1, I have a constant prior distribution for the $\theta$ parameter, i.e. pdf $p(\theta) = 1$ between 0 ...
1
vote
1answer
29 views

Posterior distribution as a distribution for a new random variable?

So in Bayesian framework one uses observed data $X=\{x_1,\dots,x_n\}$ to update the prior $p(\theta)$. My question is it justified mathematically to say that $p(\theta\mid x_1,\dots,x_n)$ corresponds ...
1
vote
0answers
17 views

Measure theoretic basis of joint distrib of parameters and data in Bayesian analysis

In Bayesian statistics you have a prior density for your parameters $\Theta$, $\pi(\theta)$ for $\theta\in\mathcal{T}\subset\mathbb{R}^k$, have the conditional distribution of the data given the ...
1
vote
1answer
46 views

Please explain to me why $ \int_b p(a|b) db \neq p(a) $

I have one question that bugs me. How is it that: $ \int_a p(a|b) da = \int_a \frac{p(a,b)}{p(b)} da = 1 $ but $ \int_b p(a|b) db = \int_b \frac{p(a,b)}{p(b)} db \neq p(a) $ I don't understand ...
0
votes
2answers
52 views

Conditional Independence - Bayesian Network

May the probability distribution $ P(A,B,C,D) $ given as: $ P(A,B,C,D) = P(A)P(B)P(C|A,B)P(D|C) $ The task is to show that this holds $ A \bot B | \emptyset $ and $A\bot D|C$. First thing I'd like ...
0
votes
1answer
177 views

Find Bayes estimator of $\theta$

I've got this exercise, which I'm trying to work off using an example, but the example seems very different so I'm not sure if what I'm really doing. I've got a loss distribution for $\theta$: ...
0
votes
1answer
87 views

Find joint probability P(X=0, Y=0)

I have this problem where I'm not too sure on how to proceed. I need to calculate $Pr(X=0 $ and $ Y=0)$ using the following information: The conditional distributions $f(x|\theta)$ and ...
0
votes
3answers
117 views

Probability question: given $P(A|B)$ and $P(B)$ how do I find $P(A)$?

I have a probability distribution for some quantity $A$ given a fixed $B$, i.e. $P(A|B)$. I also have a prior distribution $P(B)$ for $B$. I'm trying to find the distribution $P(A)$. I had thought ...
0
votes
1answer
28 views

Statistical inference (limit!)

Suppose that a random sample of size n is taken from the Bernoulli distribution with parameter θ, which is unknown, and that the prior distribution of θ is a beta distri bution for which the mean is ...
0
votes
1answer
59 views

Bayes estimator (Inference)

An urn contain 5 balls, $ \theta $ white and $ 5 - \theta $ green. The experiment consists in grab 2 balls from the urn and register the pair $(x_1, x_2)$, where $x_i = 1$ if we observe a white ball ...
0
votes
0answers
38 views

Assigning prior to $\gamma$ in composite power function $P(t) = max[\lambda t^{-\beta}, \gamma]$

I want to estimate the parameters $\lambda, \beta$ and $\gamma$ using a bayesian approach and an MCMC sampler. With the exception of $t$ all variables are random variables between $0$ and $1$. $t$ is ...
1
vote
1answer
66 views

Illustrate the invariance property of a noninformative prior

Consider $n$ i.i.d observations from a normal distribution with unknown mean, $\mu$, and unknown variance $\sigma^2$, ie, $y_i \sim i.i.d \ N(\mu, \sigma^2)$ for $i = 1, 2, \cdots, n$. Let ...
-1
votes
1answer
548 views

How to prove if P(A|B)>P(A) then P(B|A)>P(B) [closed]

How to prove that If P(A|B)>P(A) then P(B|A)>P(B)
2
votes
2answers
110 views

Comparing uniform priors

The background of the problem is this: Assume that we have a parameter vector $\Theta$ which satisfies $\Theta^\prime\Theta=1$. If we let this vector have the uniform prior, the density of the prior ...
0
votes
1answer
54 views

Checking independence of variables in a Bayesian network

I need a little help with Bayesian Networks. Consider given the following network (all variables are binary) and we need to check conditional independence of $A$ and $C$ if $X$ and $Z$ are given. Any ...
3
votes
1answer
58 views

Implied prior with relationship $y=\text{arccot}(x)$

I'm trying to solve an exercise, which I think I have almost managed to solve but not quite. Any help would be appreciated! So, what we have is a vector which we obtain by norming the vector ...
3
votes
1answer
81 views

Bayesian formula for weather exercise

If it is nice weather on one day, the probability that it is going to be nice again the next is $13/15$. If it is raining on one day, the prob. that it is going to be raining again the next day is ...
1
vote
1answer
112 views

Bernoulli trials conditional probability

Let $\Omega=\{0,1\}^\infty$ and $S_n=X_1+\cdots+X_n$ the number of “successes” or “arrivals” in $n$ steps. $p\in(0,1)$ and $\mathbb P(S_n=k)=\binom{n}{k}p^k(1-p)^{n-k}$ Let $T$ be the time until the ...
1
vote
1answer
79 views

Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
1
vote
1answer
212 views

Using Bayes Theorem intuitively without equation (tree-diagrams)

I am working on the following question and I am having some difficulty. The thing is I understand that I must apply Bayes Theorem but to be honest, I like to do problems using Bayes Theorem ...
0
votes
0answers
82 views

Mathematical notation for probability trees and their usage

A commonly used tool for visualising and solving Conditional Probability problems is the tree diagram of events and their associated probabilities. (Tree Diagram). How can one represent particular ...
1
vote
0answers
52 views

How to make this inference: Degree of a node in a graph is significantly diffenrent from poisson distribution

I am working on Gene-Gene interaction graphs. I build a graph by adding edges between genes (nodes) which show statistical interaction in predicting a quantitative parameter value (say, brain volume) ...
2
votes
1answer
208 views

Bayes Estimator

Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf: $f_{X}(x|\alpha,\beta) = ...
0
votes
1answer
86 views

Question on Bayesian Learning and Probability Theory

I have some difficulties at solving a traditional problem where we have two hats, hat A and B, where there are black and white balls in each hats but the experimenter does not know the proportion of ...
4
votes
1answer
247 views

Bayesian Inference in Measure Theory

What's the deal. How does this work, or can you point me to some references? I tried $\mu(A|B) = \mu(A \cap B) / \mu(B)$ and got stuck on $\mu(B) = 0$. Edit: Sorry for being lazy. My background is ...
2
votes
0answers
54 views

Gaussian Bayesian filtering with bound observation ($b_1<x<b_2$)

Suppose we have a Normal r.v $$ x \sim \mathcal{N}(\mu, \sigma^2) $$ and a Normal prior of $\mu$ $$ \mu \sim \mathcal{N}(\theta, \delta^2) $$ I know how to do the Bayesian update with a ...
1
vote
1answer
44 views

Inference in a probabilistic Bayes network

Given the following Bayessian Network: I wonder when is it reasonable to estimate $p(u\mid c)$ as $$ p(u\mid c) \approx p(c\mid w=w_1,\ldots,w_t)$$ I want to estimate that because I can't ...
1
vote
1answer
143 views

A few questions regarding Bayesian updating

I'm reading an introduction to Bayesian updating which includes the following example (by the way, I will add my reasoning at some parts so you guys can tell me whether I'm correct or not): Let ...