Tagged Questions
0
votes
0answers
38 views
Bayesian computational problem with R
I can't understand the iter (this is me) to solve this kind of exercise involving the approximation of the posterior density.
Consider this exercise taken from Bayesian computation with R:
We ...
0
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0answers
46 views
Estimating the radius of a circle
I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
2
votes
0answers
42 views
Probability distribution for a digit of a number
If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
3
votes
1answer
50 views
hint with Bayes rule problem
The pirate Captain Queequeg has a lazy crew and suspects they are planning to stage a mutiny. Captain Queequeg's solution is to have every member of the crew roll Queequeg's lucky die. If the roll is ...
0
votes
0answers
50 views
Bayesian networks and joint probabilities
This is my problem
My problem is modeled by a basic Bayesian Network with only two layers. So I have parent and child nodes but the children has no children. Essentially a bipartite graph. The ...
0
votes
0answers
15 views
Naive Bayesian Classifier for Object with Variable attributes
Let say our objects are connected graphs. They are to be classified into two categories, say A and B. However, for our purpose attributes for each graph is equal to the number of vertex of the graph ...
1
vote
1answer
40 views
Bayes theorem for calculation with personal probabilities
I'm completely stuck on some homework I have and can't figure it out.
The task is to calculate the probability of a bus being late conditional on the weather being snowy and bus driver being ...
0
votes
1answer
30 views
Calculating Probabilities for Substitution Ciphers using Frequency Analysis
I have been trying to put together a tool that can take in cipher text encrypted via a simple substitution cipher and calculate the most likely "key" (that is, how the plain text letters were mapped ...
0
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0answers
33 views
Hypothesis Testing Bayesian Way
I'm having trouble with the following problem:
Suppose a machine is composed of 2 components (1 and 2, independent from each other). Each component has a exponential failure probability distribution ...
0
votes
2answers
83 views
Find the posterior distribution of $\theta$
I'm having trouble solving the following problem:
Find the posterior distribution of $\theta | x$. Suppose $x$ is a random variable with distribution $f(x) = \theta x^{\theta - 1}$, you observe a ...
3
votes
1answer
39 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
32 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
0answers
5 views
Conditional distribution for a label given a scalar feature
I am trying to create a simple simulation setup for classifiers on toy data. Each data point can has a scalar feature $X$, which is uniformly distributed between -1 and 1. Depending on the feature, ...
1
vote
1answer
45 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 ...
1
vote
1answer
39 views
Difference between true positive and true negative in this case
A patient takes a lab test and the result comes back positive. It is known that the test returns a correct positive result in only 98% of the cases and a correct negative result in only 97% of the ...
1
vote
1answer
33 views
What is the distribution of parameter using Bayesian inference
I am trying something very simple to try to get more "inside" Bayesian statistics but the results are a bit odd.
Lets say I have an infinite bowl, inside of which are black and white balls. We do not ...
0
votes
1answer
74 views
question related to Bayes' rule and Bays' risk.
Let $X_1, X_2, X_3, \ldots, X_n$ be a random sample for $N(e,1)$. Let the prior p.d.f. of $e$ be $N(0,\sigma^2)$ under the square error loss function $L(e,d)={(d-e)}^2$. Find the Bayes' decision rule ...
0
votes
0answers
52 views
Mathematical Notation for Probability Trees
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 ...
0
votes
1answer
33 views
Estimating probabilities using Bayes rule?
I am working on a past exam paper. I am given a data set as follows:
Hair: {brown, red} = $\{B,R\}$
Height: {tall, short} = $\{T,S\}$
Country: {UK, Italy} = $\{U,I\}$
Our sample is:
...
1
vote
0answers
82 views
Bayesian posterior with integrals over normal densities
Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
0
votes
0answers
47 views
Conditional probability given two variables from conditional probabilities of one variable
I have a question which comes from an example of a textbook, but all I am concerned with is how we go from having probabilities $P(X|A)$ and $P(X|B)$ to having $P(X|A,B)$. In the example events $A$ ...
1
vote
1answer
31 views
Independent random variables considering expression
Having $x, y, z, c \in \mathbb{R}$, is it valid to say:
$c \propto g(x, z) h(y, z)$
The context here is to say whether or not the random variables $X$ and $Y$ are independent given the value of $Z$.
...
1
vote
2answers
74 views
Easy Probability Problem
I was told the following probability problem:
While doing a math problem today at the contest the probability of Annie, Tom and Karen getting
the problem correct first is 1/7, 1/2, and 5/14 ...
1
vote
0answers
31 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) ...
1
vote
2answers
128 views
Bayes' Theorem: Detection of bomb in a box
There is a bomb that is equally likely to be in any one of three different boxes. Let $α_i$
be the probability that that the bomb will be found upon making a quick examination ( detection ) of box i ...
2
votes
1answer
39 views
What is the meaning of “mean-field”?
In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this ...
1
vote
1answer
94 views
How would you approach this problem on the Bayes theorem?
I've been reading a book on Statistics and I could COMPLETELY understand all of its text. It basically explained the bayes theorem and what priors were, what posteriors were etc. But then in the ...
0
votes
2answers
111 views
Why is $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$?
Could anyone derive or explain why the formula $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$ is true? I understand conditional probability definition, but this formula confuses me and makes my head hurt x)
Here's ...
1
vote
0answers
27 views
Coin tossing - Two tosses, one is a head, probability other is a tail? [duplicate]
A friend of mine tossed a fair coin twice. Suppose instead that I happen to see the result of one of his tosses, and it is a head. What is the probability that the other toss is tail?
1
vote
1answer
437 views
Coin toss - probability of a tail known that one is heads
A friend of mine tossed a fair coin twice. Suppose I ask him whether he got a head in the two tosses, and he says yes. What is the probability that one toss is tail?
Now suppose instead that I happen ...
1
vote
1answer
100 views
Bayes' Rules: The probability of at least one event occurring?
There is a 60 percent chance that the event $A$ will occur. If $A$ does not occur, then
there is a 10 percent chance that $B$ will occur.
(a) What is the probability that at least one of the ...
2
votes
1answer
52 views
What is the probability of two or more from n events occuring?
A number of independent events, say $A$, $B$, $\ldots\,$, $E$, can happen with associated probabilities $P(A)$, $P(B)$, $\ldots$
For each event that happens I have to pay £10. The likelihood I have ...
0
votes
1answer
45 views
$u$~$N(0,A)$ and z$|u$~$N(u,1)$ how to show that $u|z$~$N(Bz,B)$ where $B=A/(A+1)$?
$u$~$N(0,A)$ and $z|u$~$N(u,1)$ how to show that $u|z$~$N(Bz,B)$ where $B=A/(A+1)$ ?
1
vote
2answers
122 views
Conditional probability formulas
For a compound event $E_1E_2$,
Pr($E_1E_2$) = Pr($E_1$)Pr($E_2$|$E_1$)
such that if $E_1$ and $E_2$ are independent events, we can say that:
Pr($E_1E_2$) = Pr($E_1$)Pr($E_2$)
Is there an analogous ...
0
votes
1answer
100 views
Exercise 2.8 in Mackay's Information Theory, Inference and Learning Algorithms
[Editing question per Leon's suggestions - thanks for these!]
Could someone walk me through a solution to Ex 2.8?
2.7: Bill tosses a bent coin $N$ times, obtaining a sequence of heads and tails. ...
9
votes
5answers
1k views
Bayes Theorem Example in Nate Silver's The Signal and the Noise
In his book The Signal and the Noise, Nate Silver presents this example application of Bayes's Theorem on pp. 247-248:
Consider a somber example: the September 11 attacks. Most of us would
have ...
2
votes
0answers
47 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 ...
2
votes
3answers
31 views
A bayesian way of calculating the probability $\text{Pr}(\theta \in (\theta_1, \theta_2) | y )$
I'm confused by a bayesian way of calculating the probability $\text{Pr}(\theta \in (\theta_1, \theta_2) | y )$, where $\theta$ is assumed to have a (prior) uniform distribution on $[0,1]$, and the ...
0
votes
0answers
55 views
Bayesian network over a tree - calculate CPDs of posterior
Given a Bayesian network P that factorizes over a tree graph, how can the following claim be proved?
Let E be evidence variables, and X the remaining variales.
$$
P(X|E=e)=\prod_{i\notin ...
1
vote
1answer
39 views
Inference in a probabilistic Bayes network
Given the following Bayessian Network:
I wonder when is it reasonable to estimate $p(u\mid c)$ as
$$ p(u\mid c) \approx p(c\mid w=w_1,\ldots,w_t)$$
I want to estimate that because I can't ...
2
votes
1answer
62 views
On Finding Means of Distributions
If I have a distribution which depends only on one variable, I usually find the mean by:
(Continuous Case)
$$\mu=\displaystyle \int xf_X(x).dx$$
What happens in the following cases:
Conditional ...
0
votes
0answers
99 views
Probability Question.
Every morning, I roll a die to decide how to travel to work. If I roll a $1$ or $2$, I take
the train, if I roll a $3$, I catch a bus and otherwise I cycle. The probability that I am
late for work is ...
4
votes
0answers
332 views
Trying to understand the basics of bayesian inference
This paper gives a somewhat gentle introduction to Bayesian inference: http://www.miketipping.com/papers/met-mlbayes.pdf
I got to section 2.3 without much problems but got stuck in understanding that ...
1
vote
2answers
91 views
Probabilistic classification using naive bayes
Let's say I want to determine if some person is bald using two features, age and hair color. Also I will assume that age and hair color is independent in other words use Naive Bayes classifier.
...
1
vote
1answer
63 views
Probabilities from bayesian network
I am doing problems related to bayesian network. After reading the theory part I am able to understand that by making a network or reducing a problem to some bayesian network, we are simplifying a ...
2
votes
1answer
67 views
Conditional Probability Question
There are three events, $A$, $B$ and $D$. I know that $P(D)=0.2$, $P(A)=0.34$ and $P(B)=0.43$. I have calculated that $P(D\mid A)=0.5294$ and $P(D\mid B)=0.44186$. Now I need to calculate $P(D\mid ...
1
vote
1answer
108 views
What does the error rate mean in Naive Bayes.
Can anyone explain what the Bayes error rate is in Naive Bayes, for instance in matlab:
...
2
votes
1answer
52 views
Questions on Bayesian analysis of an opinion poll (an example in a book)
I'm sorry in advance for rather long questions. This is an example in "Bayesian logical data analysis for physical sciences" by P. C. Gregory and I have some questions about the example.
In a poll ...
2
votes
0answers
84 views
Is there a name in the literature for a projectivized measure?
By a projectivized measure I mean a nonzero measure on some measurable space $X$ up to scaling. If a nonzero measure is finite, its projectivization can be identified with its normalization (to have ...
1
vote
1answer
173 views
question on probability using bayes theorem
Here goes the question:
In a certain day care class, 30% of the children have grey eyes, 50% of them have blue and the other 20%'s eyes are in other colors. One day they play a game together. In the ...
