0
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0answers
15 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
1
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1answer
31 views

Bayesian Estimate Problem

So... I'll be honest, I don't know anything about anything Bayesian, this problem being no exception (from the Society of Actuaries' Exam C sample questions): You are given: (i) The annual ...
0
votes
0answers
24 views

Jeffrey's Prior for Bivariate Lognormal

Exactly what the question says, I'm working on code for an MCMC simulation and need to set some uninformative or weakly informative priors. I haven't been able to find the prior for the sigma ...
0
votes
1answer
22 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
1
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1answer
38 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
1
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2answers
62 views

rationalwiki on “Extraordinary claims require extraordinary evidence”

I don't have a strong background in probability/statistics and I'm trying to understand the example at ...
3
votes
0answers
33 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
0
votes
0answers
32 views

Change of Variables for two level Guassian model

I have a multivariate Gaussian distribution from which two variables, u and v, are drawn. The next variables, U and V, are U = 1/(u^2+5) + N(0,sig_U) and V = v^3 + N(0,sig_V). U and V are known, ...
0
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0answers
24 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
1
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0answers
34 views

How would I analyze the accuracy of a model that predicts World Cup matches?

Say, someone made a bunch of predictions for each game between Team A and Team B, such that there's a predicted probability for each of the three possible outcomes adding up to $1.0$ : Team A winning, ...
1
vote
1answer
38 views

Confusion in Posterior Probability Calculation

I know posterior probability as, $P(\theta|x)= [(P(x|\theta)*(P(\theta))/(P(x))]$, as given in http://en.wikipedia.org/wiki/Posterior_probability I am slightly confused with the term ...
0
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0answers
29 views

determinant and trace of a huge positive definite matrix

I have a problem to compute the determinant and the trace of inverse matrix: $det(\Gamma^{-1}+I_n⊗\Phi^T\Phi)$ and $tr[(\Gamma^{-1}+I_n⊗\Phi^T\Phi)^{-1}]$ where $\Gamma$ is a huge positive definite ...
3
votes
2answers
82 views

A house is guarded by two alarms

I am trying to wrap my head around the following problem A house is guarded by two alarms. If Alarm 1 fires, p(theft) = 80% If Alarm 2 fires, p(theft) = 70% If both alarms fire at the same time, ...
2
votes
1answer
20 views

Bayesian statistics, bivariate prior distribution

I've got a simple question buy I'm not sure how to solve it. It's a bit long. Suppose you've got $n$ iid random variables $X_1$, $\dots$, $X_n$ from the normal distribution with unknown mean $M$ and ...
0
votes
0answers
23 views

Bayesian estimate for unfair die

Suppose you have a six-sided die that you suspect is not fair and toss it N times. What would be a Bayesian approach to estimating the probability of the six outcomes given that you suspect the die ...
0
votes
0answers
29 views

How do I prove and expand Bayesian Networks?

Attempting to understand Exercise 20 (pdf page 44) in the paper: (Warning: large paper; small exercise) Bayesian Reasoning and Machine Learning The party animal problem corresponds to the ...
1
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1answer
36 views

conjugate prior

A class of sampling distribution is a conjugate family of a prior distribution, if the posterior distribution belongs to the same family for all priors and all samples. Why is this phrase incorrect?
0
votes
0answers
24 views

Jeffery's prior - help

I'm studying Bayesian inference and looking at prior choices. Currently I have looked at Laplace's uniform prior choice and now I am trying to understand Jeffrey's prior. I am having trouble ...
1
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1answer
38 views

When using Bayes Rule, what are the rules for flipping the conditions and the event of interest?

Here is Bayes Rule: $$P(A\mid B) = \frac{P(B\mid A) P(A)}{P(B)}$$ This paper (http://www.cogsci.northwestern.edu/Bayes/Sivia_1996.pdf) uses Bayes rule on page 21 in the context of model selection ...
1
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1answer
56 views

Bayesian Probability Question - Parameter Estimation

I would like help on the following question and I will show my work. Here is the question in my notes and I will follow up with my work: Q: Suppose a forest is segmented into strips, referred to as ...
1
vote
0answers
42 views

Generalized Bayes Estimator

Consider a decision problem in which the model parameter, $\theta$, is any integer, the distribution for the integer observation, y, given $\theta$ is $P(y|\theta) = 1/3$ if $y \in [\theta - 1, \theta ...
1
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0answers
27 views

Why is the marginalized inverse-Wishart distribution not equal to the inverse-gamma distribution?

Given that the inverse-gamma distribution is the one-dimensional version of the inverse-Wishart distribution, why will (philosophically speaking) an inverse-Wishart distribution that originally has ...
0
votes
1answer
43 views

Give the Bayesian Posterior Mode

Suppose that $X_1, X_2, \ldots, X_n$ are IID Bernoulli random variables with success probability equal to an unknown parameter $\theta \in [0,1]$. Let $A$ and $B$ be nonnegative constants. If we ...
0
votes
1answer
31 views

In Bayesian approaches, comparing the approximated distribution and the true distribution

I'm asked to compare the distance of the approximated distribution and the true distribution in a Bayesian approach. While we used Laplace approximation to find the MAP of the target posterior ...
0
votes
0answers
18 views

Left-censoring in time series

This is from a Bayesian problem I'm working on. I have worked out \begin{align} f(y_1,...,y_T|\varphi)=f(y_1|\varphi)f(y_2|y_1,\varphi)...f(y_T|y_1,y_2,...,y_{T-1},\varphi), \end{align} and all terms ...
2
votes
3answers
109 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
2
votes
2answers
76 views

Coin toss with unknown probability – Bayesian interpretation

I have observed a coin being tossed $n$ times. I do not know whether the coin is fair or not, but in every single toss I observed, the coin came up heads. What should my belief about $p$ (the ...
1
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0answers
27 views

Estimate distance between approximated posterior and true posterior

I'm working on a paper about using graphical models to do some prediction tasks with known observations. Since the model is complicated, finding the maximum a posteriori on the true posterior ...
1
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0answers
35 views

Does this Gamma posterior make sense?

quick question about the form of a posterior distribution. Suppose that $\theta \sim Gamma(a, b)$ and that, given $\theta$, $Y$ has CDF $$F(Y\mid\theta) = 1 - e^{-\theta(e^y - 1)},\quad ...
1
vote
1answer
48 views

What am I doing wrong in calculating Fisher Information of Triangular Distribution?

I am trying to find Jeffrey's prior for the Triangular distribution which has the following probability density function: $$f(x\mid \theta) = \begin{cases} \dfrac{2x}{\theta} & : x ...
0
votes
0answers
48 views

Finding the joint posterior distribution of AR(2) process

Suppose we have AR(2) process for $\{y_t, t=3,4,..\}$ and let $a_1,a_2,\sigma^2$ be the parameters of the time series. We assume that $y_1$ and $y_2$ er independent normally distributed with mean zero ...
0
votes
0answers
24 views

Bayesian Variable and Model Selection, Books and Review Papers Desired

I'm hoping that the community will be able to suggest some literature for studying this topic. There seems to be very few books on the subject. There are some chapters in some books which provide ...
1
vote
1answer
63 views

Showing the Posterior distribution is a Gamma

Assume $X\sim \mathrm{iid}\operatorname{Pareto}(a,b)$, and $b \le \min(X)$, then $$f_n({\bf x}; \theta) = a^n b^{-n} \prod^n_{i=1}\left( \frac{b}{x_i} \right) ^{1 + a} $$ We assume b is known and ...
0
votes
1answer
29 views

Likelihood function for $\theta = 0$ (Calculate $\sum_{i=1}^4x_i ^2$ given $\sum_{i=1}^4 x_i$)

I need to find $L_x(\theta)$ for $X_1, X_2, X_3, X_4$ c.i.i.d random variables such that $X_i | \theta \sim N(\theta,1)$ when: $\theta = 0$ $\bar{x} = -0.7$ ($\bar{x} = \Large\frac{\sum_{i=1}^n ...
1
vote
0answers
58 views

Hypotesis test: $X_i | \theta \sim Exp(\theta)$ (Likelihood Ratio Test)

Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$ P.S: ...
0
votes
0answers
30 views

Hypothesis testing with Baysian methods: How many animals must I test to be sure that a disease isn't present?

I colleague has come to me with a question which I have answered for him but the only statistics I have done was what I did at school and a one semester course on Bayesian methods at university, so I ...
1
vote
0answers
48 views

Calculating Bayes factor

Example: Integer-valued data $y = (y_1, ...,y_n):$ $M_1 = Geometric(\theta_1)$ likelihood with $Beta(\alpha_1, \beta_1)$ prior on $\theta_1;$ $M_2=Poisson(\theta_2)$ likelihood with $Gamma(\alpha_2, ...
0
votes
1answer
33 views

Difference of a likelihood function for a vector and a single value

$p(x\mid C)$ is defined as the probability density of a point $x$ given that it belongs to a class $C.$ But what of $p(\mathbf{x}\mid C)$ where $\mathbf{x}$ is a vector? I'm finding hard to ...
0
votes
1answer
134 views

Gaussian with a linear combination random variable mean

A very simple (looks like...) statistical problem, however I don't even know how to name it in a formal way... Suppose in a Bayesian framework I have random variables $y, x_1,$ and $x_2$, $$f(x) = ...
0
votes
1answer
236 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
60 views

Find marginal distribution for Pareto prior

I have the following problem: The prior distribution for $\theta$ is distributed $\pi(\theta) = \frac{aP^a}{\theta^{a+1}}$, $\theta >P$ The likelihood for X is uniformly distributed, i.e. ...
0
votes
1answer
103 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
1answer
29 views

What is Bayesian Evidence?

Could someone explain this concept or give a link to the explanation of this concept please? I know what "Bayesian" is, but I don't know what "Bayesian evidence" is. A good explanation of "evidence" ...
1
vote
1answer
61 views

If X and θ are both random variables and θ is the parameter of the distribution of X, are X and θ independent?

The answer appears to be no because the distribution of X is defined conditionally by θ which is also assumed to have a distribution as opposed to be a constant. Essentially, the formulation of the ...
2
votes
0answers
105 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
0
votes
1answer
42 views

Conditional probability in continuous distribution

I'm struggling understanding how conditional probability for a continuously distributed random variable is to be calculated. The task is as follows: $f(t) = 1/8 * (4-t)$ for $0 < t <= 4 $ and ...
1
vote
1answer
374 views

Find the posterior distribution of θ

I have this problem Given the prior distribution is \begin{align}Pr(\theta=i)=\pi_i=\begin{cases} 0.5, & \text{for i=4}.\\ 0.3, & \text{for i=5}.\\ 0.2, & \text{for i=6}.\\ ...
1
vote
1answer
34 views

Why is it valid to use the PDF for a naive bayes classifier?

In my understanding of a Naive Bayes Classifier, one takes the argmax of the probabilities that example $x$ belong to class $c_i$, that is $$\text{argmax}_{c_i\in C}P(C=c_i|X=x)$$ I understand that ...
0
votes
0answers
37 views

Independence of data in the prediction interval of a Bayesian regression

The annotated screen shot below is from page 732 of Probability and Statistics, 4th ed. by DeGroot and Schervish. Please see my question in the red box. It makes sense to me that Y-hat is simply the ...
0
votes
1answer
30 views

Simple question about beta prior property

Say that person $A$ thinks that a certain proportion is $0.3$, person $B$ thinks that a certrain proportion in a population is $0.7$. We have a $Beta(4,4)$ prior. How can one mathemtically prove that ...