1
vote
1answer
37 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 ...
0
votes
0answers
27 views

Bayes Theorem with multiple observations

Let $H \in \{1,..,K\}$ be a discrete random variable and $e_1, e_2$ be observed values of 2 other random variable $E_1$ and $E_2$. We wish to calculate the vector ...
2
votes
2answers
43 views

How to find out number of possible outcomes by trying over and over?

While working on my network exploration tool project, I've ran across the problem of reliably determining number of possible exit addresses of a tunnel with single entrance. I've came up with ...
1
vote
0answers
26 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
0
votes
1answer
21 views

What is this distribution formulated with w, m and sum sign?

I have a binary classification problem, part of which is defined as follows : p(x|y=1) $\sim w (m_1 , \sum_1$) and p(x|y=0) $\sim w (m_0 , \sum_0$) Where $\sum_1$ is a covariance matrix : $$ ...
0
votes
0answers
55 views

Bayesian sequential updates of normally distributed variables

Suppose that you can observe data that are independently and identically distributed as $N(\mu, 1)$. Your prior distribution for $\mu$ is $N(m, v)$. After observing $n_1$ data with sample mean ...
0
votes
0answers
54 views

Marginal and conditional probability table without joint probability table

I've a Bayesian network, with discrete node values: for every node I've the conditional probability table $p(A|B)$, where $A$ is the node itself and $B$ is the set of the parents nodes. Now I would ...
0
votes
0answers
23 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 ...
3
votes
2answers
79 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
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 ...
0
votes
0answers
40 views

Poisson distribution probability from a single measurement

This question came up while reading a medical paper - the study showed $m_1$ out of $n_1$ people doing $X_1$ died, while only $m_2$ out of $n_2$ people died when doing $X_2$. I'm trying to ...
1
vote
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 ...
0
votes
0answers
40 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) ...
0
votes
0answers
23 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 ...
0
votes
1answer
32 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 ...
2
votes
0answers
86 views

Maximum Posterior: $ p(\bf{w}\mid\bf{x},\bf{t},\alpha,\beta) \propto p(\bf{t}\mid\bf{x},\bf{w},\beta)p(\bf{w}\mid\alpha) $ for Gaussian Distribution

At the moment I take a look at the book Pattern Recognition and Machine Learning from Christopher Bishop and as I try to understand the basics of the probability theory I get stuck trying to ...
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
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
2answers
83 views

Finding the marginal using Bayes Theorem

I am trying to find the marginal distribution f(x) when given the prior distribution $\pi(\theta)$ (Gamma $\alpha, \beta$) and conditional distribution $f(x|\theta)$ (Poisson, $\theta$). I know the ...
0
votes
0answers
14 views

How can I conceptualize the prior of a deterministic variable in Bayesian data analysis?

I have a model which includes the following priors: $\text{prec}_C \rightarrow \dfrac{1}{\sigma_C^2}$ and $\sigma \sim \text{uniform}(0,500)$ Now, as far as I understand the first is a ...
0
votes
2answers
95 views

How to I find the distribution of $\log p(X)$ given an $X$ drawn from $p$?

I have a feeling there's no general solution to this problem, but I'll ask anyway. I have a multivariate PDF $p$ and, given a random vector $X\sim p$, I'd like to find the the PDF of $\log p(X)$. ...
1
vote
1answer
161 views

Computing posterior distribution for AR(1) model

Question: For this question, note that the notation $y_{1:T} = (y_1, y_2, \cdots, y_T)$, ie, a vector of random variables. Consider the following AR(1) model: \begin{align*} y_{t+1} = \phi y_t + ...
1
vote
1answer
60 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 ...
0
votes
3answers
135 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 ...
1
vote
1answer
362 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
40 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 ...
0
votes
0answers
27 views

Deriving posterior pdf in classical linear normal regression model under noninformative prior

Question: Assume the following classical linear normal regression model: \begin{gather*} y_{i} = \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_K x_{Ki} + e_i \\ \underbrace{\boldsymbol{y}}_{n ...
0
votes
1answer
203 views

Gibbs sampling to produce posterior pdf

Suppose we have the following classical normal linear regression model: $$y_i = \beta_1 x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + e_i$$ where $e_{i} \sim iid.N(0, \sigma^2)$ for all $i = 1, 2, ...
0
votes
1answer
281 views

Natural conjugate prior for bernoulli distribution

Assume we have an i.i.d. sample of $n$ observations from a Bernoulli distribution. That is, $\displaystyle{p(y_i|\theta) = \theta^{y_i}(1-\theta)^{1-y_i}} \ \ \ \ \text{for} \ \ y_i = 0, 1$ and $i = ...
1
vote
1answer
32 views

Finding the marginal posterior distribution of future prediction, $y_{n+1}$

Assume the following bivariate regression model: $y_i = \beta x_i + u_i$ where $u_i$ is i.i.d $N(0, \sigma^2 = 9)$ for $i = 1, 2, ..., n$. Assume a noninformative prior of the form: $p(\beta) ...
0
votes
1answer
63 views

Help writing Dirichlet (multidimensional Beta) PDF correctly

I am not getting a PDF when I attempt to express the Dirichlet distribution over the random variable vector $\mathbf{\theta}=(\theta_1, ..., \theta_{27})$. Suppose a total of $2000$ observations on ...
1
vote
0answers
101 views

Bayesian updating of multivariate normal?

Let $\bf x$ be an unobserved realization of $\tilde{\bf x}\sim\mathcal{N}(\pmb\mu,\pmb\Sigma)$, where $\pmb\mu\equiv\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}$ and ...
3
votes
1answer
160 views

Maximum Entropy Distribution When Mean and Variance are Not Fixed with Positive Support

I know when the mean and variance of $\ln x$ are both fixed, then the maximum entropy probability distribution is lognormal. When the mean of a random variable is fixed the MEPD is the exponential ...
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 ...
0
votes
1answer
253 views

Find the Posterior distribution- prior: $exp(1)$, likelihood: $poisson(\lambda)($

I have a prior $\lambda \sim exp(1)$ and a likelihood $X \sim poisson(\lambda)$, and I observed in a sample of $n=5$ a mean of $3$. What is the posterior distribution of $\lambda$? Here is my ...
1
vote
0answers
56 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) ...
0
votes
1answer
224 views

Empirical Bayes estimator for a Beta-Binomial parameters

Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, ...
1
vote
0answers
29 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
3k 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 ...
2
votes
1answer
57 views

What's the posterior for mutivariate lognormal with covar known?

I know the univariate case but not the multivariate case. Suppose we have a multivariate lognormal dist: $$ \boldsymbol{X} \sim \text{lognormal }(\boldsymbol{\mu}, \boldsymbol{\Sigma}) $$ where ...
2
votes
0answers
60 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
382 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: ...
0
votes
1answer
75 views

how can I compute a posterior distribution using Bayes?

This may be a silly question, but I cannot figure out a convincing (to myself) answer to it. Suppose that you want to buy a new car. Let $v$ be the value you attach to the car. Before visiting the ...
2
votes
1answer
330 views

How do I calculate the aposteriori probability distribution for someone's answer to a poll being an approval?

Imagine I'm polling a random sample from the population and it asks them if they approve of the President or not. I also ask them some categorical demographic questions (age-bracket, race, gender, ...
2
votes
1answer
111 views

How do you take the product of Bernoulli distribution?

I have a prior distribution, $$p(\boldsymbol\theta|\pi)=\prod\limits_{i=1}^K p(\theta_i|\pi).$$ $\theta_i$ can equal $0$ or $1$, so I am using a Bernoulli distribtion so that ...