0
votes
0answers
11 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
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 ...
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
14 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
34 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
32 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
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) ...
0
votes
0answers
15 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
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 ...
2
votes
0answers
79 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
31 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
54 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
73 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
12 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
88 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
82 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
58 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
118 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
253 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
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 ...
0
votes
0answers
23 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
134 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
0answers
35 views

Uniform choice for Prior Distribution

My prior function is $\Phi\left(\mathbf{k}_\ell,W_\ell\right)=\frac{1}{N}\log p\left(\mathbf{k}_\ell,W_\ell\right)$ which is determined once I choose the Bayesian prior parameter likelihood ...
0
votes
1answer
197 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
30 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
53 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
96 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 ...
0
votes
0answers
124 views

Bayesian Updating with Gaussian Signals

My question relates to a standard bayesian result. Let $A$ be some unknown parameter normally distributed with mean $\mu$ and variance $\sigma^2$. Is we observe $X = A + \epsilon$ where $\epsilon$ ...
3
votes
1answer
133 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 ...
0
votes
0answers
163 views

How to calculate a Bayesian Inference over a Poisson Binomial Distribution

In relation to this question, how do I use Bayesian inference over a Poisson Binomial Distribution? If possible, what is the Conjugate Prior? Thanks to @Stijn, here is an elaboration of the problem: ...
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
214 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
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) ...
0
votes
1answer
206 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
28 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
55 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
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
334 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
67 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
323 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
109 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 ...