The approach and interpretation of probability associated with Bayes theorem; usually used as opposed to the frequentist approach. It can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain. A Bayesian probabilist starts with some prior ...

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Is this problem suited for Bayesian inference?

Suppose that the quality of a widget is distributed according to a score, given by a normal distribution with mean 1 and variance σ^2. A fraction, π of all widgets are defective. The cost of having an ...
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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 ...
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95 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 ...
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42 views

Computing evidence for least-squares fit

I'm at a loss trying to implement Bayesian model selection for standard least-squares polynomials fits. I have three polynomials of order $1$, $2$, and $3$, and a sequence of $(x,y)$ data points. ...
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88 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 ...
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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 ...
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92 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 ...
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21 views

Estimating conditional probability as a function of time

My question relates to estimating from a time series a time dependent conditional probability without having a prior parametric model of anything. Suppose I have two variables: r and I, and each can ...
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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} ...
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7 views

Determining the liklihood in Baye's rule for parameter estimation

I have used Bayesian statistics in classes but what I am trying to do now is different than anything I have done in class. Previously, I was given information and certain numbers adn I could ...
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37 views

Monty Hall Problem Solve Using Detailed Algebra

I have been searching the monty hall problem for two days now and I generally understand it but I am having a very hard time solving the monty hall problem using Bayes's theory. I do not know what ...
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38 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 ...
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18 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 ...
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22 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 ...
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34 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 ...
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36 views

Posterior predictive distribution in a Bernoulli process.

Suppose there are $k$ successes in a Bernoulli population $ X = \{x_1, \ldots, x_n\}$. I would like to calculate the posterior predictive distribution $f(x | X)$ where $x = \{0,1\}$. I assume the ...
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18 views

Estimating variance from a combination of processes

I am fiddling with some Bayesian probabilities for some astronomical data analysis. I have a ccd image and am testing the null hypothesis (no signal is present - all contributions due to noise) on it. ...
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83 views

Interpreting Bayes' thereom with density functions

I'm studying about Bayes' theorem and according to the theorem: $$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)}$$ This I can understand where it comes from etc. But if I use Bayes' theorem on density ...
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39 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: ...
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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 ...
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43 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, ...
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14 views

Gibbs sampler for local linear trend model

Question: Consider the local linear trend model given by: \begin{align*} y_t = \mu_t + \tau \varepsilon_t \ \cdots \ \text{Observation equation} \\ \mu_{t+1} = \phi \mu_t + \eta_t \ \cdots \ ...
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31 views

Confirm my working for the conditional posterior of $\beta$

So I have the following question from my textbook, the answer I get is slightly different from the book's answer, which I think may be wrong, could someone please confirm? Question: Suppose $y_{1:T} ...
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66 views

Expected distance traversed between 2 vertices on probabilistic graph

Let $V = \{1,2,3,...,N\}$ be the vertex set of a graph. Let $d(i,j)>=0$ represent the $(i,j)$ vertex distance between vertices $i$ and $j$, $i \in V, j \in V$. Now, define a non-negative number ...
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87 views

Bayesian random walk

Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. ...
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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 ...
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64 views

Kolmogorov's paper defining Bayesian sufficiency

I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
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96 views

Posterior density and posterior moments

I would be very grateful to get some help with the following problem. Let $X_1, ..., X_n$ be independent and uniformly distributed on the interval $(0,\theta)$ with $\theta>0$. Let the prior ...
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55 views

Laplace approximation of the likelihood Bayesian

I need help with the following question: Consider m observations $(y_1; n_1); ... ; (y_m; n_m)$, where $y_i \sim Bin(n_i; θ_i)$ are binomial variables. Assume that $θ_i \sim w_1Beta(α_1; β_1) + ...
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89 views

Find posterior mean

I have this problem, Let $X\sim U(0,\theta)$ with $\theta>0$. Assume a signal random sample $X$, the squared error loss, and the prior $\pi(\theta) = \exp(1)$ i.e. $\pi(\theta) = \theta ...
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15 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, ...
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198 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. ...
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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) ...
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22 views

Lacking intuition on the subject of graphical representations of independence

By defintion the (closed) graph G= (V, V x V) over a set of statistical variabeles is an I-map for any independence relation over $V = \{V_1, V_2, V_3, V_4 \}$, a set of statistical variables. But in ...
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14 views

Training an algorithm (computing probability)

I am in the process of training a machine learning algorithm. I am trying to save a probability value as I train to keep track of improvements. I am dealing with binary decision problem. The output ...
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13 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 ...
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9 views

Computing Object Classification with bayesian statistics

Say I want to know if there is a zebra $\theta$, in an image $x$. According to Bayes statistics applies to image recognition, I should be computing: ...
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17 views

Bayesian ranking system acting up

My Bayesian ranking system seems to be acting quirky.. and I'm wondering about my implementation. This is basically my formula: ...
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16 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 ...
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23 views

Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data. •Data∼Gamma(α,β) •Parameters α∼Gamma(kα,θα) β∼Gamma(kβ,θβ) I used Winbugs (code ...
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58 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 ...
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EKF to fuse gyroscope and accelerometer readings

I found it interesting to implement EKF for fusing gyroscope and accelerometer data. Trying reach my goal i discovered a lab with some theory explaned, also it has nice app for the phone to stream ...
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14 views

Bayesian Network understanding

I am confused by the definition of Bayesian Network. It's well know that graph $G$ of Bayesian Network can be viewed in two very different ways: As a data structure that provides the skeleton for ...
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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 ...
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40 views

Monty Hall Problem Extended Using Bayes's Theory

I know there is a question on the website concerning the extension of the monty hall problem. The question is provided with very good answers given by the participants on the website which I would ...
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25 views

How many numbers for the full joint?

Suppose you have 3 binary nodes A, B, C. A and B are independent given C. How many numbers do we need for the full joint? How many numbers do we need for the Baysesian Net? I know the answers to ...
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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 ...
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21 views

Bayes updating (Beta Distribution)

I have been trying to use Bayes' theorem to update a Beta distribution B(alpha, beta). For each iteration of randomly drawn values from some Beta distribution, I multiply my 2D joint likelihood ...
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16 views

Estimating the magnitude of a change in a non-stationary stochastic process

In this paper by Adams and MacKay, they present an algorithm for the online detection of change-points in a stochastic process subject to some hypotheses. Their algorithm gives both the predictive ...
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Folded normal distribution

I am using bayesian stats for image classification and I have a variety of input variables. These inputs are normally distributed except for two. One has a truncated normal distribution and the other ...