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Questions tagged [bayesian]

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 probability, and evaluates the evidence in favour of a hypothesis by combining the prior with the likelihood function of the observed data.

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Bayesian Posterior Mode VS MLE [on hold]

Suppose that $X_1, X_2, \dots $ are i.i.d Bernoulli random variables with success probability equal to an unknown probability $θ∈[0,1]$. Considering the example of flipping a coin with $n=1$, provide ...
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Probability the event occurs knowing that I received no information

First I want to thank you if you pay attention to my post. I apologize if it seems elementary to you, note that I searched a lot an answer before posting. I'm going to try not to be vague, do not ...
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Finding estimator that minimizes the weighted mean-square error

I have continuous multivariate random variable $x$ in $R^n$ with known prior $p(x)$ over the latent random variable. I observe $z$ and want to come up with a estimator for $x$ that minimizes the ...
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Bayesian bootstrap : is this correct?

So my problem is as follow : I have a given string of characters, and I would like to quantify the uncertainty linked to the probability of each letter types in the string, based on there observed ...
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Dual Bayesian Interpretation

I am reading a book which says there are two ways of interpreting Bayes. “In the Bayesian approach, parameters can be viewed from two perspectives. Either we view the parameters as truly varying,...
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Bayes' theorem and likelihood that an observation belongs to one of x clusters

I am trying to write a script is inspired by the following website. Here they examine mixture models, where they treat the distribution of batting averages as a mixture of two beta-binomial ...
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Bayesian LASSO: A step within the Gibbs sampler

I'm intending to implement a Bayesian LASSO inside the Gibbs sampler I use to estimate a multivariate time-series model, but I have a doubt about how to draw this step. The prior is a Double-...
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How is the continuous input probabalistic generative model derived from single class model?

So for the single valued model we have: $p(C_1|\textbf{x}) = \frac{p(\textbf{x}|C_1)p(C_1)}{p(\textbf{x}|C_1)p(C_1)+p(\textbf{x}|C_2)p(C_2)}$ If we rearrange the terms, we can write this as a ...
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Finding an unconditional joint probability in a Bayesian Belief Network

I have a Bayesian network as drawn in the picture: We can see that $B$ and $C$ are conditionally independent given $A$. My goal is to find $P(B\cap C)$. My first thought was to use the Law of Total ...
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Bayesian HDIs for paired samples vs independent samples

According to Kruschke (http://www.users.csbsju.edu/~mgass/robert.pdf), if I have two different groups and collect their response times to a certain task, to determine if the two groups are ...
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How is L2 regularization derived?

I just proved to myself why the regularization is added rather than multiplied to loss function. I did so by taking the MLE formula... $$argmax\sum log(P(x_{i}|\Theta ))$$ and since we know that ...
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Simplify maximum a posteriori

I have the following example of a maximum a posteriori... $$\prod_{n=1}^N \mathcal{N}(y_n|\beta x_n,\sigma^2) \mathcal{N}(\beta|0,\lambda^{-1})$$ Where I am multiplying the prior by the likelihood ...
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How to Apply Bayesian Statistics to Normal Data

I have data that is normally distributed with mean 1 and variance of about 0.006. Each data point is itself a ratio, where the numerator and denominator essentially represent the sample size for that ...
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Different order of insertion - different Bayesian network ? how to prove formally?

I have some Bayesian network which i constructed from some data, say it consists of nodes A, B, C and D and that was the initial order of insertion. If i ...
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Bayes decision rule: Bayes Risk.

Suppose that we replace the deterministic decision function $\alpha(x)$ with a randomized rule, the probability $P(\alpha_i|x)$ of taking action $\alpha_i$ upon observing $x$. (a) Show that the ...
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Understanding Beta distribution in prior belief distribution

I was looking at the Bernoulli Distribution and its relation to the Prior Belief Distribution. The equation is written as $$ \frac{x^{\alpha - 1}(1-x)^{\beta -1}}{B(\alpha, \beta)}. $$ I've also ...
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Computing posterior density

If I have two observations $X=n,X=m$, how do I then compute the posterior density? I can think of 2 ways but I don't know which one is the right one: 1) First compute posterior given $X=n$ $$p(\theta|...
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Pure Strategy Equilibrium in Second Price Action

Bayesian Nash Equilibrium is to always bid the true value. Is this the only equilibrium? Are there any other pure strategy bayesian nash equilibria for second price auction?
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Mean of the posterior distribution in bayesian linear regression with infinitely broad prior

Currently reading from Christopher Bishop's Pattern Recognition and Machine Learning book about parameter distribution under a bayesian linear regression. On page 153, the author deduces that the ...
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Bayesian network probability diamond shape

I am looking to find $P(A=0,D=1)$. We will have something like $(0.5)(.......)$ The dots are the following: $P(D=1|C=0,B=0)*P(C=0,B=0|A=1) + \dots + P(D=1|C=1,B=1)*P(C=1,B=1|A=1)$ How am I ...
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Why is the likelihood for bayesian liner regression in that form?

Let's say we have a linear model $Y = bX + e$ where $b$ represents the parameter. Normally, the posterior is written like this for data $X_n$ and $Y_n: P(B|Y_n, X_n) = P(Y_n|X_n,b)P(b|X_n)/P(Y_n|...
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Bayesian classification question

$x^n=(x_1,...,x_n)$ i.id. $f\in\: \mathcal{F}= \{N(\mathbf{\mu_j},\Omega):j=1,2\}$, the prior is $\pi(1)=\pi(2)=1/2$, with0-1 loss. The gola is to classify $x^n$ as a whole to one of the distribution ...
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I want to have a positive uniform prior for $\mu$ in a Poisson distribution, what $gamma(\alpha, \beta)$ do I use? [closed]

I can't find any examples in my textbook for this, nor can I find anything online. Everything just talks about how beta(1,1) is equivalent to a uniform prior, which is fine, but I specifically need a ...
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Bayesian inference considering known correlation

Let's suppose there are many states that we want to observe: $x_i, i\in \{1...N\}$, and suppose that for each state $x_i$ observation equation $p(y|x_i)$ is given. In this case, given an observation $...
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What is this probability notation?

I came across a conditional probability function that looks like this. $p(y|x; \{ \sigma + \tau \})$ But I can't seem to find what the notation after the ; and ...
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Bayesian Statistics exercise?

I am having issues trying to solve this exercise in Bayesian analysis. The waiting time in minutes until being serviced by a phone call center follows an Exponential(λ) model, with E[y|λ] = 1/λ. Out ...
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Prior with support not $(0,1)$

Suppose the prior is uniformly distributed on $\left(\frac{1}{3},\frac{1}{2} \right)$ and the likelihood is binomial. Now if the prior was uniform on (0,1) the posterior would be a beta distribution. ...
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Dynamic Bayesian Networks and cycles

I am aware that Bayesian Networks and Dynamic Bayesian Networks do not allow cycles. However, there is something I can't figure out and which is simple: what is a cycle in a DBN? Consider two nodes, $...
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Let M be the event that A's blood type matches the guilty party's. Problem says that A does match guilty party's blood type. Why isn't P(M) = 1?

A crime is committed by one of two suspects, A and B. Initially, there is equal evidence against both of them. In further investigation at the crime scene, it is found that the guilty party had a ...
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Proportionality term in Normal-Gamma distribution

I am currently learning from Christopher Bihops's Pattern Recognition and Machine Learning book about posterior distributions for the Normal distribution whenever both $\mu$ and $\tau$ (the precision ...
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What density distribution could be used for time delays?

I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on: The definition of the problem is: Each truck has ...
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Interpretation of Bayes Theorem Example

A protein that has the objective of assisting pig growth is given to pigs in their lunch. A researcher observes that few pigs may have developed tumours as a result of their eating which comprises the ...
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Fusing Sensor Data from Identical Source

I'm trying to find info on methods for fusing multiple readings of the same binary data, given some performance metric such as 'rate of success'. E.g. if N motion detecting sensors are monitoring the ...
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How to interpret the proof that information cascades will form?

I am reading the 1992 paper of Bikchandani, Hirshleifer and Welch on information cascades. They claim and prove that, given an environment of sequential decision making, an information cascade will ...
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Sequential analysis without knowing the hypothetical probability distribution?

When learning sequential probability ratio test, I get the impression that one should know exactly what the hypothesis is, and what the likelihood function is, in order to calculate and accumulate the ...
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Product of two conditional distributions in Bayesian modelling

I am currently reading an introduction to Bayesian Modelling. Im trying to understand why the following is true: $$p(w|x,t,\alpha,\beta)\propto p(t|x,w,\beta)p(w|\alpha)$$ where: $w$ is are the ...
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Calculating a discrete maximum entropy prior

Given a discrete random variable $\theta$ that takes values on $\{\theta_i\}_{1\leq i \leq m}$, with probability $\pi(\theta_i)$ the entropy is defined as $$ \mathcal{E}(\pi) = -\sum_i \pi(\theta_i)...
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Agreeing to disagree “simple” example

I'm looking at Aumann's work "Agreeing to Disagree", and trying to understand the very first numerical example. So, the paper starts with definitions [L]et $(\Omega, \mathcal{B}, p)$ be a ...
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Bayesian statistics… posterior distribution of sigma squared unconditional on theta

There is a section in my textbook that leaves out a lot of steps when showing that The posterior distribution of sigma squared not conditioned on theta is an inverse gamma. I've included a picture of ...
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Independence and Bayes' rule in EWC (Overcoming catastrophic forgetting)

I have a problem in understanding this passage, i.e. $$ \begin{eqnarray} \log p(\theta|D) &=& \log p(D|\theta) + \log p(\theta) - \log p(D) \ldots \label{1}\tag{1}\\ \log p(\theta|D) &=&...
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change of variable for integral to calculate posterior distribution

I'm working through an example which can be found here (p. 36), if someone is interested. I have an integral of the form: $$P(x|\mu)=\int d\sigma P(x|\mu, \sigma)P(\sigma)=\int d\sigma \frac{1}{\sqrt{...
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Conditional Posterior witha Truncated Mutlivariate Prior

My Math knowledge is only limited and I have difficulties to get my head around the conditional posterior to use in a MCMC alogrithm. I want to predict Y, with a non-linear Model. I want to draw my ...
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sampling mechanism for a data pair (X1, X2) where X2 depends on X1

If X2 is dependent on X1, how to generate the random sample of (X1,X2)? One scenario is that we know the prior distribution of X1 and functional relationship between X1 and X2, how to generate the ...
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Can a Bayesian regression problem be regarded two sub consecutive regression?

Assume I have $n$ observations for y and X, which leads to the following regression, $y=X*\beta+e, e\sim N(0,\sigma^2)$ Assume priors: $\sigma^2\sim \frac{\nu_0 s_0^2}{\chi^2_{\nu_0}}$ and $\beta\...
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Maximum bound using Bayes Theorem

A protein that has the objective of assisting pig growth is given to pigs in their lunch. A researcher observes that few pigs may have developed tumours as a result of their eating which comprises the ...
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Objective function to determine posterior distribution.

I am trying to understand the effect of the density of a random variable $a$ which itself is a function of several other random variables $a = f(x,y,z)$, the following question is in accordance with ...
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How to automatically select the grid of $\lambda$ for Bayesian Model Averaging (BMA) ridge regression

I want to use Bayesian Model Averaging (BMA) to do ridge regression on $Y\sim X$. I have learned that one can use cross validation to find the best $\lambda$ for ridge regression. But how can we ...
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Find posterior distribution given beta prior

The output of a certain integrated-circuit production line is checked daily by inspecting a sample of $100$ units. Over a long period of time, the process has maintained a yield of $80$ percent, that ...
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How to spread probability in a “gradient”, i.e. so that it decreases with distance from a point on a dimension?

I have data tasks which involve a little bit of math - which is really not my strong suit - hope you can help. Background Respondent has been given a question that they answer with options on a ...
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Bayes theorem algebra question

A protein that has the objective of assisting pig growth is given to pigs in their lunch. A researcher observes that few pigs may have developed tumours as a result of their eating which comprises the ...