Questions tagged [bayesian]

Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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Bayesian update vs optimization in multivariate case

Say I have a multivariate normal vector $r$~$N(\mu , \Sigma )$ and I observe that $ y \equiv Pr + \epsilon = Q$ where $P$ is a matrix and $Q$ a vector and $\epsilon$~$N(0 , \Omega )$. Now I ...
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Conditional probability table from deterministic relationships of two discetizied distributions - for Bayesian Networks

Consider a simple Bayesian Network of three variables A, B, and C. All of the variables are discrete variables between (0,1] that are discretized as below: ...
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17 views

Estimate distribution of aleatoric variable using Bayesian inference

Given a model as follows: $$y = cx + e$$ where y is the model output, x is the model input, c is an unknown variable and e is a Gaussian model error with zero mean: $$e \sim N(0,\sigma)$$ Data is ...
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415 views

Why does the likelihood function of a binomial distribution not include the combinatorics term? [duplicate]

So the likelihood function for a binomial distribution is: Why is the likelihood function above not multiplied by a combinatorics term: n! / (x! * (n - x)!) If the likelihood function is ...
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1answer
30 views

How do I approach this Bayesian question?

I am auditing a Bayesian Statistics course and I am facing problem in understanding the following question. Suppose you are given a coin and told that the coin is either biased towards heads (p = 0.6)...
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Marginal In Bayesian Optimization Expression

I am reading a paper and presentation on batch Bayesian optimization and came across the following formula. Question 1: Does the expression inside the redbox evaluate to $p(\mathcal I_{t,0})$? ...
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1answer
36 views

Probability that a set of values came from a distribution

I have a probability distribution. And I have a set of values. I need to figure out how to calculate the probability that these values were generated by the same model as the distribution. I found ...
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105 views

In the example of guess a specified number between 1 and 20 (both inclusive), what is the sample space?

This post is discussing Bayesian reasoning in the context of guess a specified number between 1 and 20 (both inclusive). Consider the following example: I’m thinking of a number between 1 and 20 (...
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Covariance Matrix of Bayesian Network [on hold]

I have a bayesian network that the edges are likelihood estimations from features {x1,...,xn}. how can to estimate a covariance matrix from bayesian net between x? Normally, we use from a correlation ...
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11 views

How do we determine marginal Pr(Data) in Bayesian Analysis?

I am trying to learn Bayesian Analysis and I am really confused as to how to calculate the required equations/values. From a very high level standpoint, I understand the concept. We basically use ...
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17 views

Conditional expectation of hierarchical model parameters via marginalisation

Firstly apologies, I have fairly limited mathematical skills so there is a good chance that my question is simple or obvious. I have a model in which I want to calculate the conditional expectation ...
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21 views

Bayesian Optimization does not improve RMSE of XGBoost

I have some serious problems with Bayesian optimization of an XGBoost model. The optimal hyperparameters resulting from Bayesian Optimization lead to an RMSE that is higher than through ...
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Making inference about correlated values after some observations

I'm working on making an inference out of some observations of correlated values. Suppose there are two values $x_1$ and $x_2$ which are independently drawn according to a CDF $F$. There also are ...
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8 views

Bayesian case control binary regression

tl;dr I would like to evaluate the posterior distribution for the parameters $\theta$ of a binary regression model $P(y_i=1|x_i, \theta)=\rho(x_i, \theta)$ given the features $x_A$ of all positive ...
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Why does the marginal likelihood integral have no closed-form solution?

In Bayesian inference we end up with the formula: $$ P(\mathbf{w|t,X)}= \frac{P(\mathbf{t|w,X)}P(\mathbf{w)}}{\int P(\mathbf{t|w,X}) P(\mathbf{w}) d\mathbf{w}}$$ Assume the prior $P(w)$ is a ...
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24 views

Are the **likelihood** in Bayes Rule the same to the one in [Maximum likelihood estimation][1]?

"Think Bayes by Allen B. Downey" calls the P(X | A) part likelihood in Bayes Rule \begin{align} P( A | X ) = & \frac{ P(X | A) P(A) } {P(X) } \\\\[5pt] \end{...
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Can I use a Bayesian method to test for enrichment instead of the hypergeometric test?

When I test for enrichment with the hypergeometric test, I determine the probability of having obtained a number of successes in a sample given the total number of possible successes/failures in the ...
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Understanding priors and full conditional posteriors

I have a question regarding this problem that we are discussing in one of my classes. a) I understand that the prior is a beta distribution with the given parameters below, and that if I have a ...
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5 views

Interpretation of tauBs in output of jointModelBayes [closed]

I am using the JMbayes package for R to fit joint models between a longitudinal and time-to-event outcome. The model output lists a variable for "tauBs" however I am uncertain as to what this refers. ...
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1answer
12 views

Posterior distribution and multiple parameters

In this problem I am doing I am trying to model count data with the negative binomial distribution. $k|(r,p)$ ~ $negBinom(r,p)$ Where we have the following priors: $ r $ ~ $Exp(2/3)$ and $p$ ~ $...
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47 views

Could someone please explain why it is called **marginal density** in the context of Bayes' Theorem?

This post calls the P(X) part marginal density in Bayes Rule \begin{align} P( A | X ) = & \frac{ P(X | A) P(A) } {P(X) } \\\\[5pt] \end{align} Could someone ...
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What does it mean when a distribution is “INDEXED” by something? [duplicate]

I am doing some reading and am seeing phrases such as the following. For context, I am learning about Bayes Inference and prior/posterior distributions so theta is the parameter: In such problems, ...
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Bayesian Linear Regression, trouble with posterior. Variance equal identity

I am trying to solve the following problem. If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $ \pi(\beta|y)$ and show that $E(\beta|y)$ is a function ...
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1answer
31 views

Literature on Bayesian stuff with Normal Distribution?

I am writing something on Bayesian Analysis involving the normal distribution. I know that the conjugate prior is the so-called normalized Gamma inverse distribution, I know the update rule for the ...
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What is the prior of $\ell_{2,1}$ loss in Multi-Task learning?

We all know Laplacian prior is the prior for Lasso, as the MAP of a Bayesian setting. Multi-task lasso is a generalized lasso for multi-task problems, which encourages group-wise sparisty. However, ...
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When did MCMC become commonplace?

Does anyone know around what year MCMC became commonplace (i.e., a popular method for Bayesian inference)? A link to the number of published MCMC (journal) articles over time would be especially ...
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Difference between a Bayes classifier with diagonal multivariate gaussian class conditionals and a Naive Bayes classifier?

In a Bayes classifier, let's say we want to fit a multivariate Gaussian distribution for the class-conditional probabilities and we restrict its covariance matrix to be a diagonal matrix. In a Naive ...
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3answers
46 views

Bayes: How many heads given n flips?

I’m trying to wrap my head around a rather simple coin flip experiment. Say I have studied a coin and obtained a posterior for the probability of getting a head when flipping it. How do I then use ...
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1answer
30 views

Probability of location based on cell tower signal strength

I'm tackling the following problem: Your cell phone is constantly trying to keep track of where you are. At any given point in time, for all nearby locations, your phone stores a probability that ...
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44 views

Using information involving multiple model parameters as a prior

I am estimating a relatively simple linear equation of the following form: $$ y = \beta_0 + \beta_1p + \beta_2t +\epsilon $$ I would like to take a bayesian modelling approach, and have existing ...
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2answers
71 views

Why do we draw parameters to draw from a marginal distribution that does not contain the parameters?

I'm sure this is dreadfully simple to everyone who sees it, but I'm completely lost. From what I've read, I think this concept must be so fundamental to Bayesian statistics that no one bothers to ...
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1answer
21 views

Problem using Bayes Rule [closed]

I own a dog that I don't always remember to feed in the morning before work. By the time I get home from work, I remember whether or not I fed the dog. I feed the dog 60% of the time. If I feed the ...
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39 views

Bayesian updating of a probability density for evidence on its cumulative distribution

Suppose that I have a continuous variable E as a result of a simulation, which has a probability distribution as in the figure below: As seen from the cumulative plot, ...
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How is the total error equation derived?

In this video Ng said that there is a Bayes optimal error(also called irreducible error), and in this article I learned that the total error is equal to the squared bias plus the variance plus the ...
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Dirichlet Process mixture model with independent features

I'm trying to construct a Dirichlet process mixture model for clustering where the samples have independent features. In other words, to evaluate the likelihood of sample $x_i$, I would compute $\...
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17 views

Bayesian hierarchical model with varying scales [on hold]

I would like to have a bayesian hierarchical regression model. Suppose that we have multiple data sets, which adhere to a hierarchy. Let us call the response variable in dataset 1 and 2 as $Y_1, Y_2$. ...
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14 views

Between Subject Design with multiple Trials per participant

Between the subject experiment with two groups: Each group has a pair of 10 participants. (Player 1 and Player 2) Pairs in both groups play a game where one factor is manipulated. Hypothesis: The ...
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2answers
34 views

What is the conceptual difference between posterior and likelihood? [duplicate]

I have trouble discerning conceptually between these two notions. I am aware of their formal relations, proprieties and what not, but I just can't wrap my head around what they "mean", if that even ...
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22 views

bayesian estimation of difference between 2 non-normal groups

Lets say we have 2 sets of groups with random variable X as shown. Features of X based on real dataset: They are all positive numbers have really long right tail and almost no left tail Cant share ...
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1answer
33 views

The significance of 'significant'

I am writing up a paper where I report the results of a set of analyses using Bayesian parameter estimating but am really struggling to come up with synonyms for significant. Take this sentence. "...
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18 views

Samples from distributions that change slowly in time [closed]

Suppose I receive a sequence of random samples $\{(t_i,x_i)\}_{i=1}^\ell$, where $x_i\sim\rho_{t_i}(x)$. That is, $\rho_t$ is a time-varying distribution over $x$, and occasionally I receive a single ...
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48 views

Dirichlet Process Concentration Parameter - collapse at zero

Background I've implemented the blocked gibbs sampler for sampling from the posterior of a dirichlet process mixture model as described on p.552 of Bayesian Data Analysis, placing a Gamma prior on the ...
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1answer
63 views

How is that likelihood is fixed by the model

When talking about conjugate distributions on a online video, it applies the Bayes theorem as: $$ Pr(\theta|X) = \frac{Pr(X|\theta) Pr(\theta)}{Pr(X)} $$ and says that $Pr(X|\theta)$ is fixed by our ...
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1answer
20 views

“Adaptation incomplete” in rjags. How bad is it?

I'm new to jags and Bayesian inference, but I've run a fairly complex model in jags via rjags. However, I get the warning "adaptation incomplete". As far as I understand it, this means that the ...
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19 views

Bayesian model when the raw data isn't available? [closed]

I have data of the form $X = (X_0, X_1, ..., X_T)$ where $X$ is a sequence of categorical variables ($k$-dimensional) captured over time $t = 0,...,T$. The variables are fully observed. I am trying to ...
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1answer
34 views

Are there some measures of information content of Bayesian evidence?

Let us have a binary random variable $X$ with values $a,b$ and a prior distribution $P(a), P(b)$. Now, let us suppose that we learn a piece of evidence $E$. $E$ is a random variable with two values, $...
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1answer
41 views

Updating posterior probability as more data is given

There are a couple of questions with similar title, but I don't think my doubt is addressed by those, so asking a new question. I have some random variable $X$ and another $D$ such that initially the ...
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34 views

Conditional Independence Assumption

In Bayesian inference, but also in MLE, the conditional independence assumption (CIA) is a usual assumption. Typically, it is used to simplify the likelihood of the model, and to allow the ...
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28 views

$X\sim \mathcal{N}(\theta,\sigma^2)$, $\pi(\theta,\sigma^2)\propto 1/\sigma^2$, $Y\sim \mathcal{N}(\rho X,\sigma^2)$, $\rho$ fixed. $f(y|x)$?

like in the title I have the following question. Let $X\sim \mathcal{N}(\theta,\sigma^2)$ with the improper prior $\pi(\theta,\sigma^2)\propto 1/\sigma^2$ and consider $Y\sim \mathcal{N}(\rho X,\...
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analogy on Tenenbaum's phd thesis (on prior, likelihood, posterior)

This is from the book: Machine Learning from a Probabilistic Perspective page 69 and 70. There is a very interesting analogy/explanation on how to visualise Prior, Likelihood and Posterior ...