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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1answer
23 views

MMSE estimator with dirac delta prior pdf

The question is as follows, it's mainly part 3 that I was having problem with. A discrete-valued parameter with the prior pdf $$p(x) = > \sum_{i=1}^2p_i\delta(x-i)$$ is measured with the additive ...
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3answers
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Bayesian analysis used merely as a computational tool?

I have sometimes seen some statisticians used bayesian analysis and related techniques such as MCMC simply as a tool when a frequentist approach is not satisfying, typically for example when the ...
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Beginner: Understanding difference between pmf, conditional pmf and likelihood

I have a point of confusion regarding the three types functions. I have looked at some other posts here and blogs and scripts and YouTube videos. But I still don't get it. Let's look at the coin toss ...
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What can I do to make Student-T priors in a brm Bayes model less tall? [closed]

What can I do to make Student-T priors in a brm Bayes model less tall? Particularly this is a linear model: $a+bx$ And my current priors are: ...
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How to select a variance for student-t, when it approximates coefficient of linear model?

How to select a variance for student-t, when it approximates coefficient of linear model? Especially in context of Bayesian priors. Or as in: https://github.com/stan-dev/stan/wiki/Prior-Choice-...
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Is there any relationship between confidence ratio in association rule and Bayesian rule?

I am currently studying Association Rule and somehow I thought about if there is any relationship between confidence ratio in Association Rule and Bayesian Rule. My knowledge in Bayesian Rule is not ...
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1answer
26 views

If one fits a linear model with just one $x_i$ then does this mean df=1-2=-1?

If one fits a linear model with just one $x_i$ then does this mean that the number of degrees of freedom $=1-2=-1$? For a linear model the degrees of freedom is: $${\rm df}=n-k$$ where $k$ is number ...
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1answer
15 views

Find marginal distribution of X (Bayesian setting)

$X|\theta$ follows $N(\theta,w)$ and $\theta$ follows $N(\mu,\sigma^2)$ Both follow a normal distribution but with different mean and variance. I assume it is a Bayesian setting. How to find the ...
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2answers
23 views

What methods exist to identify the optimal splits in some exogenous variable, such that a dependent variable is maximized?

I've come across an interesting problem recently, and I'm wondering if I'm missing some obvious approach here. The problem statement is thus: Imagine I run an online business, and I'm interested in ...
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17 views

Running Regression estimation using rstan [migrated]

I am using stan through rstan package in R. Below is my model. This model has an interaction term as ...
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1answer
20 views

Regarding Gibbs sampling and HMC in fitting Bayesian model, their differences and advantages

I have a question regarding the two MCMC algorithms, Gibbs sampling and Hamiltonian Monte Carlo (HMC) for performing the Bayesian analysis. If using Gibbs sampling, my understanding is that we need to ...
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1answer
58 views

Why do we need the concept of Risk in Bayesian Decision theory?

I'm studying Bayesian decision theory as introduction to machine learning and I see the concept of Risk in a lot of places. In the course I read, they define risk as: Risk is the expected error ...
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1answer
36 views

Statistical conclusions from incompatible measurement results

I assume that the following situation is not uncommon in scientific practice: Two research groups analyse two samples. The reported results concern the same physical quantity at the same location ...
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21 views

Bayesian multivariate regression with common coefficients

In a hierarchical model I'm working on, I have $K$ different $N\times P$ predictor matrices, each denoted $X_k$ and $K$ length $N$ outcome vectors each denoted $y_k$. Essentially, I have a ...
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0answers
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Can stochastic gradient descent for Bayesian Inference? [duplicate]

I was looking at the Bayesian MAP estimate formula which is the "argmax(likelihood * prior)". Can this be calculated using stochastic gradient descent? Gradient descent requires knowing the ...
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1answer
43 views

Coherence of conditional probabilities

Dennis Lindley's paper The Philosophy of Statistics in 2001 includes the following 'simple' example of statistical coherence: "A set of uncertainty statements is said to be coherent if they ...
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1answer
45 views

Conjugate Prior for Alpha Power Inverse Weibull Distribution

Let $X$ has Alpha Power Inverse Weibull (APIW) distribution with pdf $f(x) = \frac{\log \alpha}{\alpha - 1} \lambda \beta x^{-(\beta+1)} e^{-\lambda x^{-\beta}} \alpha^{e^{-\lambda x^{-\beta}}}, \; x&...
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1answer
81 views

Can Bayesian Model Averaging be Optimal when the Hypothesis Space does not contain the true hypothesis?

I am utterly confused. I have been reading about the optimality of Bayes classifier and Bayesian model averaging all the time, but when I try to dig deeper, I just get more confused. On the one hand, ...
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Difference between Bayesian T-Test and Bayesian Informative Hypotheses Evaluation (BAIN) on JASP?

0 1 I have three friends in an honours project and we are trying to perform a Bayesian Analysis. I am doing a Welch’s T-Test due to very uneven group sizes, one is doing a linear regression and ...
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1answer
83 views

Bayesian Estimation: MCMC vs MAP

I'm still relatively new to understanding the bayesian mentality. MCMC (e.g metropolis hasting) finds out the posterior distribution of the parameters of interest. MCMC requires taking many samples ...
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0answers
64 views

What distribution has the following likelihood function?

I'm working with a model that uses the Beta-Binomial natural conjugate family. In other words, the prior over the variable of interest $\theta \sim Beta(\alpha_0,\beta_0)$ distribution over $[0,1]$ ...
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1answer
22 views

Rationale behind ignoring the "denomintator" in Bayes Rule [duplicate]

In the context of MCMC sampling, we often say that the posterior distribution is only proportional to the numerator of Bayes Law. We tend to say that the "denominator" (i.e. the normalizing ...
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28 views

The Role of Summary Statistics

I am reading about this algorithm called "ABC" (Approximate Bayesian Computation). https://cran.r-project.org/web/packages/abc/vignettes/abcvignette.pdf (page 3) Over here, it makes mention ...
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1answer
76 views

Bayesian MAP Estimates

Suppose you have a simple linear regression problem (y = bo + b1x) and you decide to use Bayesian Estimation to estimate the value pf bo and b1. Using Bayesian Estimation, you obtain a list of ...
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0answers
20 views

Parameter extraction and validation of robust Bayesian regression

I am getting wrong parameter estimates of my regression model. Compared to the pre-build posterior plot my manual estimate is wrong - because it is outside/next to the range it sould be. This is my ...
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1answer
26 views

How to manually adjust predicted probabilities from lm based on prior lognormal distribution parameters?

@drob showed a great example of adjusting batting averages using a beta-based prior distribution. He used a prior calculated Beta distribution to adjust batting averages individually, and it’s as ...
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1answer
14 views

JAGS error in the estimation of a simple INAR model

I am having a hard time try to figure out how to translate a simple INAR(1) model in JAGS. \begin{equation} Y_t = \alpha \circ Y_{t-1} + e_t \end{equation} where $\circ$ is the binomial thinning ...
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1answer
29 views

Learning HMM parameters by counting?

In 8.4.3 of the book Speech and Language Processing: An introduction to natural language processing, the two matrices transition probabilities and emission probabilities can be learned by counting as ...
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0answers
18 views

Bayesian regression for the sum of Gaussians

I'm pretty new to Bayesian statistics and I want to use Bayesian regression on a 2D data set (frequency on x-axis and measurement data on the y-axis) to quantify the uncertainties. The model is a ...
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2answers
37 views

If the prior and likelihood not be conjugate, how to get conditional distribution to sample from using Gibbs sampling?

I know that when prior is conjugate with the posterior, by writing the loglikelihood and log prior and eliminate the non-independent terms for each parameter one can get the conditional distribution ...
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2answers
26 views

Is it possible to decompose the model evidence?

Assume I want to apply Bayes theorem with some state variable $x$ (scalar or vector, doesn't matter) and an observation vector $\mathbf{y}=[y_1,...,y_N]^T$:: $$p(x|\mathbf{y})=\frac{p(x)p(\mathbf{y}|x)...
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Showing the Posterior of Distribution of Normal Data with Uniform Prior on mu and log sigma has a t-distribution with (n-1) df

we were told in class that if you have data x_1, x_2, ..., x_n that are iid normal with mean = mu and variance = sigma^2 and then put a uniform prior on mu (from -M1, to M2) and log sigma (from -M2 to ...
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67 views

Bayes Theorem and (some) examples of equivicating the likelihood function with the posterior probability

So a common talking point re. statistics in parlance goes something like this: "Group A makes up x% of the population, yet makes up y% of [insert subgroup here]" where $x\% \leq 50\%$ and $y\...
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0answers
13 views

How to measure the strength of association between two variables where majority of pair are assosiated?

I am quite new to the stat so facing a huge problem in result extraction, I have a large dataset running ~19000 (genes) x1500 (patients). I would like to see the dependence between two variables (one ...
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1answer
141 views

Why is naive Bayes overconfident?

In the fourth edition of "Artificial Intelligence: a modern approach" by Russel and Norvig, they write in section 12.6, regarding the Naive Bayes Model for text classification, the following:...
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9 views

Estimating the population distribution of a quantity from multiple finite-error experiments

I'm trying to understand how one can estimate the "true variability" of a quantity, given a finite number of experiments, where each experiment collects a finite number of data points. ...
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0answers
33 views

How to create bayesian test for ROI/ROAS metric

I am interested in using A/B Bayesian testing to compare ROAS for marketing scenarios. I've seen references dealing with cases such as: Using a beta distribution prior to model a binomial-distributed ...
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0answers
17 views

Abuse of notation : same function name for different distributions

Is this too much of an abuse of notation to use the same letter, e.g. $f$, to designate the joint - $f(x,y)$, marginal - $f(x), f(y)$, and conditional - $f(x|y), f(y|x)$ - probability/cumulative ...
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1answer
36 views

Making sure that the design matrix is positive (-semi) definite

In bayesian linear regression, how to make sure that the design matrix produced by a neural network $ \Phi$ is positive definite? Because to computing the covariance matrix on the weight requires ...
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0answers
15 views

Reducing Size of Credible Interval Bayesian Regression Pymc3

I am interested in ways to reduce the width/size of a credible interval in a Bayesian regression. Suppose you have a simple Bayesian linear regression $y \sim \mathcal{N}(\mu, \sigma)$, formulated ...
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1answer
31 views

Bayesian updating of a constant probability using one data point

A reformulation of a question that came up in a model: Imagine a toy store that sells $K$ toys, where our prior is that each toy has equal probability $1/K$ of being purchased by a customer. Then you ...
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13 views

Modelling time both as a fixed effect and as part of an autocorrelation structure?

I want to build a model to assess whether a species is declining in three different national parks. My dependent variable is count data of the species and I have date, park, season and food as ...
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2answers
33 views

Compute likelihood of state given multiple observations?

I am trying to use Bayes formula to compute the likelihood of a given state given a collection of independent but not sequenced observations - knowing the priors and knowing the probabilities of being ...
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0answers
26 views

Rate of convergence for Beta posterior

Recently I started studying posterior distribution rates of convergence. In order to check what I understand I tried to formulate an example. Let $X_{1}, X_{2},..., X_{n}\sim Bin(N,p)$, we then ...
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0answers
26 views

Summarization and resources for Bayesian decision theory

Looking for textbooks and/or resources to get familiar with Bayesian decision making. I have the book, Statistical Rethinking, by Richard McElreath and I've found this to be a really great resource ...
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0answers
21 views

Help in understanding zero inflated neg binomial model summary

I'm writing this topic because I would need to get some more information about model conversion in brms (zero-inflated_negbinomial) model. Let's say I have this model result : Where I want to model ...
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1answer
46 views

Help with rstan models [closed]

I would need help in order to write a specific Stan model. The biological question The idea of the model is modeling the number of Bones (NbBones : discret ...
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0answers
12 views

Centre-scaled data and Approximate Bayesian Computation (ABC)

I have 200000 simulations and I want to use Approximate Bayesian Computation (ABC) to determine the best 1000, based on specific targets. These simulations have 12 parameters (my priors, dependent ...
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0answers
140 views

Showing calibration of Bayesian credible intervals

I'd like to try and understand how one can prove that a particular strategy for assessing correctness of computational methods for Bayesian inference is sound. For a number $M$ of simulations, ...
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0answers
8 views

Estimating true prevalence from apparent prevalence using priors for sensitivity and specificity in a Bayesian model

I want to use the estimates for the sensitivity and specificity of a diagnostic test as priors in a Bayesian model to estimate the true prevalence in a given population. The sensitivity and ...

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