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

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Bayesian Updating

Yesterday I was given a data set $(a_1,\ldots,a_n)$ (i.e., $n$ i.i.d. realizations) and computed a desired empirical conditional probability $P(A_n|B_n)$ where $A_n,B_n$ are events in the data. ...
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Implementing bayesian networks in python for gaze estimation using visual saliency

I am developing an appearance based gaze estimation system based on opencv and python. I have currently developed a prototype which can estimate the gaze based on active calibration, which is ...
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20 views

Upcoming areas in theoretical Bayesian Machine Learning [on hold]

I will soon be going for a postdoc position at a new university. One thing that the panelist asked me to think of is if I were to start, what would I work on. Now the thing is so far with my PhD ...
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Best way to test that one mean is greater than all the others

Suppose I have $k$ samples of different sizes, each of a different univariate variable. I want to test the significance of the hypothesis that the population of $k_0$ has a mean greater than all of ...
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Iterative Bayesian updating when some numerically estimated likelihoods are equal to zero

I am using Bayesian updating to derive the concentration of a contaminant from the results of a sequence of field sampling events. The laboratory methods used to analyze the water samples indicate ...
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Dirichlet predictive distribution

I need to derive a posterior predictive density for observation T+1 given that the posterior distribution of $(x_{1}, x_{2}, 1-x_{1}-x_{2})$ is a Dirichlet distribution with parameter $(\alpha_{1}, ...
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59 views

Can anybody help with this state space model for filtering with random matrix connecting observation and state

I need an urgent help in an issue with a state space model for filtering. My state model is like: $\mathbf{d}_k = \mathbf{d}_{k-1} + \boldsymbol{\eta}_k$ with $\boldsymbol{\eta}_k \sim \mathcal{N}(0, ...
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16 views

Help Simple Conditional Counts Example

Let's say we have a sequence $S$ \begin{align} t \quad 0 \quad 1 \quad 2 \quad 3 \quad 4 \\ S_t \quad 1 \quad 1 \quad 0 \quad 0 \quad 1 \end{align} And we want to predict $S_{t+1}$ from by ...
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Bayes Theorem - Both Events Need Nonzero Probability?

Bayes' theorem: $$ P(A|B) = \frac{P(B|A)P(A)}{P(B)}. $$ Clearly, $P(B)>0$ is required. However, $$ P(B|A) := \frac{P(B \cap A)}{P(A)}, $$ so if $P(A)=0$ we would have $$ P(A|B) = ...
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25 views

What would be more flexible alternative to the Dirichlet distribution that is still “nice”?

Let's assume that I have some measurements about the total mass of certain compound. These measurements are very accurate so let's assume that the mass is just 1 (unit) in each case. Then I know that ...
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10 views

Noninformative prior for variance, understanding and coding

I have three questions regarding the understanding behind and implementation of a noninformative prior for variance. I'm attempting to build a Metropolis sampler and I'm trying to sample from a ...
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30 views

Observed function of hidden random variables

Let's say a worker can perform 4 types of tasks in a day: A,B,C,D. Each of which tasks takes time that is distributed according to some probability distribution, say $$ T_A \sim Gamma(\alpha_A, ...
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23 views

Using empirical priors in PyMC

I'm using PyMC to sample the posterior distribution and I've run into a roadblock with using priors from samples, not models. My situation is as follows: I have some empirical data for a parameter ...
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Parameter estimation using bayesian estimates in $R^k$ Euclidean space?

I am facing difficulty in identifying how the formula given by Eq(2) in the paper ...
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48 views

Understanding Likelihood Function

Just learning Bayesian techniques through Insua et.al.'s "Bayesian Analysis of Stochastic Process Models." On page 18 they give an example of a gambler estimating the parameter $p$ in a binomial ...
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11 views

Bayesian SEM in AMOS - suggest resources?

I'm looking for resources regarding Bayesian SEM estimation using AMOS built-on procedures (I have ordered categorical data). Google gives some basic resources on theory, but not so much on ...
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19 views

Calculating Bayesian Probability for Multiple “Tests”

I'm wanting to calculate the Bayesian probability of a Stock price going up or down based on multiple indicators. There is an abundance of clear literature on calculating the Bayesian probability for ...
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283 views

Calculating SD for normal distribution with only mean and 5% and 95% quantile values

As part of a Bayesian method to estimate the divergence times of species, priors have to be set with values based on previous literature or known fossil dates. These priors can have different ...
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22 views

Computing marginal posterior in a multivariate setting

I am computing posteriors using individual level data and would like to know if my formulation in the end is right. Let the sequence of choices made by individual $i$ be $y = y_1, y_2 ......y_j$ in ...
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55 views

Can anyone give a simple example of when Bayesian and frequentist methods give exactly the same answer? [closed]

Is this even possible given the differences in the philosophies of the two methods?
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Prior distribution on/of a parameter

Is it correct to write the prior distribution on $\mu$ ? I think it is correct because it refers to the fact that "we put a distribution on (the state space of) $\mu$". However, I would not say the ...
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43 views

A simple question about MAP and MLE

I recently got this simple question from a friend. But I am quite confused about it. Suppose we toss a coin $N$ times, and got heads $m$ times. Assume the binomial distribution with $p$ which is the ...
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27 views

AR(1) model - which prior to use?

I want to use the following univariate model: $y_t = \mu_t + \epsilon_t, \ \epsilon_t \sim N(0,1)$ $\mu_t = \phi \mu_{t-1} + \omega_t, \ \omega_t \sim N(0,\sigma_\omega^2)$ That is, $\mu_t$ follows ...
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How to manually do Bayesian fixed effects meta-analysis?

Assume $Y_i \sim Normal(\mu, \sigma_i)$ for $i = 1, 2, 3, \ldots, k$ independent studies. Here I assume $\sigma_i$ is known. The prior distribution is $\mu \sim Normal(\mu_0, \sigma_0)$. What will be ...
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23 views

Relationship between LASSO and MAP

What is the relationship between the LASSO regression and MAP? Is the Bayesian interpretation of the LASSO that x_est is the MAP estimate of x under the prior Laplace pdf? If there is also a ...
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75 views

Interpretation of Bayesian vs Frequentist statement

Although I am completely new to Bayesian Analysis I struggle sometimes when trying to investigate some intersections between Bayesian and Frequentist analysis. I would like to discuss the different ...
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Variational Bayes with non-symmetric priors

I recently developed a Variational Bayes (VB) model with the intention encoding priors on a latent variable that was previously estimated using EM. In particular the VB model has two latent variables, ...
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17 views

How to optimize costly, smooth, multidimensional, varying scale function with flat regions and slight noise

I am trying to optimize hyperparameters of a complex model. Each iteration takes roughly 30s (during which the lower level model is run many times). I believe the underlying function to be generally ...
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15 views

Generalised Bayes minimax estimators

How do you prove that generalised minimax Bayes estimators are more numerical stable than the relative least squares estimators
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66 views

Equal weight between prior probabilities

While constructing a model hierarchy for Bayesian analysis, I have two parameters: $\theta_0$ ~ Uniform(80, 90) $\theta_1$ ~ Normal(0.093, 0.002) I take the $ln$ of the pdf for the parameter's ...
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61 views

ROC curve for two-sided cut-off

I am very, very confused about ROC curves. I have a Bayesian model which outputs a prevalence on a continuous scale between 0 and 1. I have a classification I would like to use that classifies that ...
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79 views

Bayes theorem: normalisation denominator and likelihood

I have been racking my brains trying to understand Bayes theorem. So, the way I have understood is that the likelihood is the probability of observing the particular outcome given a set of parameter ...
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What is the problem with empirical priors?

In literature I sometimes stumple upon the remark, that choosing priors that depend on the data itself (for example Zellners g-prior) can be criticized from a theoretical point of view. Where exactly ...
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What do/did you do to remember Bayes' rule?

I think a good way to remember the formula is to think of the formula like this: The probability that some event A has a particular outcome given an independent event B's outcome = the probability of ...
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how to find the aleatory uncertainty in parameter using Bayes?

Generally, the uncertainty can be categorized into aleatory and epistemic according to whether it can be reduced or not. In Bayesian statistics, one "true fixed parameter" is presumed as discussions ...
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Hyperprior Noninformative Beta Binomial Model

I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model ...
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36 views

Metropolis-Hastings acceptance rate confusion

I ran a Bayesian model that have about 2700 parameters. Among these parameters, Adaptive Metropolis algorithm was implemented to estimate ~790 parameters in the I-group and Metropolis algorithm was ...
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Independence of “residuals” in a Bayesian multilevel hierarchical model

So i'm having some problems realising what model checks I should do after fitting a bayesian model other than convergence diagnostics. Lets say i'm fitting a hierarchical bayesian regression model, I ...
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18 views

How to uniquely define the credible interval?

Consider we are in the Bayesian paradigm and consider that we know the posterior $P(\tau|d)$ of the single parameter $\tau$, whose max of the posterior is given by $\tau_{\text{best}}$. The credible ...
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28 views

MAP Parameter Estimates with Bayesian Logistic Regression

I am trying to reproduce the CLG algorithm for the Laplace prior given in Genkin et al to find the MAP estimates for a logistic regression model. I am using Python (Anaconda 2.2) with Numpy to ...
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Expected Euclidean distance between normal prior and posterior as sample size changes

Suppose the prior for some random variable $x$ is normal with mean $\mu$ and variance $v$. Denote the prior density by $f(x|\mu,v)$. Given normal likelihood with known variance $v$, the posterior for ...
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Jeffreys Prior for normal distribution with unknown mean and variance

I am reading up on prior distributions and I calculated Jeffreys prior for a sample of normally distributed random variables with unknown mean and unknown variance. According to my calculations, the ...
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Save and Restore current state in PYMC [migrated]

Recently, I launched a Bayesian model run that are written in PYMC. Due to power outage, the results generated during halfway of the run are gone. So, the logical step is to look for ways to save the ...
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Stepwise regression for Bayesian models

Why isn't stepwise regression, like backward elimination, used for Bayesian models? What is generally used to find insignificant variables in bayesian methods? Or does one simply not worry about ...
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52 views

Significance of explanatory variables in Bayesian models

I was wondering if there is a general way to handle parameters of which posterior distributions include zero. Should one remove these parameters and refit the model? E.g. You fit a regression model ...
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Questions about the paper of “inferring causal impact using bayesian structural time series models”

I am simulating this model in the paper Brodersen et al. (2015) and found I have difficulties on figuring out how to choose the starting point of Bayesian inference. To be explicitly, to my ...
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1answer
51 views

Using Bayesian inference to fit functions

I'm trying to understand Bayes Inference better and was wondering if it is possible to use it to fit a function $f(t)$ where I only know the value $f(t)$ for a few values of $t$. For each value of $t$ ...
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Bayesian model selection, functional form of variance

I'm working on a linear regression model of the form, $$y = X\beta + \epsilon(X) $$ where each $$\epsilon_i \sim N(0, \sigma^2_i)$$ My variance term is depends on a subset of the regressors $X$. ...
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Selecting correct priors for Bayesian A/B Testing

I'm running an A/B test. The test is a funnel that, at some step in the funnel, sends half the population to experience A, the other half to experience B. Traditionally everyone saw experience A at ...
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Pseudoreplications and the methodes used to explore the correlations

In my experiment I have measured growth of different trees on predefined circular plots (x, y, z, a). On each plot all trees were measured. For each location I have one treatment information. Now ...