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Questions tagged [prior]

In Bayesian statistics a prior distribution formalizes information or knowledge (often subjective), available before a sample is seen, in the form of a probability distribution. A distribution with large spread is used when little is known about the parameter(s), while a more narrow prior distribution represents a greater degree of information.

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How to obtain likelihood ($P(B/R)$ given the prior $P(R)$ and the posterior $P(R/B)$

I am working on a topic related to multiple-choice response. I would like to measure the efficiency of the information source (or a student’s information search) and I believe Bayesian statistics is ...
Francisco 's user avatar
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Can I use the Mean Squared Prediction Error to select the prior SD in a CausalImpact model?

I'm using the CausalImpact package (in R), and (as I expect is typical) the findings are very sensitive to the prior being used. I have an OK understanding, I think, of what the prior is doing in this ...
André CB's user avatar
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Estimating Markov Chain Probabilities with Limited Data

Suppose I have some data on transitions between states of a Discrete Time Markov Chain. Let's say that transitions between some events are observed more frequently from others. For example, in a 3 ...
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Does the example given correspond to a prior predictive check?

Could someone explain to me precisely what is meant by prior predictive check, in Bayesian inference? In some documents, one uses observed data (“in which we ...
Andrew's user avatar
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How to decide the parameters of a Gamma distribution for a Gamma-Poisson model?

In Bayesian inference, the Gamma-Poisson model uses usually a Gamma($\alpha$,$\beta$) prior on the $\lambda$ parameter of the Poisson distribution. Are there any rules for setting appropriate values ​​...
Andrew's user avatar
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Strange Variance Term for Normal Prior $w^2\sigma^2$

I've attached two screenshots, one with the question and one with the answer. It seems to me that the prior is wrong and it should include $w^2$ not $w^2\sigma^2$ I apologise for, including such a ...
CormJack's user avatar
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1 answer
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How to choose between gamma and Gaussian given a choice of gauges?

I'm trying to make the choice between the gamma and Gaussian distributions as a prior distribution for some data. When I learned statistics a while ago, I was given the rule of thumb: if your data ...
Corbin's user avatar
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1 answer
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BIC with non-negligible priors

I want to do model selection based on the best-fit/MAP/marginal posterior I find from an MCMC and likelihood maximization. I have a likelihood $\mathcal{L}(X|\theta)$, some informative priors $\pi(\...
ojima's user avatar
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Understanding PRIOR option in SCORE statement for PROC LOGISTIC (SAS)

Say I have a binary response which I want to model with logistic regression on covariates $x$. Fitting a model with PROC LOGISTIC will fit MLE coefficients for the model $$ \text{logit}(\pi) = \alpha +...
cpahanson's user avatar
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1 answer
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Avoid singular fits in mixed models in R with blme - checking layman's priors

While fitting linear mixed models, I would like to avoid zero random-effects (ranef(model)) and cluster-level SD estimates (...
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Turning a list of cost into categorical probability mass distribution

Background Given a noisy dataset $D$, I have to solve a classification problem where the possible anserwer is $i\in\{1,\dots,N\}$. So far I can get pretty decent result with an algorithm that, based ...
matteogost's user avatar
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How should uncertainties be treated when scaling data for optimisation

I have a large dataset for which I am using Bayesian statistics for parameter estimation and model selection (using MultiNest for more detail). This involves setting a prior over which the nested ...
shram's user avatar
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9 votes
2 answers
411 views

How is data generated when using an improper prior

Let $X$ be an $\mathcal{X}$ valued random variable. We are doing Bayesian statistics. Suppose that $\theta$ is a $\Theta$ valued random variable with known prior distribution $\Pi$ and that the ...
温泽海's user avatar
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Can we solve by hand the early exit multi-class classification problem? [closed]

Problem: Find a solution $\hat{\varepsilon}$ of the following minimization problem \begin{align*} &\min_{\varepsilon \in \mathbb{R}^M} \sum_{h=1}^M \varepsilon^h \hat{R}^h+\beta \sum_{h=1}^M \...
ohana's user avatar
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Random sequence generator algorithm non informative piror distribution

I want to conduct a Bayesian statistical analysis of a sequence generation phenomenon. The sequences generated contain elements from a known alphabet. Working on that, I have tried to define the prior ...
Guilhem Nespoulous's user avatar
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How to interpret a noninformative joint prior?

I am currently working on a homework assignment and have the following question: $\theta_1$ and $\theta_2$ are parameters of interest and $y_1$ and $y_2$ are the likelihood functions which are $\text{...
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How to derive conditional destribution of MVN variable

I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147): $$Y \sim MVN(X\beta, \sigma^2I)$$ $$\beta|\sigma^2 \...
BlankerHans's user avatar
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1 answer
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Full conditional posteriors

so up to now I dealt with posteriors in the form of: $$p(\theta|x) \propto p(x|\theta) p(\theta)$$ No we started to model a linear regression with the bayesian approach: $$Y \sim MVN(X\beta, \sigma^2I)...
BlankerHans's user avatar
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How to sample from the prior predictive distribution with the BSTS R package?

Assuming I have to use the bsts package in R, I'm trying to understand the degree to which my prior distribution choices (implicit or otherwise) are consistent with ...
user3215964's user avatar
2 votes
0 answers
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Credible intervals with parameter near boundary

When doing Bayesian inference on a parameter that is bounded, often we use priors that approach 0 as the parameter approaches the boundary. For example, when estimating $(\mu, \sigma^2)$ for normal ...
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How to select a proper prior to control the time dependent structure of variable?

I am new in analyzing RCT data and not familiar with the techniques that are always used in RCT analysis. I am analyzing a dataset of a study: An RCT study with 50 participants; the data was collected ...
doraemon's user avatar
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8 votes
2 answers
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Do we ever use the prior predictive distributions of Bayesian Statistics?

As my question states, I am wondering if there is any chance we use the prior predictive distribution. I am studying Bayesian Statistics and have understood what it is. It is a must to go through in ...
mathccino's user avatar
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Understanding the Binomial likelihood notation

Let $X \sim Bin(n,\pi)$. I don't understand why the binomial likelihood is then given by $f(x|\theta)=\binom{n}{x} \theta^x (1-\theta)^{n-x}$. Shouldn't it be $B(x|\pi,n)=P(X=k)=\binom{n}{k} \pi^k (1-\...
BlankerHans's user avatar
3 votes
1 answer
162 views

Bayesian linear regression: How to enforce constraint on the sum of coefficients?

I have a linear regression problem in which my $X$ matrix is not full rank. Here is a small example: $$X = \left[\begin{array}{rrrr} -1 & 0 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 &...
ischmidt20's user avatar
6 votes
1 answer
156 views

Trouble understanding priors

The following comes from a book called Bayesian Statistics by Ben Lambert: Assuming the following model for $r$ disease-positive people out of $n$ people: $$Pr(Z=r, \theta) = {n\choose r}\theta^r(1-\...
HMPtwo's user avatar
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3 votes
1 answer
107 views

Choosing Bayesian Priors [duplicate]

I am fairly new to Bayesian Modeling, however I am experimenting with such framework in order to produce several estimates. The part I am struggling the most with is the selection of prior ...
Marco De Virgilis's user avatar
4 votes
3 answers
108 views

How subjective are prior beliefs?

I find the term 'prior-beliefs' to be a little bit vague, what is acceptable to class as a prior belief in Bayesian analysis? For example, I could not look at any data and decide to myself "I ...
Ewan McGregor's user avatar
8 votes
1 answer
204 views

How to match my prior beliefs to beta distribution?

I have some data that I believe comes from the binomial distribution. I also have some old data from a past-experiment that I would like to base my prior beliefs on. The old data observations are: $$6,...
Ewan McGregor's user avatar
2 votes
1 answer
59 views

Using old posterior as new prior given new data [duplicate]

Suppose I have some data, and use this data to create a posterior distribution. Now suppose I have some new data that I believe is from the same population as the data before. Can I now use my old ...
Ewan McGregor's user avatar
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Confusion forming prior beliefs

Let's say, for the sake of ease, I have a sample of data that I believe comes from the binomial distribution. Let's say I have 100 observations in my sample. Previous to this experiment, I have ...
Ewan McGregor's user avatar
7 votes
1 answer
460 views

Why would I pick a beta prior for binomial data? More generally, how would I pick any prior distribution? [duplicate]

Obviously, being a conjugate prior is useful as it saves performing tricky integrals or using a simulation, but why would I want to use a beta distribution? More generally, how does one pick an ...
TerryStone's user avatar
1 vote
1 answer
72 views

Picking parameters for beta prior

I have some data that I believe come from a binomially distributed population. A beta prior seems like an appropriate choice, but I don't have any very strong prior beliefs. I could use a less ...
TerryStone's user avatar
0 votes
1 answer
50 views

Estimator of variance for a binomially distributed sample

I want to run some Bayesian analysis on some data. Suppose we have a sample that we believe comes from a binomial population. We have $m$ observations $$X_i \sim \text{Bin}(n,p) \quad \quad \quad \...
TerryStone's user avatar
3 votes
1 answer
61 views

Posterior probability for $\theta$ with a discrete prior

I'm trying to find a posterior probability for this model but I can't find the solution. Help would be appreciated! Prior distribution: $\theta$ follows a discrete probability function: $\mathbb{P}(\...
Alexandre Beaudry's user avatar
3 votes
1 answer
48 views

Derivation of acceptance probability from Linero, Yang (2018)

I am wondering how this paper Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity by Linero & Yang (2018) derived the acceptance probability for $\sigma$. The authors give $\...
xiaoshuxiaowei's user avatar
4 votes
1 answer
80 views

Bayes estimator of possion distribution with Pareto prior

Consider a random sample of size $n$ following the possion distribution with parameter $\ln \theta$, that is $$ f(x|\theta)=\frac{(\ln\theta)^x}{\theta x!}, x=0,1,2,\cdots $$ and the prior of the ...
Javier's user avatar
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2 votes
1 answer
63 views

Centering Priors on MLEs vs. Using MLEs as Initial Conditions for MCMC [duplicate]

Here: Centering prior distributions on MLE/OLS estimates I ask about centering priors on MLEs in the context of a logistic regression (in my case with only categorical predictors), which I've seen a ...
compbiostats's user avatar
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101 views

posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision

What is the posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision. Thus, what is the distribution of x given: \begin{equation} x \sim \mathcal{N}(x; \...
Snowy Baboon's user avatar
3 votes
0 answers
38 views

prior distribution for iid gaussian, with a known variance

I have been reading Pattern Recognition and Machine Learning by Bishop, and I have a question regarding the prior distribution of an iid Gaussian with known variance. The relationship $\dfrac{n}{\...
cgo's user avatar
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2 votes
0 answers
145 views

Why not use a half-normal distribution as a prior for variance parameters in Bayesian estimation?

Typically, distributions with fat tails, such as the inverse Gamma or the half-Cauchy are used as prior distributions for the variance parameters. I am trying to understand why do we need a fat-tailed ...
J. Doe's user avatar
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0 answers
30 views

Prior probability distribution when we have a single estimate of the mean and no estimate of the variance

Say we have some real parameter $p$ we'd like to determine experimentally. If we have a single estimate of $p$ but no associated uncertainty, what prior probability distribution(s) can/should we use ...
user3716267's user avatar
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0 answers
43 views

What should be the appropriate choice of prior for a dummy variable in a Bayesian Linear regression?

I have a dataset where the dependent variable is Sales. The independent variables are Media Spends 1, Media Spends 2 and Covid dummy. I am trying to build a Bayesian Linear Regression model. The covid ...
T_S's user avatar
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0 answers
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Choice of Prior

What should be the appropriate prior in case of a dummy variable and an interest rate variable which has values like 4.5, 4.6, 4.9 etc in a Bayesian linear regression?
T_S's user avatar
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2 votes
1 answer
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Is bayesian updating framework a valid concept?

When I google search for the term, only 6 pages showed up. There is no authoritative paper on this, except https://arxiv.org/abs/1306.6430 which argues for using informatics concepts to generalize a ...
Chloe's user avatar
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What to attribute the difference of observed probabilty in a bayesian model to simply sampling from data?

Comparing the probability from data that a zombie has iq > 18 the bayesian approach ...
Rishav Dhariwal's user avatar
1 vote
0 answers
29 views

Do Bernardo Priors still Encounter Paradoxes?

I heard that Bernardo Priors are better versions of Jeffrey's prior that work in multi-dimensions & match frequentist confidence intervals. Apparently they also dodge many paradoxes of other ...
profPlum's user avatar
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1 vote
1 answer
96 views

Empirical Bayes grid approximation: How to prevent the prior from overwhelming the likelihood when there are many more samples in the prior?

I am trying to construct a grid approximation of the probability that an event will occur on the discrete interval 0 - 10, the spacing is 0.01. The probability that any single event on the interval is....
Roger Erismann's user avatar
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0 answers
43 views

Performing a Bayesian (or Bayesian-like) linear regression

I'm trying to model the relationship between two variables. Without going into details on how I get this expectation, my belief on the relationship between the variables looks like this: However, as ...
gammapoint's user avatar
7 votes
1 answer
151 views

Terminology for distribution of posterior mean before seeing data

Imagine we are performing Bayesian inference with normal-normal conjugate priors. We have some prior: $$ \mu \sim N(\mu_0, \sigma_0^2). $$ We know we will collect some normally distributed data $x$ ...
jjj's user avatar
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0 answers
52 views

Role of the prior in Linear Discriminant Analysis

I am trying to understand probabilistic generative models a bit better. I am looking at linear discriminant analysis for $C = 2$ classes. According to what I've seen, the likelihood function can be ...
user3813234's user avatar

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