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Bayes Theorem and Total Probability

I think I have worked this out... $P(275-300 \mid CC) =$ $\frac{P(CC \mid 275-300) \cdot P(275-300)}{P(CC \mid 275-300) \cdot P(275-300) + P(CC \mid 305-325) \cdot P(305-325) + P(CC \mid 330-350) \...
jerH's user avatar
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3 votes
Accepted

Bayes Theorem Question with compound events

The calculation is going to depend on whether you interpret "The two systems operate independently" as conditional independence of fault detection $P(A \cap B \mid F) = P(A\mid F) \cdot P(...
Henry's user avatar
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1 vote
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How to decompose the conditional posterior prob?

It appears that you are taking $\sigma^2, \tau^2,$ and $\theta$ to be known. You have $y\mid \mu \sim \operatorname N(\mu, \sigma^2).$ So you have a likelihood function \begin{align} L_{\mu\,\mid\,y}(...
Michael Hardy's user avatar
1 vote
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ABC (Approximate Bayesian Computation) Sampling, Simulating data from Complex models

This question is addressed in many ABC-related papers and in the Handbook of Approximate Bayesian Computation as well. The difference between a well-defined probability model $P_\theta$ and a closed-...
Xi'an's user avatar
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1 vote

Difficulty Obtaining Bayesfactor in Baysian Correlation in R (correlationBF from BayesFactor)

In this case the Bayes factor is a "very large number", so it seem correlationBF runs into a numerical precision issue during the computation. The default ...
dipetkov's user avatar
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1 vote

How do we get rid of $p(x|\theta)$ in MLE

under which assumption do we remove $p(x|\theta)$? If $x$ is independent from $\theta$ then $p(x|\theta) = p(x)$ and can be absorbed in the constant of proportionality. $$p(\theta|x,y) \propto p(y|x,...
Sextus Empiricus's user avatar
3 votes
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How do we get rid of $p(x|\theta)$ in MLE

The model you're describing, $p(y|x,\theta)p(x|\theta)p(\theta)$, is a stochastic regressor model. You can get away with ignoring $p(x|\theta)$ when $\theta$ partitions into $(\theta_y, \theta_x)$ ...
Durden's user avatar
  • 1,351
3 votes

How do we get rid of $p(x|\theta)$ in MLE

We typically assume either that $x$ is fixed or that $x$ depends on different parameters from $y|x$ and it can be conditioned on. That is, if $y=x\beta+\epsilon$, with $\epsilon\sim N(0,\sigma^2)$, it'...
Thomas Lumley's user avatar
0 votes

Multivariate time series with additional features

The {mvgam} R package was designed specifically for these kinds of problems. It handles both univariate and multivariate time series, and is set up to allow for ...
Nicholas Clark's user avatar
2 votes

Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

OP says: “data are fixed” means the observed data $x$ is fixed and “data are random” means the random variable $X$ is random. The “data” in the two statements are not referring to the same thing. ...
civilstat's user avatar
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12 votes
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Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

The usage of the expressions 'data are fixed' or 'parameters are fixed' in the linked references should not be taken literally. Data is just as well considered random in Bayesian analyses, how else ...
Sextus Empiricus's user avatar
6 votes

Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

After reading some materials, it seems that “data are fixed” means the observed data 𝑥 is fixed and “data are random” means the random variable 𝑋 is random. The “data” in the two statements are not ...
Cliff AB's user avatar
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5 votes

Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

In both Bayesian and frequentist stats, your current sample $x$ is fixed while potential other samples (i.e., future realizations of $X$) are random. Frequentists rely on the idea of resampling for ...
Durden's user avatar
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6 votes

Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

I suppose what they mean (it may help to cite your sources) is that Bayesian inferences involve posterior probabilities conditioned on the observed data. Frequentist inferences are typically ...
Scortchi - Reinstate Monica's user avatar
6 votes

Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

No they are not talking about the same thing. This is readily seen in sequential experimentation where frequentist statistics takes an $\alpha$ (type I assertion probability) penalty for multiple ...
Frank Harrell's user avatar
-1 votes

Are “Data are fixed” in Bayesian viewpoint and “Data are random” in frequentist viewpoint talking about the same thing mathematically?

What are data? Data come in all shapes and sizes. But what are data actually? My height and weight can be measured, and their values are read using a scale and a ruler. The values can be recorded with ...
Romke Bontekoe's user avatar
1 vote
Accepted

Assumptions in definition of "Log Pointwise Predictive Density" (LPD)

There were some details I was missing I will hopefully add here. Based on the comments, I'll notionally use $y'$ in place of $y_i$ to show more clearly this is new data. This is fully answered in page ...
JoshCowley's user avatar
0 votes

Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

There are already some decent answers, and I offer an additional perspective. Contrary to what many people think, it is actually perfectly possible and not at all uncommon to use human judgement for ...
0 votes

Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

I think the motivation for leaving a review matters most, success or failure in finding something that the reviewer thinks is hard to find. To that end, Amazon has added specific attributes for ...
3 votes

Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

There are already excellent answers by Björn and jpa. I will just give a basic philosophical thought that may help to put in perspective the many possibilities. There is no unique truly best way of ...
2 votes

Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

There are two separate probabilities involved: A person that is shopping for this type of product ends up buying this particular product. A person that bought this product leaves a positive review of ...
10 votes
Accepted

Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

Clearly, several online platforms like Trip Advisor etc. have implemented a sorting system that trades these two things off. If they all did it the same way I'd guess one might consider that a ...
3 votes
Accepted

Simple example of Log-Sum-Exp trick for continuous case

Introducing $c$ as $\max{\log[p(x)]}$, we have \begin{align} p(x)&=\exp\{\log[p(x)]\}\\ &=\exp\{\log[p(x)]-c+c\}\\ &=e^c\exp\{\log[p(x)]-c\}\\ &=\exp\{\log[p(x)]-c\}\Big/\int_\mathfrak{...
Xi'an's user avatar
  • 106k
1 vote

Priors in a bayesian model? Equivalent GLMM

Brief Answer I would heavily recommend reading up on Bayes before using Bayes. There is a serious danger of the defaults (Gelman & Yao, 2021; Moyé, 2008; Smid & Winter, 2020), which are the ...
Shawn Hemelstrand's user avatar
2 votes

Priors in a bayesian model? Equivalent GLMM

The posterior and likelihood, and their marginals, are equivalent when you use uniform priors. So the maximum likelihood estimate and maximum a posteriori estimate are equivalent when you use uniform ...
Sextus Empiricus's user avatar
2 votes

Understanding the Jeffreys-Zellner-Siow (JZS) prior in Bayesian t-tests

As a start, $M_0,M_1$ specified in p. 229 are not any particular distributions but rather symbolize the different distributions under $H_0:\mu=0$ and $H_1:\mu\ne0$. In the section relevant for ...
Spätzle's user avatar
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0 votes

Interpretation of posterior predictive distributions

The possible answer might be is that the predictions of heights at specific weights include the uncertainty in the intercept: **Pred110 = mcmcMat[,"beta0"] + 110 * mcmcMat[,"beta1"]...
Rustam's user avatar
  • 73
14 votes

Do we believe in existence of true prior distribution in Bayesian Statistics?

I'll comment on both the idea of a "true parameter/parametric model" and a "true prior" as the question is somewhat ambiguous about which one of these is of interest here. First ...
Christian Hennig's user avatar
4 votes

Do we believe in existence of true prior distribution in Bayesian Statistics?

do we believe in the existence of a true distribution of $\theta$, $P_{\theta}$ (such that we view posterior as an estimation of this true $P_{\theta}$) A posterior distribution does not aim to ...
Sextus Empiricus's user avatar
29 votes

Do we believe in existence of true prior distribution in Bayesian Statistics?

Do not confuse a priori knowledge of the true parameter value with having a true prior over the parameter space The answer to this question depends on the underlying philosophical interpretation of ...
Ben's user avatar
  • 125k
3 votes

Do we believe in existence of true prior distribution in Bayesian Statistics?

The question is interesting, albeit somewhat ill-posed. Bayesians are generally comfortable with the idea of some point $\theta_0$ in parameter space $\Theta$ being the true parameter of a given ...
Durden's user avatar
  • 1,351
1 vote

Definition of $\text{do}$ operator

$x$ cannot be outside the support of $X$ Both equations deal with the theoretical law of the random variables, so any concrete values play no role here. That said, perhaps the question concerns the ...
Scriddie's user avatar
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2 votes

Can you specify correlated coefficients in Stan models?

The trick here is to use the multi_norm family of log probability density functions in the model block of your stan program. Here is an example of putting a joint ...
Demetri Pananos's user avatar
1 vote

Can the "true" prior lead to better posterior estimation?

Recapitulation of the question Let's consider the following data generating process (considering only the mean $\bar{X}$, instead of multiple $X_i$, since the mean is a sufficient statistic and ...
Sextus Empiricus's user avatar
2 votes

Interpretation of posterior predictive distributions

Can I say “there is a 95% probability that a person with a weight of 110 kg is taller than a person with a weight of 100 kg”? You can't, but if you have the posterior you can (in principle) obtain ...
Doctor Milt's user avatar
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