# Tag Info

Accepted

### XKCD's modified Bayes theorem: actually kinda reasonable?

Well by distributing the $P(H)$ term, we obtain $$P(H|X) = \frac{P(X|H)P(H)}{P(X)} P(C) + P(H) [1 - P(C)],$$ which we can interpret as the Law of Total Probability applied to the event $C =$ "you ...
• 3,387
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### Differences between prior distribution and prior predictive distribution?

Predictive here means predictive for observations. The prior distribution is a distribution for the parameters whereas the prior predictive distribution is a distribution for the observations. If $X$ ...
• 3,363

### XKCD's modified Bayes theorem: actually kinda reasonable?

Believe it or not, this type of model does pop up every now and then in very serious statistical models, especially when dealing with data fusion, i.e., trying to combine inference from multiple ...
• 20.7k
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### Why LKJcorr is a good prior for correlation matrix?

The LKJ distribution is an extension of the work of H. Joe (1). Joe proposed a procedure to generate correlation matrices uniformly over the space of all positive definite correlation matrices. The ...
• 90.4k

### Use of Bayesian hierarchical model

In my opinion, there are two different aspects to your question: when should I use a hierarchical model? when should I perform a Bayesian analysis? When should I use a hierarchical model? An ...
• 4,671

### Differences between prior distribution and prior predictive distribution?

Let $Y$ be a random variable representing the (maybe future) data. We have a (parametric) model for $Y$ with $Y \sim f(y \mid \theta), \theta \in \Theta$, $\Theta$ the parameter space. Then we have a ...
• 76.8k
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### Downsides of inverse Wishart prior in hierarchical models

Here are some relevant resources (full disclosure: the first link is to a paper of mine): http://newprairiepress.org/agstatconference/2014/proceedings/8 http://www.themattsimpson.com/2012/08/20/prior-...
• 4,671
Accepted

### What precisely does it mean to borrow information?

This is a term that is specifically from empirical Bayes (EB), in fact the concept that it refers to does not exist in true Bayesian inference. The original term was "borrowing strength", which was ...
• 12.5k
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### Is it wrong to use sufficent statistc estimated from the data as a prior in Bayesian data analysis?

Yes, setting $\mu_0=\bar y$ absolutely is circular. The Bayesian model cannot be justified unless you genuinely do have information about $\mu$ that is separate to the data. Based on what you have ...
• 12.5k
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### Relation between Bayesian analysis and Bayesian hierarchical analysis?

In my view, hierarchical modeling in a Bayesian setting mainly refers to the building of a complex prior structure. Consider a parameter of interest $\theta_{0}$ and your observation $(x_i)$. Now, ...
• 5,023
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### Use of Bayesian hierarchical model

To the best of my knowledge, there is no opposition between Bayesian models (BM) and hierarchical Bayesian models (HBM) (see e.g. Relation between Bayesian analysis and Bayesian hierarchical analysis?)...
• 5,023
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• 104k

### What are some statistical tests for exchangeability of a data set?

The theorem in question tells us that exchangeability is equivalent to being conditionally IID. Hence, in practice, data analysts consider the same things when deciding whether observations are ...
• 20k

### Comparison between Bayes estimators

Your answer for the squared error loss part is wrong. $$\pi(\theta|x) \propto f(x|\theta) \pi(\theta) = 2\theta x^{\theta-1}I_{(0,1/2)}(\theta).$$ This is a $Beta(\theta,1)$ distribution in $x$, ...
• 15.5k
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### Over-parameterization in Bayesian Hierarchical Model

Important to note first: Bayesian inference does NOT automatically guard against overfitting. Adding additional variables will pretty much result in the same problems as in an non-Bayesian analysis. ...
• 7,839

### Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, and updating our confidence accordingly

What you describe is the concept of dynamic predictions from mixed models. Initially, when you have no information for a doctor you only use the fixed effects in the prediction, i.e., you put his/...
• 20.8k
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### Are analytically tractable posterior distributions exclusively the result of a conjugate relationship in Bayesian hierarchical models?

Conjugate priors are not necessarily tractable (Robert, 1994): take for instance a Beta distribution$$f(x|\alpha,\beta) = B(\alpha,\beta)^{-1} x^{\alpha-1}(1-x)^{\beta-1}\mathbb I_{(0,1)}(x)$$as the ...
• 104k

### Formulating posterior predictive distribution from hierarchical model

It appears that the stated distribution is for $x_t^{(j)} | \eta_t^{(j)}$ and the random variable $y_{i,t}$ is being ignored for now. It also appears that the authors are being a bit loose in their ...
• 123k
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### Matt's trick (reparametrization) makes my models slower, not faster

It is not unheard of for the centered parameterization to be better. This post on the Stan forums goes into the exact same issue. There it is suggested that [...] centered actually works better when ...
• 4,251
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### How to interpret the standard deviation of the slope random effect in a multilevel model

I am assuming you have fit a model in which you have estimated an overall average (fixed) effect of time and allowed the effect of time to vary by region (random slope). The model then estimates the ...
• 1,192
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### Is there a difference between hierarchical GAM (HGAM) and Mixed effect GAM (GAMM)?

They are the same thing; we just prefer the terminology "hierarchical" over "mixed", because the salient practical feature of these models is that they can model variation in the ...
• 47.1k

### "Unidentified" hierarchical model in brms/stan - where to go from here?

From your model summary I see that you have 4209 obervations in total and 3091 persons. That is, most persons only have 1 corresponding data point and therefore it will be difficult to estimate the ...
Accepted

### Multi-level Bayesian hierarchical regression using rjags

You want a distribution for each quarter (given a state), each state (given a region), and each region. That means you'll need at least some state parameters indexed by s (in your model b0, b1, ...
Accepted

### How can we convert values proportional to probabilities to Bernoulli probabilities?

Since $p(1)=p$ and $p(0)=1-p$ are both proportional to a known expression* (the unscaled probabilities, $u(i)=c.p(i)$, with the same unknown constant of proportionality, $c$) and you know the $p(i)$ ...
• 281k

### Ergodicity of MCMC in a hierarchical model

Supose the posterior distribution is denoted by $\pi$ defined on a subset of $\mathbb{R}^d$. Then the Markov chain that samples from this distribution is a general state space Markov chain. Here are ...
• 15.5k