I am trying to compute the Bayes factor via *bridge sampling* from Stan models.

According to the vignettes by Gronau (2021),

… to compute the (log) marginal likelihood for a Stan model, we need to specify the model in a certain way. Instad of using “~” signs for specifying distributions, we need to directly use the (log) density functions. The reason for this is that when using the “~” sign, constant terms are dropped which are not needed for sampling from the posterior. However, for computing the marginal likelihood, these constants need to be retained. For instance, instead of writing `y ~ normal(mu, sigma)`

we would need to write `target += normal_lpdf(y | mu, sigma)`

In a model block, the Jeffreys prior for the **error standard deviation** \sigma_\epsilon is coded as

```
model {
target += -log(sigma); // Jeffreys prior
}
```

Moreover, the Jeffreys prior for the **grand mean** \mu implicitly follows a uniform distribution. My question (to compute the log marginal likelihood) is: Do I need to write it out in some `_lpdf`

syntax or leave it undefined?

```
model {
// mu ~ implicit uniform prior // Jeffreys prior
}
```

A concrete example can be view at RPubs - Bayes Factor for One-Way ANOVA

Thanks for any comments.

`brms`

models are compatible with `bridgesampling`

by default and they generate code which uses implicit uniform priors, so I’d think you were fine. But the bridgesampling devs are also on discourse, so I’ll ping them to clarify just in case: @Quentin @Henrik_Singmann

In general though, I’d argue that it’s best practice to make priors explicit in models, to guard against any changes to default behaviours in the future

Thank you, Andrew.

The grand mean \mu, in this case, has been assigned an improper uniform prior (unbounded) (Stan User’s Guide, version 2.31). I doubt we can write it explicit in Stan, can we?

If you’re referring to the implicit uniform, then note that:

```
parameters {
real mu;
}
model {}
```

Is the same as:

```
parameters {
real mu;
}
model {
mu ~ uniform(negative_infinity(), positive_infinity());
}
```

@avehtari what’s your perspective on whether or not to explicitly specify the implicit uniform prior?

Terrific! Thanks again, Andrew.

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Marginal likelihood is not defined when prior is improper, and `uniform(negative_infinity(), positive_infinity())`

is improper, so it is irrelevant in this thread whether it is defined implicitly or explicitly.

They are not fine with improper priors. The brms documentation example explicitly defines proper priors (Log Marginal Likelihood via Bridge Sampling — bridge_sampler.brmsfit • brms)

@DexterW you need to define proper priors to have valid (log) marginal likelihood computation.

4 Likes

Ah of course, thanks Aki!

Thank you, Aki.

I have tested: (1) undefined, (2) large-range flat, (3) large-spread normal.

`// mu ~ implicit uniform prior`

`target += uniform_lpdf(mu | -1000, 1000);`

`target += normal_lpdf(mu | 0, 100);`

The resulting Bayes factor values are very close. I guess it is because the same Jeffreys prior is assumed on \mu for both competing models of the Bayes factor.