Calculating the Posterior Predictive Density

[Noah Silverman]

As noted in my previous posts, I'm looking for ways to evaluate goodness-of-fit for out-of-sample testing data.

The CPO and PIT are great, but impossible to evaluate for NA values in INLA.

Reading through an old paper "Posterior and Cross-validatory Predictive Checks: A Comparison of MCMC and INLA", the authors discuss model criticism and comparison.

On page 94 (5th page of actual PDF), the authors describe the Posterior Predictive Density.

Intuitively this makes sense.  Instead of just comparing the center of predicted values to the observed values, we can compare at the entire predictive distribution to the observed value.

However, I'm a bit stuck on  how to actually calculate this in R. 

I can't figure out how to calculate the first term of the left hand side

The dnorm() command in R requires both a mean and standard deviation.  INLA supplies 48 points from the marginal fitted distribution.  How do I calculate the above values with those points? 

Alternately:  Perhaps I'm going about this the "long way", when there may be existing tools or function in R.  Any suggestions would be very welcome.

Thank You,

[Helpdesk (Haavard Rue)]

this

\pi( y_i | y)

is 'po' and is computed if you request it.

> r=inla(y ~ 1, data=data.frame(y=1:2), control.compute=list(po=TRUE))
> r$po$po
[1] 0.4086249594 0.4086249594

On Tue, 2022-09-13 at 21:14 -0700, Noah Silverman wrote:
> As noted in my previous posts, I'm looking for ways to evaluate
> goodness-of-fit for out-of-sample testing data.
>
> The CPO and PIT are great, but impossible to evaluate for NA values in
> INLA.
>
> Reading through an old paper "Posterior and Cross-validatory
> Predictive Checks: A Comparison of MCMC and INLA", the authors discuss
> model criticism and comparison.
>
> On page 94 (5th page of actual PDF), the authors describe the
> Posterior Predictive Density.
>

> Screen Shot 2022-09-14 at 12.06.32.png
>
> Intuitively this makes sense.  Instead of just comparing the center of
> predicted values to the observed values, we can compare at the entire
> predictive distribution to the observed value.
>
> However, I'm a bit stuck on  how to actually calculate this in R. 
>
> I can't figure out how to calculate the first term of the left hand
> side

> Screen Shot 2022-09-14 at 12.08.52.png
>
> The dnorm() command in R requires both a mean and standard deviation. 
> INLA supplies 48 points from the marginal fitted distribution.  How do
> I calculate the above values with those points? 
>
> Alternately:  Perhaps I'm going about this the "long way", when there
> may be existing tools or function in R.  Any suggestions would be very
> welcome.
>
>
> Thank You,
>
>
>
>
>
>

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[Noah Silverman]

Thank You!

Is this the same as inla.dmarginal(y_i,  m1$marginals.fitted.values[[i]]) ??

[Helpdesk (Haavard Rue)]

it is

\int \pi(y_i | y, \eta_i) \pi(eta_i | y) d\eta_i

> https://groups.google.com/d/msgid/r-inla-discussion-group/9f0f65c9-6cee-4cab-9d04-c0cc6797511en%40googlegroups.com
> .

[Noah Silverman]

Got it.  Thank you so much!

Out of curiousity, is that a simple function somewhere in the INLA code?  Might help me a bit to see how it is actually constructed.

Thanks!!!

[Helpdesk (Haavard Rue)]

this part is done in C. you can also use the 'posterior.sample' apprach
as suggested by Finn; see the post 'CPO and PIT for predictions'

> https://groups.google.com/d/msgid/r-inla-discussion-group/1abcd89d-aeed-4b51-8543-984d19c08030n%40googlegroups.com
> .