currently our templates do not include this basic staple.

In EBEB 2016 Mike West said he didn't find the fact that LP is invariant under symmetric transforms (and not invariant under non-invertible transforms) very interesting.

I'm not sure.

Can ask Raftery.

@fccoelho @lsbastos @DVMath , your 2 cents?

Write a script to assert if our formulas are correct.

Professor Raftery suggested using integrated likelihoods. Problem is: once we calculate them, what are going to do with them? Calculate prior odds?

Mrs @lsbastos and @fccoelho, o poder e' de voces!

Ass. Capitao Planeta

define LP before the section on its properties

Given in the appendix.
They seem correct, but it'd be nice to work out a simple example, like Beta or Gaussian to see whether the formulae lead to the right answer. Should do for entropy and KL.

We should think of a simple PSA for the hierarchical priors on this example and maybe the bowhead example as well.

After the writing is concluded, just move the Appendix to a different file [as per BA guidelines], taking care of preserving the cross-referencing.

In their BA paper, Rufo et al. (2012) derive their results for the exponential family. Should we follow them and derive [the properties of] our hierarchical prior for the expo family too? @lsbastos & @fccoelho ?

We extended the algorithm in Poole & Raftery to include a variable alpha. Need to describe that in the manuscript.

https://iris.uniroma1.it/retrieve/handle/11573/1156672/798985/Tesi_dottorato_Marchetti.pdf

Lots of good stuff in it

• Meta analysis;
• Savchuk;
• Simulated Savchuk: (i) simulated (ii) more data (ii) higher variance/lower variance;
• Bowhead: (i) modified SpIR (ii) simulation study;

We talk about KL(pi || f_i) everywhere but we need KL(f_i || pi).

If we have any hope for the hierarchical prior to be applicable in the real-world, we will need to devise a sampler that can incorporate the alphas and sample the model at the same time.

Currently we have working implementations in Stan, but that is just because the pooled prior happens to have a closed-form solution. The sampler has to be able to sample from the pooled prior even when it does not have a closed-form solution [which is the vast majority of interesting cases].

This article contains some useful information on a potential clog in this engine, the IMIS algorithm, that could be used to construct the pooled prior [or posterior, thanks to external Bayesianity].

http://discourse.mc-stan.org/t/two-parameter-distribution-over-the-simplex/3737/7?u=maxbiostat

Should we keep both Beta and Gamma?

@lsbastos , @fccoelho ?

@maxbiostat, what is the size limit of this abstract? is is and extended abstract or a short paper?

...and re-implemented.

Reason: Rufo et al, 2012, propose to minimise KL(pool || expert_i), which I think makes more sense than what we propose, that is KL(expert_i || pool).

When we say family we may be implying some sort of connection between the experts' opinions, where none need exist. So the right name for $F(\boldsymbol\theta)$ is 'set of distributions'

We need a smarter way to take in integration limits

http://www.mdpi.com/1099-4300/20/3/209

Following the suggestions of Eduardo Mendes (FGV), we should improve the experiments in the simulated example section to include draws from the generating process.

Claudio Struchiner (FGV/Fiocruz) also suggested doing something similar for the melding examples, specially the boarding school one.

Please add C. to my name in the manuscript, making it Flávio C. Coelho

• Aitchinson: (i) description (ii) moment matching;

We need to address the connection between our approach and the "power prior" approach of Ibrahim & Chen (2000) and Ibrahim et al. (2015).

For the final draft.

https://arxiv.org/abs/1803.01984

For background, see this discussion. I proposed LP could be a way of combining many posteriors. Aki Vehtari points to his paper on stacking distributions.

My question is: should someone look into comparing a choice of the weights in LP that minimises, say, the KL distance to the true model, a bit like Rufo, Martin & Perez did? My idea would be to take the examples from Aki's paper and directly compare LP and stacking.

Not something for now, just maybe a project to keep in mind for the future, @fccoelho @lsbastos @DVMath

there's a minus sign that shouldn't be there.

Re-do the calcs

'CODE' vs 'code' is screwing up the repo

Currently some of the normal (Gaussian) examples are just copy-paste from the binomial ones. Gotta fix that.

If no, how to present this information? Table? @fccoelho @DVMath @lsbastos

@lsbastos , you derived the expression for the modified weights when alpha varies.I wrote the section on SpIR. Can you please check that equation \ref{eq:SpIRweights} is correct?

Would be interesting to prove the result in general.

For all of the examples considered so far (Beta, Gamma, Normal) the entropy (and sometimes the KL too) optimisation problem is quite unstable, tending to one of the "corners" of the simplex. This happens because the entropy function is "dominated" by a parameter and thus any solution that assigns alpha_j = 1 for some j such that parameter_j = max(parameters) will be the best one.

To see this, consider the normal example. Since the entropy depends only on the variance, the distribution with larger variance will tend to dominate the optimisation problem, leading to the trivial solution (0, ..., 1, 0, ...). where the j-th position is a 1 and everybody else gets weight 0.

It would thus be interesting to analyse the entropy functions for the distributions considered to get insight into how much a parameter dominates the entropy.

The final solution, I think, is propose a constrained optimisation problem, where, for example, one would obtain the maximum entropy distribution with a given mean.

As previously noted, neither the max_ent nor the KL(pool | expert_i) approaches lead to unique solutions.
In a future study, we could look into some of the ideas here (about linear mixtures) to create a unique mapping between the component distributions and the weights.

We need to look at the two tables and the figure and tell a consistent story about the differences/similarities between them.
@DVMath and @fccoelho are invited to give their pitacos as well.

See Professor Genest's corrections to LC note. We're missing volume and edition numbers, stuff like that.

Needed to ensure smooth publication in the proceedings

https://pdfs.semanticscholar.org/a57f/80cac0249fe3aba3719470e2813892945124.pdf

@fccoelho , @lsbastos and @DVMath :

I propose to use the model in Carvalho, Bastos & Struchiner (2015)

It has three levels:

• 0. The hyperparameters of the thermodynamic functions;
• 1. The thermodynamic functions r(T) and K(T);
• 2. The population size P(t, T)

Problems:
Figure out a sensible propagation strategy: it probably doesn't make any sense to propagate the uncertainty on the hyperparameters caused by uncertainty on P(t,T), although it is certainly desirable to study how the uncertainty about the hyperparameters influences the uncertainty on P(t,T).

An initial idea, thus, is to study only the prior on P(t,T) obtained by combining:

Induced prior on P(t,T) from the hyperparameters + induced prior on P(t,T) from the priors on the functions r(T) and K(T) and the actual prior on P(t,T).
This would be nice because we would be combining uncertainty from three levels and also because we could in principle apply our automatic methods to choose the weights (alpha).

As a bonus, we could look into the combined prior for r(T) and K(T) combining only:
(i) the induced priors from the hyperparameters and (ii) the actual priors on r(T) and K(T).

https://www.sciencedirect.com/science/article/pii/S0167587709001834