Compare Psudo-SDERGM and SDERGM for toy model dirBin0Rec0

[domenicodigangi]

• [x] Write pseudo likelihood filter and estimates
• [x] Write exact likelihood filter and estimates
• [x] compare as misspecified filters for dgps similar to the paper
  • [x] AR
  • [x] Sin
• [x] Compare rmse for one dgp considering networks of incresing size in sparse and dense regimes
  • [x] Write function that returns ergm parameters given the value of stats and network size. (needed to explore the two regimes)
  • [x] What is the appropriate scaling for the two statistics? Try with the same scaling of Avg number of rec pairs as in Erdos-Reny
  • [x] For each N, find a way to define the parameter's dynamics, and the density regime is satisfied and the dgp parameters do not imply mean val parameters close to physical bounds.

Altought the physical bounds are always satisfied there are issues when the natural paramters get close to them. One possible way to define the time evol for eta and theta is the following :

• choose a scaling for alpha

  • alpha~const dense scaling
  • alpha~1/(N-1) sparse scaling
• choose a scaling for beta, s.t. it remains well within the bounds : 0 + c3<beta<alpha/2 - c4 . Where we assumed alpha<1/2, and choose c3 and c4 to guarantee that beta stays well within the bounds. E.g. we would like to have at least a few rec pair on average, i.e. beta = min_n_pairs/(N^2-N)
• define a dynamics for alpha and beta, that remains well within the bounds. Then transform the obtained paths for alpha and beta in paths for theta and eta and use the latters for the dgp

[domenicodigangi]

written tested and compared on SIN and AR

[domenicodigangi]

Tried some scalings that return weird results for SIN filter of large N

[domenicodigangi]

Notes on the model and the relation between parameters in the mean value (alpha and beta) and natural parametrizations (eta theta)

[domenicodigangi]

Altought the physical bounds are always satisfied there are issues when the natural paramters get close to them. One possible way to define the time evol for eta and theta is the following :

• choose a scaling for alpha
  • alpha~const dense scaling
  • alpha~1/(N-1) sparse scaling
• choose a scaling for beta, s.t. it remains well within the bounds : 0 + c3<beta<alpha/2 - c4 . Where we assumed alpha<1/2, and choose c3 and c4 to guarantee that beta stays well within the bounds. E.g. we would like to have at least a few rec pair on average, i.e. beta = min_n_pairs/(N^2-N)
• define a dynamics for alpha and beta, that remains well within the bounds. Then transform the obtained paths for alpha and beta in paths for theta and eta and use the latters for the dgp

[domenicodigangi]

The approach of defining the dynamics in the space of mean values, i.e. alpha and beta (mean density and reciprocity), transform to natural par space and use the latter as dgp, seems to work. Indeed the mapping induces some highly nonlinear transformation effectively forcing theta and eta to be counterciclycal, and non precisely sin shaped (sin with random phase was the original dgp as shown in the attached figure)

[domenicodigangi]

Attached recap email mail_toy_model_ML_vs_PML_filter.pdf

[domenicodigangi]

Update da call 4/12/2020:

• Le note sulla regione di esplorabilitá vanno riscritte in latex
• Su input di Fabrizio decidiamo di capire se l'intuizione sulla regione di osservabilitá a N finito é valida con test nel caso statico.

[domenicodigangi]

Ho aggiunto le formule delle note al pdf in overleaf

[domenicodigangi]

Riguardo i problemi riscontrati dai filtri (both mle e pmle) in caso di dgp definito in modo naive: , penso che l'intuizione sulla regione di osservabilitá a N (numero di nodi) finito sia confermata da alcune simulazioni fatte. Ricapitolando:

• Tutti i valori di eta e theta sono concessi, nel senso che risultano in modelli con pmf ben definita (normalizzata e con probabilitá reali e positive).

• Nella figura precedente osserviamo che il filtro non riesce a seguire il DGP in alcune regioni.

• Io penso che questo avvenga quando la pmf é tale che una frazione signicativa di samples risulta avere zero coppie reciproche (una delle due sufficient statistics).

• Per supportare questa spiegazione, plotto la frazione di samples con esattamente zero coppie di links reciproche intorno al punto in cui il filtro mostrava dei problemi, per vari valori di N

• Quando il numero di links reciproci é esattamente zero, la MLE diverge e lo score (calcolato nella MLE) é esattamente zero. Non mi stupisce che il filtro non riesca a seguire il DGP quando le osservazioni sono cosi "patologiche"

• Penso che possiamo accontentarci di un filtro che funziona bene quando la MLE esiste ed é finita