Page 381 - Special Topic Session (STS) - Volume 2

STS506 L. Leticia R. et al.
the posterior distribution (|) of the parameters based on the
observed data . In this work we focus in the ABC-MCMC (Marjoram, et
al. 2003) that can be implemented without specifying likelihood function
but when it is easy to sample ∈ from (∙ |) , for any in the
parametric space . Since this method is “likelihood free”, it is gaining a
lot of attention in a wide range of scientific disciplines where their
complex models have intractable likelihoods, or they are too expensive
to calculate.
The ABC methods use the modified posterior density on ×
(, |) ∝ ()(|) (,) ,

(3)
where (,) are the pseudo-observations that are “closer up to ” to the
true observations (Marin, et, al. 2012). Formally (, ) = { ∈
: ((), ()) < }, where (∙) summarizes the observations, and is a
distance function.
A modification to (3) replaces the indicator by a kernel function ℎ
evaluated on a joint statistic for and . That is
(, |) ∝ ()(|) ((, )),
ℎ
ℎ
where ℎ is the kernel bandwidth and (, ) is a discrepancy measure between
observations. Then the Metropolis-Hasting can be modified to perform ABC-
MCMC inference, considering its acceptance probability for candidate ∈
equal to
ℎ (|)( () |) () ℎ (( ,))( () |)
(|) = { , 1} = { ,1},
ℎ ( () |)(| () ) () ℎ (( () ,))(| () )
where () , are pseudo observations from (∙ |()) and (∙ |), respectively,

and is the proposal distribution.
In the SIR epidemic model in networks, the parameter of interest is = (,
). For arbitrary networks, the likelihood is intractable but using the agent-
based and event-driven algorithm used in Ramírez- Ramírez, et al. (2013), we
can obtain pseudo observations for any , > 0.

2.3 Recurrent Neural Networks
Deep Neural Networks are gaining popularity because they have shown
good results in diverse problems like in the areas of object and speech
recognition (see Krizhevsky et al.2012 and Dahl et al., 2012 for examples). They
are machine learning technique oriented to learn high-level abstractions by
using hierarchical architectures. This architecture is based upon the classic
approximation theorem by Cybenko that states that perceptrons with a hidden
layer of finite size and sigmoid activation functions can approximate complex
continuous functions and was implemented using back-propagation
algorithm in multilayer perceptrons to overcome the classic XOR problem.

370 | I S I W S C 2 0 1 9