74
D. Krpelík and T. Basu
parameters. We call these the conjugated families. For a particular choice of the
exponential model for the observed samples in Example 2.7, the conjugated family
of distributions of θ is the gamma distributions. This class also induces a closed
form for the posterior predictive distribution (Eq. (2.28)), the Lomax distribution. If
no conjugate form can be found for the Bayesian inference, the problem needs to be
solved numerically, generally using Monte Carlo algorithms [28].
2.6.2 Robust Bayesian Inference
One application of the IP theory was to provide means for sensitivity analysis
for various decision-making problems under uncertainty. In the case of Bayesian
inference, it was labelled robust Bayesian analysis [6]. In the Bayesian framework,
we can analyse the sensitivity on both the prior distribution and/or the observation
model, the likelihood function. A straightforward solution is to consider sets of
functions (priors and likelihoods) instead of just a single one in the analysis. The
set of prior distributions would define an F-probability with the respective credal
set. The credal set for the posterior F-probability would be given by the set of all
updated priors. All the assertions of interested would then be given by extremisation
over the posterior credal set (Eq. (2.10)).
Example 2.8 Assume the same observations as in Examples 2.6 and 2.7
and the same set of admissible sampling models, P. This leads to the same
likelihood function (Eq. (2.26)). Now, assume that we cannot properly specify
one prior distribution for the Bayesian analysis as in Example 2.7. Instead, let
us consider a set of prior distributions, again conjugated with our likelihood
(i.e. gamma distributions).
As emphasised in Sect. 2.2.4, while consulting the IP model, we need to
consider answers from all the singular models in the credal set. In the case of
reconstructing the predictive CDFs, we construct the bounds for all the CDFs
from the set of all the updated prior distributions. Therefore,
F (x) := min
π ∈Π 0
1
Z π ( x)
x
0
θ
p(x n+1 |θ)L(θ ; ;
x)π(θ)dθdx n+1 .
An example of a robust Bayesian inference is depicted in Fig. 2.8, where the
lower and upper bounds are given for both the prior and posterior predictive
distributions.
A powerful result of the robust Bayesian inference is a closed-form solution
for the imprecise Dirichlet model [40]. The power lies in two features. First, the
model is constructed for multinomial sample distributions. It can therefore be used
D. Krpelík and T. Basu
parameters. We call these the conjugated families. For a particular choice of the
exponential model for the observed samples in Example 2.7, the conjugated family
of distributions of θ is the gamma distributions. This class also induces a closed
form for the posterior predictive distribution (Eq. (2.28)), the Lomax distribution. If
no conjugate form can be found for the Bayesian inference, the problem needs to be
solved numerically, generally using Monte Carlo algorithms [28].
2.6.2 Robust Bayesian Inference
One application of the IP theory was to provide means for sensitivity analysis
for various decision-making problems under uncertainty. In the case of Bayesian
inference, it was labelled robust Bayesian analysis [6]. In the Bayesian framework,
we can analyse the sensitivity on both the prior distribution and/or the observation
model, the likelihood function. A straightforward solution is to consider sets of
functions (priors and likelihoods) instead of just a single one in the analysis. The
set of prior distributions would define an F-probability with the respective credal
set. The credal set for the posterior F-probability would be given by the set of all
updated priors. All the assertions of interested would then be given by extremisation
over the posterior credal set (Eq. (2.10)).
Example 2.8 Assume the same observations as in Examples 2.6 and 2.7
and the same set of admissible sampling models, P. This leads to the same
likelihood function (Eq. (2.26)). Now, assume that we cannot properly specify
one prior distribution for the Bayesian analysis as in Example 2.7. Instead, let
us consider a set of prior distributions, again conjugated with our likelihood
(i.e. gamma distributions).
As emphasised in Sect. 2.2.4, while consulting the IP model, we need to
consider answers from all the singular models in the credal set. In the case of
reconstructing the predictive CDFs, we construct the bounds for all the CDFs
from the set of all the updated prior distributions. Therefore,
F (x) := min
π ∈Π 0
1
Z π ( x)
x
0
θ
p(x n+1 |θ)L(θ ; ;
x)π(θ)dθdx n+1 .
An example of a robust Bayesian inference is depicted in Fig. 2.8, where the
lower and upper bounds are given for both the prior and posterior predictive
distributions.
A powerful result of the robust Bayesian inference is a closed-form solution
for the imprecise Dirichlet model [40]. The power lies in two features. First, the
model is constructed for multinomial sample distributions. It can therefore be used