![](bg7.png)
days
posterior
0 50 100 150 200 250 300 350
0.0 0.2 0.4 0.6 0.8 1.0
Figure 2: Posterior Probability Plot of p
θ
t
= 1
x
x
x
t
with
Informative Priors.
Table 9: AIC values for Proposed and Stationary Models
with Informative Priors.
Items Proposed Stationary
AIC -505.776 -522.896
6 CONCLUSIONS
This paper proposes a class of nonstationary Bernoulli
distribution and the Bayes optimal prediction algo-
rithm under the known nonstationary hyper parame-
ter. The proposed class has only one extra hyper pa-
rameter compared to the stationary Bernoulli distribu-
tion, and it is proved that the posterior distribution of
the Bernoulli parameter is obtained analytically. Fur-
thermore, the predictive performance of the proposed
algorithm is evaluated using real binary data. As a
result, a certain advantage for predictive performance
is discovered by comparing the results to those of the
stationary Bernoulli model; however, this point can-
not be explained in terms of model selection theory.
As important factor in the abovementioned ad-
vantage is the additional nonstationary hyper param-
eter in the proposed model. Because the empirical
Bayesian approach is used in this study and the ad-
ditional hyper parameter is estimated by the approx-
imate maximum likelihood estimation, the objective
likelihood function should be analyzed in detail. This
point will be left for future work.
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APPENDIX
A: Proof of Lemma 2.1
Suppose t = 1, A
1
= a
1
and C
1
= c
1
are defined as,
A
1
∼ Gamma(α
1
, 1) , (34)
C
1
∼ Beta [kα
1
, (1 − k) α
1
] , (35)
according to Definition 2.5 and Definition 2.6, respec-
tively.
Since A
2
= C
1
A
1
from Definition 2.3, and A
t
and
C
t
are conditional independent from Definition 2.4,
On the Prediction of a Nonstationary Bernoulli Distribution based on Bayes Decision Theory
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