## Rao-blackwellized estimates for the multivariate bayesian inference

#### Citation

Purutçuoğlu, V. ve Wit, E. (2009). Rao-blackwellized estimates for the multivariate bayesian inference. Maltepe Üniversitesi. s. 380.#### Abstract

In Bayesian inference the summary statistics of the parameter estimates are generally chosen as the mean and standard deviation of the posterior distribution after the burn-in period. If we know the conditional distribution of each
parameter cj (j = 1, . . . , r) given other parameters ck (k 6= j), the mean of the estimates can be improved via Raoblackwellization estimator which is the conditional probability of cj averaged over the conditional probability of ck’s via
E(cj ) = 1
n
Pn
k=1 E(cj |Θ
(k)
−j
). Here Θ(k) denotes the r-dimensional parameter vector Θ = (c1, . . . , cr) at the kth iteration
after the burn-in period, thereby Θ(k)
−j
states all the components of Θ(k)
except for cj . Finally n represents the number
of samples chosen from the conditional posterior distribution of the parameter after burn-in. From empirical results, it is
shown that due to the consequence of the Rao-blackwell theorem, the given equation above typically gives better estimate
than the mean of the posterior distribution of cj in terms of accuracy.
In Rao-blackwellization where the direct computation of Θ component is not possible, whereas the full conditional
distribution of cj is known, the sampler for cj can be generated via Gibbs sampling from the MCMC outputs (Gelfand
and Smith, 1990; Boys et al., 2000). On the other hand if the conditional distribution of Θ is unknown, hereby Gibbs is
not applicable, an ε-neighbourhood of Θ is defined in such a way that a sufficiently small sphere with radius ε can cover n
samples of cj (Tanner and Wong, 1987). Although this idea works for small set of variables, it is difficult to find such an
ε-radial sphere for high dimensional multivariate estimation unless we reduce the dimension of Θ.
For calculating a Rao-blackwellized estimate for the model parameters in which the Gibbs and ε-neighbourhood techniques cannot be plausible, we propose an alternative solution in such a way that the unknown normalizing constant of the
transition probability from the full conditional distribution of parameters is calculated by numerical integration. By using
the available MCMC results, we approximate a specific normalizing constant for each cj considering the possible interval of
Θ without reducing any dimension. Then these findings are combined with the computations of the conditional distribution
of Θ which are updated by either Gibbs or Metropolis-Hasting algorithms.
To evaluate the performance of our approach, we implement it to estimate the Rao-blackwellized estimates of model
parameters of a large biological network where the known methods cannot be utilized.

#### Source

International Conference of Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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