Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 3. Hierarchical multinomial Dirichlet model incorporated uncertainty about order restrictions

3.1 Model specification

We consider adding uncertainty to the model to increase the robustness and flexibility. Let $L_{pos} = ℓ$ be the mode position of cell probabilities. The extension of the hierarchical multinomial Dirichlet model, denoted as $M_{4},$ is

n_{i} | θ_{i}, L_{pos} = ℓ \overset{ind}{~} Multinomial (n_{i \cdot}, θ_{i}), i = 1, \dots, I, ℓ = 1, \dots, K,

θ_{i} | μ, τ, L_{pos} = ℓ \overset{ind}{~} Dirichlet (μ τ), i = 1, \dots, I, θ_{i} \in C_{l},

π (μ, τ | L_{pos} = ℓ) = \frac{K (m_{l} - 1)! (K - m_{ℓ})!}{{(1 + τ)}^{2}}, μ_{j} > 0, \sum_{j = 1}^{K} μ_{j} = 1, μ \in C_{μ_{ℓ}},

where

C_{ℓ} = {θ_{i} : θ_{i 1} \leq \dots \leq θ_{i m_{ℓ}} \geq \dots \geq θ_{i K}},

C_{μ_{ℓ}} = {μ : μ_{1} \leq \dots \leq μ_{m_{ℓ}} \geq \dots \geq μ_{K}},

and

P (L_{pos} = ℓ) = w_{ℓ}, \sum_{ℓ = 1}^{K} w_{ℓ} = 1, ℓ = 1, \dots, L .

Modes are the same for all areas but we are uncertain about where they are.

Then the joint posterior distribution of $θ,$ $μ,$ and $τ,$ is

\begin{array}{l} π (θ, μ, τ | n) & \propto \sum_{L_{pos} = 1}^{L} w_{L_{pos}} \prod_{i = 1}^{I} {\prod_{j = 1}^{K} θ_{i j}^{n_{i j}} \frac{\prod_{j = 1}^{K} θ_{i j}^{μ_{j} τ - 1} I_{C_{L_{pos}}} I_{C_{μ_{L_{pos}}}}}{D (μ τ) C (μ τ)}} \frac{1}{{(1 + τ)}^{2}} \\ \propto \sum_{L_{pos} = 1}^{L} w_{L_{pos}} \prod_{i = 1}^{I} {\frac{\prod_{j = 1}^{K} θ_{i j}^{n_{i j} + μ_{j} τ - 1} I_{C_{L_{pos}}} I_{C_{μ_{L_{pos}}}}}{D (μ τ) C (μ τ)}} \frac{1}{{(1 + τ)}^{2}}, \end{array}

where $I_{C_{L_{pos}}}$ and $I_{C_{μ_{L_{pos}}}}$ are the indicator functions under that order restriction.

3.2 Estimation of $P (L_{pos} = ℓ | n)$

To generate samples of $θ,$ $μ$ and $τ,$ we have to deal with the uncertainty indicator $L_{pos} .$ In $M_{4},$ variable $L_{pos}$ has prior $P (L_{pos} = ℓ) = w_{l}$ and posterior

P (L_{pos} = ℓ | n) = \frac{w_{L_{pos}} \int_{μ} \int_{τ} \prod_{i = 1}^{I} {\frac{\prod_{j = 1}^{K} θ_{i j}^{n_{i j} + μ_{j} τ - 1} I_{C_{L_{pos}}} I_{C_{μ_{L_{pos}}}}}{D (μ τ) C (μ τ)}} \frac{1}{{(1 + τ)}^{2}} d τ d μ}{\sum_{L_{pos} = 1}^{L} w_{L_{pos}} \int_{μ} \int_{τ} \prod_{i = 1}^{I} {\frac{\prod_{j = 1}^{K} θ_{i j}^{n_{i j} + μ_{j} τ - 1} I_{C_{L_{pos}}} I_{C_{μ_{L_{pos}}}}}{D (μ τ) C (μ τ)}} \frac{1}{{(1 + τ)}^{2}} d τ d μ} .

Chen and Nandram (2021) notice the order restrictions will significantly increase the computational difficulty, especially for the marginal likelihood. There is an accuracy-efficiency trade-off. We notice that for each iteration from M1 model, there are two patterns of unimodal structure in $θ$ for different counties. One is that the normal BMI level has the highest cell probability among five levels, which can be considered as an unimodal structure and the mode is at the second position. Another is that the overweight BMI level has the highest probability, which can be considered as an unimodal structure and the mode is at the third position. We can approximate the posterior sample using information obtained from M1.

Estimation Method:

Apply M1 model to the data and acquire posterior samples of $θ .$
For each iteration, count areas whose first cell probabilities are the largest among other cells.
In the same iteration, count areas whose second cell probabilities are the largest.
Count areas whose third cell probabilities are the largest, until the last cell.
Compute the ratio of different cases. For example, we may only have 13 counties whose normal BMI level probabilities are the largest and 22 counties whose overweight probabilities are the largest. Then we have the ratio is 13/22.
Compute the average of ratios for overall iterations. Use the average as approximated mixture probabilities.

For example, in our application BMI, 37.2% of $θ$ has mode at the second position, 62.8% of $θ$ has mode at the third position. Then we can have $(\hat{L_{pos} = 2} | n) \approx$ 0.372 and $P (\hat{L_{pos} = 3} | n) \approx$ 0.628 as probabilities to mix samples from M2 (mode at 2^nd) and samples from M3 (mode at 3^rd) together.

Then ${\hat{CPO}}_{i (M_{4})} \approx {[{\sum_{ℓ = 1_{_{_{}}}}^{K} }^{^{^{}}} P (\hat{L_{pos} = ℓ} | n) \frac{1}{{\hat{CPO}}_{i (L_{pos} = ℓ)}}]}^{- 1},$

where ${\hat{CPO}}_{i (L_{pos} = ℓ)}$ are computed in Section 2.1. In the following numerical example,

{\hat{CPO}}_{i (M_{4})} \approx {[P (\hat{L_{pos} = 2} | n) \frac{1}{{\hat{CPO}}_{i (M_{2})}} + P (\hat{L_{pos} = 3} | n) \frac{1}{{\hat{CPO}}_{i (M_{3})}}]}^{- 1},

without extra computation, taking advantage of known CPOs and the estimated

P (L_{pos} = ℓ | n)

from the previous section, we can easily acquire the CPO of

M_{4},

as in Appendix A.3.

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Date modified:: 2022-06-21

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Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 3. Hierarchical multinomial Dirichlet model incorporated uncertainty about order restrictions

3.1 Model specification

3.2 Estimation of $P (L_{pos} = ℓ | n)$

Estimation Method:

Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction Section 3. Hierarchical multinomial Dirichlet model incorporated uncertainty about order restrictions

3.1 Model specification

Estimation Method:

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 3. Hierarchical multinomial Dirichlet model incorporated uncertainty about order restrictions