Bayes optimal estimation and its approximation algorithm for difference with and without treatment under IRSLC model

Abstract

We consider verifying the effect of the treatment under the situation in which a response is given when a treatment is applied to units with features. In estimating the effect, there are problems such as the treatment can be given only once to the unit and the features of the unit cannot be controlled. For such problems, conventional studies have some mathematical models. However, in this paper, we propose the different data generative model in which there are latent classes of units with the same response, each latent class contains units with similar features. We call this model the identical response structure latent class model (IRSLC model). Under the proposed model, we calculate the Bayes optimal decision and its approximation algorithm for the difference with and without the treatment for the entire population. We conducted experiments using the synthetic data of the model assumed by the proposed method or the conventional method. Then, we compared our method with previous studies to confirm the characteristics of the proposed model.

Appendix A: Proof of proposition 2

By substituting the approximate posterior distribution $q^{(T)}$$({\varvec{z}}_k^N, {\varvec{\theta }}_k, k) =q^{(T)}(k) q^{(T)}({\varvec{z}}^N_k,{\varvec{\theta }}_k\mid k)$ in the posterior distribution $p({\varvec{z}}_k^N,{\varvec{\theta }}_k,k\mid {\varvec{D}})$, we have

$$\begin{aligned} d^*({\varvec{D}})&=\sum _{k=1}^{k_{max}} \int _{\Theta _k} p({\varvec{\theta }}_k, k \mid {\varvec{D}}) \nonumber \\&\quad \quad \cdot \sum _{l_k = 1}^{k} \pi _{l_k} \left( {\varvec{s}}_{wl_k}^\top {\varvec{m}}_{l_k}+\check{s}_{wl_k,1}\right) d {\varvec{\theta }}_k \end{aligned}$$

(A1)

$$\begin{aligned}&=\sum _{k=1}^{k_{max}}\sum _{{\varvec{z}}^{N}_k\in \mathcal Z_k^{N}}\int _{\Theta _k} \sum _{l_k = 1}^{k} \pi _{l_k} \left( {\varvec{s}}_{wl_k}^\top {\varvec{m}}_{l_k}+\check{s}_{wl_k,1}\right) \nonumber \\&\quad \cdot p({\varvec{z}}^{N}_k,{\varvec{\theta }}_k,k\mid {\varvec{D}})d{\varvec{\theta }}_k \end{aligned}$$

(A2)

$$\begin{aligned}&\approx \sum _{k=1}^{k_{max}}\sum _{{\varvec{z}}^{N}_k\in \mathcal Z^{N}}\int _{\Theta _k} \sum _{l_k = 1}^{k} \pi _{l_k} \left( {\varvec{s}}_{wl_k}^\top {\varvec{m}}_{l_k}+\check{s}_{wl_k,1}\right) \nonumber \\&\quad \cdot q^{(T)}({\varvec{z}}^{N}_k,{\varvec{\theta }}_k,k)d{\varvec{\theta }}_k \end{aligned}$$

(A3)

$$\begin{aligned}&=\sum _{k=1}^{k_{\max }} \int \int \int \int \left\{ \sum _{l_k = 1}^{k} \pi _{l_k} ({\varvec{s}}_{wl_k}^\top {\varvec{m}}_{l_k}+\check{s}_{wl_k,1})\right\} \nonumber \\&\quad \cdot q^{(T)}\left( \check{{\varvec{S}}}_{k},\check{{\varvec{S}}}_{\textrm{wk}}\right) q^{(T)}(k) q^{(T)}(\varvec{\pi }_k)q^{(T)}(\varvec{M_k},\varvec{\Lambda _k}) \nonumber \\&\quad d \check{{\varvec{S}}}_k d \check{{\varvec{S}}}_{\textrm{wk}}d {\varvec{\pi }}_k d{\varvec{\Lambda }}_k d{\varvec{M}}_k \end{aligned}$$

(A4)

$$\begin{aligned}&= \sum _{k=1}^{k_{\max }} q^{(T)}(k) \nonumber \\&\quad \cdot \int \int \int \int \left\{ \sum _{l_k = 1}^{k} \pi _{l_k} ({\varvec{s}}_{wl_k}^\top {\varvec{m}}_{l_k}+\check{s}_{wl_k,1})\right\} \nonumber \\&\quad \cdot \textrm{Dir}\left( {\varvec{\pi }}_k\mid {\varvec{\alpha }}_k^{(T)}\right) \mathcal N\left( {\varvec{m}}_{l_k}\mid {\varvec{\mu }}_{l_k}^{(T)},(\gamma _{l_k}^{(T)}{\varvec{L}}_{l_k})^{-1}\right) \nonumber \\&\quad \cdot \mathcal W\left( {\varvec{L}}_{l_k}\mid {\varvec{W}}_{l_k}^{(T)},\nu _{l_k}^{(T)}\right) \nonumber \\&\quad \cdot \prod _{l_k=1}^k \mathcal {N}\left( \left[ \begin{array}{c}\check{{\varvec{s}}}_{l_{k}} \\ \check{{\varvec{s}}}_{wl_{k}}\end{array}\right] \mid \left[ \begin{array}{c}\check{{\varvec{\beta }}}_{l_{k}}^{(T)} \\ \check{{\varvec{\beta }}}_{wl_{k}}^{(T)}\end{array}\right] , \left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) ^{-1}\right) \nonumber \\&\quad d \check{{\varvec{S}}}_k d \check{{\varvec{S}}}_{\textrm{wk}}d {\varvec{\pi }}_k d{\varvec{\Lambda }}_k d{\varvec{M}}_k . \end{aligned}$$

(A5)

To calculate (A5), we define

$$\begin{aligned} \left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) ^{-1}: =\left[ \begin{array}{ll}\left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) _{(11)}^{-1} &{} \left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) _{(12)}^{-1} \\ \left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) _{(21)}^{-1} &{} \left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) _{(22)}^{-1}\end{array}\right] . \end{aligned}$$

(A6)

Then, we can rewrite (A5) as

$$\begin{aligned}&\sum _{k=1}^{k_{\max }} q^{(T)}(k) \int \int \int \int \left\{ \sum _{l_k = 1}^{k} \pi _{l_k} ({\varvec{s}}_{wl_k}^\top {\varvec{m}}_{l_k}+\check{s}_{wl_k,1})\right\} \nonumber \\&\quad \cdot \prod _{l_k=1}^k\mathcal N\left( \check{{\varvec{s}}}_{\textrm{wk}}\mid \check{{\varvec{\beta }}}_{wl_{k}}^{(T)}, \left( {\widetilde{{\varvec{\Lambda }}}_{l_{k}}}^{(T)}\right) _{(22)}^{-1}\right) \textrm{Dir}\left( {\varvec{\pi }}_k\mid {\varvec{\alpha }}_k^{(T)}\right) \nonumber \\&\quad \cdot \mathcal N\left( {\varvec{m}}_{l_k}\mid {\varvec{\mu }}_{l_k}^{(T)},(\gamma _{l_k}^{(T)}{\varvec{L}}_{l_k})^{-1}\right) \nonumber \\&\quad \cdot \mathcal W\left( {\varvec{L}}_{l_k}\mid {\varvec{W}}_{l_k}^{(T)},\nu _{l_k}^{(T)}\right) d \check{{\varvec{S}}}_{\textrm{wk}}d {\varvec{\pi }}_k d{\varvec{\Lambda }}_k d{\varvec{M}}_k \end{aligned}$$

(A7)

$$\begin{aligned}&=\sum _{k=1}^{k_{\max }} q^{(T)}(k) \sum _{l_k=1}^{k}\frac{\alpha _{l_k}^{(T)}}{\hat{\alpha }_k^{(T)}}\left\{ \left( {{\varvec{\beta }}}_{w{l_k}}^{(T)}\right) ^\top {\varvec{\mu }}_{l_k}^{(T)} + \check{\beta }_{w{l_k,1}}^{(T)}\right\} , \end{aligned}$$

(A8)

where $(\check{\beta }_{w{l_k,1}}^{(T)},\dots , \check{\beta }_{w{l_k,p+1}}^{(T)})$ are elements of $\check{{\varvec{\beta }}}_{w{l_k}}^{(T)}$ and

$$\begin{aligned} \hat{\alpha }_k^{(T)}&:= \sum _{{l_k}=1}^k\alpha _{l_k}^{(T)}, \end{aligned}$$

(A9)

$$\begin{aligned} {{\varvec{\beta }}}_{w{l_k}}^{(T)}&:=\left( \check{\beta }_{w{l_k,2}}^{(T)},\dots , \check{\beta }_{w{l_k},p+1}^{(T)}\right) . \end{aligned}$$

(A10)