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42a7341fe75822d745ddec94f91a0a15c565485f | subsection | 9 | 63 | Approximate Theory and Method | This
motivates our approximate methods.We note that given {\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2,{y\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}, the i are independent with
\begin{equation}
\pi (\mu _i \mid {\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2,{y)\crcr \vbox to.... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
3293,
167729,
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6820,... | [
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0.119384765625,
0.057647705078125,
0.153564453125,
0.092529296875,
0.12... | |
adb0877f4b08d339b98ae5e7e244a523678d1277 | subsection | 10 | 63 | Approximation to the Posterior Density | In this section we discuss the approximation to the joint posterior density
in (REF ).Let f({\tau )\crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }} = e^{h({\tau )\crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }}} denote the density of a vector of parameters
{\tau \crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }}. Let g\c... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.2020263671875,
0.098388671875,
0.1833... | |
63b86f6be8d350be96c640c39c0627ca5c80ebff | subsection | 11 | 63 | Approximation to the Posterior Density | Then, \tau \crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss } approximately has a multivariate normal distribution,{\tau \stackrel{}{\crcr }\vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }}{}{\sim }\mbox{Normal} ({\tau ^\crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }}*-H^{-1}{g,\crcr \vbox to.2ex{\hbox{$g\tilde{}$}\vss }}-H^{-1}... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.004730224609375,
0.1905517578125,
0.0042114... | |
b991df9707015e96fef4cd7ea95dfd08b1d01235 | subsection | 12 | 63 | Approximation to the Posterior Density | That is,y_{ij}|\mu _i,{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)} \stackrel{ind}{\sim } \mbox{Bernoulli} \left\lbrace \frac{e^{{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\prime }{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}+\mu _i}}{1+e^{{x_\crcr \vbox to.2ex{\hbox{$x\tilde... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.11169433593... | |
34c2c34f69bc523ac9eb580223effbd4cadc6003 | subsection | 13 | 63 | Approximation to the Posterior Density | First, we find a
convenient point to expand the log-likelihood in a multivariate Taylor's series
expansion. In Appendix B, we show how to obtain quasi-modes for {\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)} and
\mu _i, i=1,\ldots ,\ell , of the log-likelihood function.First, we use the empirical logist... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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d1f1497862b9b35c9432c0f8b3ff30aeae92c1d6 | subsection | 14 | 63 | Approximation to the Posterior Density | We use the log-likelihood of the \mu _i with
the regression coefficients replaced by {\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}^\ast , and solve its first derivative for zeros using a first-order
Taylor's series expansion to get{\mu _i}^* = \log \left[ \frac{\frac{1}{n_i}\sum _{j=1}^{n_i}e^{-{x_\crc... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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71b7c46b1318716cd77ef17ba36b4af265564427 | subsection | 15 | 63 | Approximation to the Posterior Density | The partial derivatives can be expressed in terms of response yij and covariates x_x\tilde{}ij as
\frac{\partial \Delta }{\partial {\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}}=\sum _{i=1}^\ell \sum _{j=1}^{n_i} ( {x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij} y_{ij}-\frac{{x_\crcr \vbox to.2ex{... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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3d1b7f26f882042f90efdc342a4036073c581c25 | subsection | 16 | 63 | Approximation to the Posterior Density | For the convenience of computation, denote {g=\crcr \vbox to.2ex{\hbox{$g\tilde{}$}\vss }}\left(\begin{array}{c} {g_\crcr \vbox to.2ex{\hbox{$g\tilde{}$}\vss }}1\\{g_\crcr \vbox to.2ex{\hbox{$g\tilde{}$}\vss }}2\end{array}\right) and H=-\left(\begin{array}{c}
D\quad C^{\prime }\\C \quad B\end{array}\right), where{g_\cr... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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f0b0295510663d886b6980929a4dc90d3e2c0aa0 | subsection | 17 | 63 | Approximation to the Posterior Density | Then,
A, B and C of the negative Hessian matrix can be written as,B = \frac{\partial ^2\Delta }{\partial {\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}^2}=\sum _{i=1}^\ell \sum _{j=1}^{n_i}
p_{ij}(1-p_{ij}){x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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d9cbca8dee8fefef90cb221a85e8bfcaf9e4b2d4 | subsection | 18 | 63 | Approximation to the Posterior Density | Then, the Schur complement isS = \sum _{i=1}^\ell \sum _{j=1}^{n_i} p_{ij}(1-p_{ij}) \sum _{j=1}^{n_i} \omega _{ij}
{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij} {x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^\prime - \sum _{i}^\ell \sum _{j=1}^{n_i} p_{ij}(1-p_{ij})
\sum _{j=1}^{n_i} \omega _{ij} {x_\crcr \... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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13cf8d89d4f42735cb969a5379d1545b7866b69e | subsection | 19 | 63 | Approximation to the Posterior Density | By Lemma 2.1 the posterior density is approximately
a multivariate normal density. | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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] | |
3220d395713cf957729168c95c6cf8a6fba4c248 | subsection | 20 | 63 | Approximation to the Posterior Density | We provide the approximate mean and variance to completely specify the
multivariate normal density.By Lemma 2.1, evaluating all appropriate quantities at {\tau ^\crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }}\ast , the posterior mean is\left(\begin{array}{c}{\mu _\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\mu }\\... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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266fd0090121def361b639d0e391f969c2a82cfa | subsection | 21 | 63 | Integrated Nested Normal Approximation | In this section, we obtain the integrated nested normal approximation (INNA).
INNA, which does not require posterior modes, is competitive to the integrated nested
Laplace approximation (INLA) that requires posterior modes.Next, using the normal priors for the \mu _i and Theorem 1, we have the following approximate hie... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.033355712890625,
0.2393798828125,
0.2790527343... | |
fc75a39848e21ed4e966ae783dff0f364b29ddd1 | subsection | 22 | 63 | Integrated Nested Normal Approximation | Then, using Bayes' theorem again, the approximate joint posterior density for the parameters {\mu ,\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\beta \crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }} and 2 is
\pi _a({\mu ,\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
47009,
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ae5eca1173bc4a2583d7df591058821ee7648b67 | subsection | 23 | 63 | Integrated Nested Normal Approximation | \begin{equation}
\times \frac{|D|^{1/2}}{{|\delta ^2I|}^{1/2}|G|^{1/2}}\times \frac{1}{(1+\delta ^2)^2}\times e^{-\frac{1}{2}\left[{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}-{\mu _\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\beta }\right]^{\prime }G^{-1}\left[{\beta _\crcr \vbox to.2ex{\hbox{$\... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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bfba56b5aa2ead0ad75f1a7e6f37ce005589505e | subsection | 24 | 63 | Integrated Nested Normal Approximation | \end{equation}
}Next, we state the main result of the paper in Theorem 2.2.Theorem 3.2 Using the multiplication rule, the joint posterior density,
\pi ({\mu ,\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2 \mid {y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vs... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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1d1d0ac357103422b02adf417cdbea682b17223e | subsection | 25 | 63 | Integrated Nested Normal Approximation | Because (D+\frac{1}{\delta ^2}I) is diagonal, given {\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2,{y\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}, the i are independent. This is an
important result because parallel computation can be done for i, which accommodates time-consuming and massive storage ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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3f5381d8c91a0c0a0b3ad9c27fa8c73547c1580b | subsection | 26 | 63 | Integrated Nested Normal Approximation | Second, because {\mu \crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }} has a multivariate normal distribution, we can integrate out \mu \crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss } from the joint posterior density \pi _a({\mu ,\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }} {\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tild... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.219970... | |
403148f2e1f30147749335dfd6710bafab5dc2fd | subsection | 27 | 63 | Integrated Nested Normal Approximation | \right.
and
}
\pi ({\beta |\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2,{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }} \propto e^{-\frac{1}{2} \left[{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}-{\mu _\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\beta }\right]^{\prime }G^{-1}\left[{\b... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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ebfaf77424d85065ecf7228ebafa5931af5c650c | subsection | 28 | 63 | Integrated Nested Normal Approximation | \right.
}\end{equation}Let
\Delta _{(0)}=CD^{-1}(D^{-1}+\delta ^2I)^{-1}D^{-1}C^{\prime }+G^{-1},
\delta ^2_0={j^{\prime }\crcr \vbox to.2ex{\hbox{$j\tilde{}$}\vss }}(D^{-1}+\delta ^2I)^{-1}{j,\crcr \vbox to.2ex{\hbox{$j\tilde{}$}\vss }}
{\gamma =\crcr \vbox to.2ex{\hbox{$\gamma \tilde{}$}\vss }}CD^{-1}(D^{-1}+\de... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
41872,
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c0e451968f1755051d2c6c0a730b511dcb44e457 | subsection | 29 | 63 | Integrated Nested Normal Approximation | After extensive algebraic manipulation, we can show that
{\beta |\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2,{y\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }} has an approximate multivariate normal density,
\begin{equation}
\left(\begin{array}{c}
\beta _0\\{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.0422668457... | |
1c23f4e5e0db8a8027e1e6e8a15b7ee82dba64cf | subsection | 30 | 63 | Integrated Nested Normal Approximation | Samples are obtained by first
drawing from \pi _a(\delta ^2|{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}, then \pi _a({\beta \mid \crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2, {y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }} and finally
\pi _a({\mu \mid \crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\b... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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36b432ab020c5172b4e1f255a3841cd320a17fad | subsection | 31 | 63 | Integrated Nested Normal Approximation | Then,
posterior density, \pi (\eta |{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}, is\pi (\eta |{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}\propto \left\lbrace \frac{1}{|\delta ^2D+I|^{1/2}}\left|\left(\begin{array}{c}
\delta ^2_0 \quad {\gamma ^{\prime }\crcr \vbox to.2ex{\hbox{$\gamma \tilde{}$}\vss }}\\
{\gamm... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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50b62e34487b666cf7987f0f7f02898417d2029a | subsection | 32 | 63 | Integrated Nested Normal Approximation | Equation (REF ) is the basis of our INNA approximation. This is simply the multiplication rule of probability;
simply draw \eta from (REF ) and retransform to \delta ^2 to get \pi _a(\delta ^2 \mid {y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }},
draw {\beta \crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }} from a(\bet... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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3ebcab434a8d09cd9636004a17535c4c4eefa6b6 | subsection | 33 | 63 | Comparison of the Two Methods | As a summary, we compare the approximate and exact methods. The exact method is given in Appendix A.First, we note that the exact method actually uses the approximate method. We
use a Metropolis step with \pi _a({\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2|{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.... | |
e9e59ee2c2dce780a9c06da6aeca3cb304ed7f6c | subsection | 34 | 63 | Comparison of the Two Methods | Third, for prediction two Bayesian bootstraps are used to get the nonsampled household
sizes and the nonsampled covariates (\approx two million people). This is done within wards. | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.... | |
bfd43dcf8861ed5b181ac314d1e4d8dd67980bd8 | subsection | 35 | 63 | Illustrative Example | In Section 3.1, we briefly describe the Nepal Living Standards Survey (NLSS II) and in Section 3.2, we use the health status
data with five covariates to compare our approximate Bayesian logistic regression method with the exact one. | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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cd62c9d4b1609bdf5f892900f39b0e257606f902 | subsection | 36 | 63 | Nepal Living Standards Survey | We use data from the Nepal Living Standards Survey (NLSS II, Central Bureau of Statistics, 2003-2004) to illustrate INNA with logistic regression. NLSS is a national household survey in Nepal, actually population based (i.e., interviews are done for individual household members). NLSS follows the World Bank's Living St... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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cdd08e6e739d58f9035e7d9e10feba07ab3ec8e5 | subsection | 37 | 63 | Nepal Living Standards Survey | We use Bayesian bootstraps (Rubin 1981) for unknown household sizes and nonsampled covariates; the bootstrapping is done within wards. The 2001 Census can potentially provide these two pieces of information, but there is a mis-match between the households in the census and the NLSS (a record linkage can be performed). ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0... | |
da8e0d0f3b74d559f93617f5a741da0a92c42409 | subsection | 38 | 63 | Numerical Comparisons | We predict the household proportions of members in good health for 60,221 households. This analysis is based on 3,912 sample
households from 326 wards (PSUs). Our primary purpose is to compare the approximate method with the exact method when there
are random effects at the household level. We want to show that one can... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0edc5c1fdd13324b7695bc2669bb2507efcc542f | subsection | 39 | 63 | Concluding Remarks | We make three statistical comments.
First, the approximate method is necessary when there are a large number (millions) of households (clusters or areas).
Second, it is difficult to use the census data effectively but it is desirable (matching problem). Third,
it is possible to obtain similar approximations for spatial... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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95b92df183a40d00607251fdd2a08bf98c1e271e | subsection | 40 | 63 | Concluding Remarks | Letn \ge n_e = (\sum _{i=1}^n \omega _i)^2/(\sum _{i=1}^n \omega _i^2),
~~\tilde{\omega }_i = n_e \frac{\omega _i}{\sum _{i=1}^n \omega _i}, i=1,\ldots ,n;see Potthof, Woodbury and Manton (1992) for pioneering work on equivalent sample sizes.For (y_i, \tilde{\omega }_i,{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}i),... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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8063471da440862d790c96b413ae19e13f2a8723 | subsection | 41 | 63 | Concluding Remarks | For small areas, with
(y_{ij}, \tilde{\omega }_{ij},{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}), ~j=1,\ldots ,n_i, ~i=1,\ldots ,\ell , we have
p(y_{ij} \mid {\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }} {\nu }_i) =
\frac{e^{y_{ij} ({\tilde{x}_\crcr \vbox to.2ex{\hbox{$\tilde{x}\tilde{}$}\vss }}{ij... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.09979248046875,
0... | |
5b953f502497d34e72ef6c7db4a55c769534d3b7 | subsection | 42 | 63 | Concluding Remarks | We assume that
y_{ijk}|{\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\nu _i, \mu _{ij} \stackrel{ind}{\sim } \mbox{Bernoulli}
\left\lbrace \frac{e^{{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ijk}^{\prime }{\beta +\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\nu _i + \mu _{ij}}}{1+e^{{x_\crcr \vbox ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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567bb2484025eab98cd67de26c48a9480d6e67e8 | subsection | 43 | 63 | Concluding Remarks | For logistic regression, this research is currently in progress. }{}}\begin{}
\section {Exact Method for Logistic Regression}
\end{}Recall the Bayesian logistic model with covariates that we worked on with INNA method
y_{ij}|\mu _i,{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)} \stackrel{ind}{\sim } ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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... | |
1a665a871fc4bd8e425b54c6b4bd303178263fe4 | subsection | 44 | 63 | Concluding Remarks | \end{equation}
According to Bayes^{\prime } theorem, the joint posterior density of the parameters ({\mu ,\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}{\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2|{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }} is
\pi ({\mu ,\crcr \vbox to.2ex{\hbox{$\mu \tilde{... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
3611,
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90,
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... | |
b70265866a39c4fbcab05ecaf55afa6cf826a214 | subsection | 45 | 63 | Concluding Remarks | Putting a logconcave prior on the i does
not change the logconavity of (\beta ,\beta \tilde{}\mu \mid \mu \tilde{}2,y)y\tilde{} because the product
of two logconcave densities is another logconcave density. In addition, logconcave densities
have sub-exponential tails and their moment generating functions exist (see Dha... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.1473388671875... | |
4c73135cefbdebad69cf6677281da7919ad128cf | subsection | 46 | 63 | Concluding Remarks | We apply Metropolis Hastings sampler to draw samples for parameters {\beta \crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}, 2 and \mu \crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }.The idea of exact method is to get full conditional posterior distributions for all of the parameters in the model, and then get a large ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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f0838e11f14c2a0467be358b9cfe0b25fc36a630 | subsection | 47 | 63 | Concluding Remarks | Divide the integration domain to m equal intervals [t_{k-1},t_k], k=1,...,m. Let z_i=\frac{\mu _i-\beta _0}{\delta } with standard normal distribution. | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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74893babb36660dfa5473db386d61b22f4d8f7a1 | subsection | 48 | 63 | Concluding Remarks | We get an approximate density (very accurate though),\pi ({\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2|{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}\propto \frac{1}{(1+\delta ^2)^2}\left(\frac{1}{\sqrt{\delta ^2}}\right)^\ell \prod _{i=1}^\ell \left\lbrace \sum _{k=1}^{m}\int _{t_{k-1}}^{t_k}\fr... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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132... | [
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0... | |
6fefc64134e0692b84a46f31fbb94f56c0d86ac6 | subsection | 49 | 63 | Concluding Remarks | We have the following deduction\pi ({\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2|{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}\approx \frac{1}{(1+\delta ^2)^2}
\prod _{i=1}^\ell \left\lbrace \sum _{k=1}^{m}\frac{e^{\sum \limits _{j=1}^{n_i}({x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\p... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.108154296875... | |
ac1f1e68462649dcc8e5fcbdd2e0438e175bba48 | subsection | 50 | 63 | Concluding Remarks | Then the joint posterior density for {\beta \crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }} and 2 can be expressed as
\begin{equation}
\pi ({\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\delta ^2|{y)\crcr \vbox to.2ex{\hbox{$y\tilde{}$}\vss }}\approx \frac{1}{(1+\delta ^2)^2}
\prod _{i=1}^\ell \left\lbrace... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
47009,
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e59c4d8cc1f8bc2284295e7da20cf972366497a9 | subsection | 51 | 63 | Concluding Remarks | So we take {\beta \sim \crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\mbox{Normal}({\hat{\beta },\crcr \vbox to.2ex{\hbox{$\hat{\beta }\tilde{}$}\vss }} \sigma ^2 \hat{\Sigma }), \eta /\sigma ^2 \sim \mbox{Gamma}(\eta /2, 1/2).
Tuning of the Metropolis sampler is obtained by varying \eta (e.g., \eta =8 corresponds t... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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32dd8b1c1164994602f5d1497abb0a9b27a41b4a | subsection | 52 | 63 | Quasi-Modes for Logistic Regression | Now we have to specify {\beta ^\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}*_{(0)}, {\mu ^\crcr \vbox to.2ex{\hbox{$\mu \tilde{}$}\vss }}*, {g\crcr \vbox to.2ex{\hbox{$g\tilde{}$}\vss }} and H. Consider the log likelihood function
f({\tau )\crcr \vbox to.2ex{\hbox{$\tau \tilde{}$}\vss }}
=\log h({\tau )\crcr \vbo... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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2eb9aa6824fffec07a57a0eab08313e078a20f10 | subsection | 53 | 63 | Quasi-Modes for Logistic Regression | Plug {\hat{\mu }_i}^* in the log likelihood function () and consider it as a function of {\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)} only,g({\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)})
=\sum _{i=1}^\ell \sum _{j=1}^{n_i} \left\lbrace ({x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.001... | |
ff8344cc02b5096b99bf4f94af720e2127d80c34 | subsection | 54 | 63 | Quasi-Modes for Logistic Regression | We use first order Taylor series approximation to simplify the above function. Since the first order Taylor expansion of (1+e^{-({x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\prime }{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}+\hat{\mu }_i^*)} )^{-1} equals (1-e^{-({x_\crcr \vbox to.2ex{\hbox{... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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... | |
81f331799ea639456821d919e739f812d51fde5a | subsection | 55 | 63 | Quasi-Modes for Logistic Regression | Thus (REF ) approximately equals\sum _{i=1}^\ell \sum _{j=1}^{n_i} \left\lbrace {x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij} y_{ij}-{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij} (1-(1-({x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\prime }{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.018234... | |
405c62fdb9d1e5246e0136161ac8a72e1ba69496 | subsection | 56 | 63 | Quasi-Modes for Logistic Regression | Solve for g^{\prime }({\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)})=0, and we can get the approximate posterior mode of {\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}^*= \left[ \sum _{i=1}^\ell \sum _{j=1}^{n_i}{x_\crcr \vbo... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.0196990966796875... | |
65cc02ae04e1476c0117c9271a50f89688b6ccf8 | subsection | 57 | 63 | Quasi-Modes for Logistic Regression | The first derivative function of q(i) over i is
\begin{equation}
q^{\prime }(\mu _i)=\sum _{j=1}^{n_i} \left\lbrace y_{ij}-\frac{e^{({x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\prime }{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}^*+\mu _i)}}{1+e^{{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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c5b3c2ff3f620cd1bc579f611122fcda4e5d5d03 | subsection | 58 | 63 | Quasi-Modes for Logistic Regression | Solve for q'(i)=0 , then the approximate posterior mode (quasi-mode) of i can be obtained as {\mu _i}^*=\log \left[ \frac{\sum _{j=1}^{n_i}e^{-{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\prime }{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)}^*}}{n_i(1-\bar{y}_i)} \right] .
Notice that the term ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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a9e5c9945b7b2ea46ed99e75bfd71de22a2ca89c | subsection | 59 | 63 | Empirical Logistic Transform (ELT) | We consider the empirical logistic transform (ELT) without covariates for binary
data. See Cox and Snell (1972) for the empirical logistic transform (ELT) that
accommodates binary data.
Letting y denote a binomial random variable with success probability p, the empirical logistic transform, Z, isZ=\mbox{log} (\frac{Y+\... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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4079d02261f7972932ea3f50ad96f0a7a51d2f07 | subsection | 60 | 63 | Empirical Logistic Transform (ELT) | Faes, C., Ormerod, J. T. and Wand, M. P. (2011).
Variational Bayesian Inference for Parametric and
Nonparametric Regression With Missing Data.
Journal of the American Statistical Association, 106, 959-971.
Fay, R.E. and Herriot, R.A. (1979).
Estimates of Income for Small Places: an Application of James-Stein Procedures... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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... | |
32306fc6ec28f9ab60ff47e862ece8ac2845de2b | subsection | 61 | 63 | Empirical Logistic Transform (ELT) | Journal of the American Statistical Association, 105, 120-135.
Nandram, B. and Erhardt, E. (2005).
Fitting Bayesian Two-Stage Generalized Linear Models
Using Random Samples via the SIR Algorithm.
Sankhya, 66, 733-755.
Ormerod, J. T., and Wand, M. P. (2010).
Explaining Variational Approximations.
The American Statistici... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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ac1a3b91cf32635e65434fa6eac0376eabdaa57b | subsection | 62 | 63 | Empirical Logistic Transform (ELT) | Journal of Multivariate Analysis,127, 36-55.
[table]skip=0pt
[Table: Categorical tables for 60,221 households by posterior coefficient of variation of model (2) with random effects at theward level projected to the households and the model (1) at household level][Figure: Comparison of the posterior means (PM) of the ho... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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cac981698b6ee19c3c6b5ab039a31183274fd4a6 | abstract | 0 | 14 | Abstract | Self-replication is a key aspect of biological life that has been largely
overlooked in Artificial Intelligence systems. Here we describe how to build
and train self-replicating neural networks. The network replicates itself by
learning to output its own weights. The network is designed using a loss
function that can b... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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565f3d9b39fd3667161d0154b2357481c6dc0a55 | subsection | 1 | 14 | Introduction | The concept of an artificial self-replicating machine was first proposed by John von Neumann in the 1940s prior to the discovery of DNA's role as the physical mechanism for biological replication. Specifically, Von Neumann demonstrated a configuration of initial states and transformation rules for a cellular automaton ... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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869b9c405fc16a403ebed09a6fab5d4d97cbd9b0 | subsection | 2 | 14 | Introduction | Analogously, we can construct a self-improving mechanism for artificial intelligence via natural selection if AI agents had the ability to replicate and improve themselves without additional machinery.\item Neural networks are capable of learning powerful representations across many different domains of data \cite{repr... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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1c7328425497fd7c5e83e4b4dc729590a4755a0d | subsection | 3 | 14 | Introduction | To circumvent this, we need an indirect way of referring to $\Theta$.\subsubsection{Indirect Reference}HyperNEAT \cite{hyperneat} is a neuro-evolution method that describes a neural network by identifying every topological connection with a coordinate and a weight. We pursue the same strategy in building a quine. Inste... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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6b83f948159c7bdc4cee638275add2b77327da49 | subsection | 4 | 14 | Introduction | We demonstrate a visualization of this in Figure \ref{fig:no-one-hot}: contiguous weights might be very different, but contiguous outputs cannot be very different.\begin{figure}[h]\begin{center}\includegraphics[width=0.25\textwidth]{no-one-hot.png}\end{center}\caption{Log-normalized illustration of the weights and weig... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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53443a40b76f1f3cf4fbc66952150e38b0360d12 | subsection | 5 | 14 | Introduction | Our primary aim in this paper is to demonstrate a proof of concept for a neural network quine, which makes MNIST a suitable auxiliary task as it is considered an easy problem for modern machine learning algorithms.\section{Training the Network}\subsection{Network Architecture}Before describing how the neural network qu... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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0398507541fd981d1865564948885a948b2df313 | subsection | 6 | 14 | Introduction | In our case, the loss function is a moving target, since $\Theta_c$ changes after each gradient update. Updating the loss function after every mini-batch update is expensive. To avoid that, we split the set of possible coordinates into random mini-batches of size 10, and update the loss function after every training ep... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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7040c37f52150f7dc579152277765942a7850fa3 | subsection | 7 | 14 | Introduction | We note that regeneration is sensitive to choices of weight initialization and activation function.\begin{program}\mbox{Pseudo-code for Regeneration:}\BEGIN \\ |Initialize set of parameters|\ \Theta_C|Initialize number of generation epochs|\ G|Initialize number of optimization epochs|\ T\FOR g:=0 \TO G \DO//\ Optimizat... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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b15df6a0bf0d032c24b79172f259ce15621555c6 | subsection | 8 | 14 | Introduction | For example, \cite{lipson} estimate the self-replicating quotient of Penrose Tiling \cite{penrose} to be below $0.69$ and that of animals to be at least $46.05$. This framework is useful for distinguishing between trivial and non-trivial replicators. To compute this metric for our network, we need to compute the chance... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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50be5a6e832cb86731a0a2fe7bf7a7a5248e8797 | subsection | 9 | 14 | Introduction | This corresponds to a self-replicating quotient of $10.06$ and an average weight prediction margin of $0.0065$, which is an order of magnitude better than the best solution found previously.\begin{figure}[h]\begin{center}\includegraphics[width=0.5\textwidth]{vquine_gen.png}\end{center}\caption{Training a vanilla quine ... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
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7fca1a7383e7d40d6119e8cf72b841af7a3f4d3f | subsection | 10 | 14 | Introduction | (It is not immediately obvious from Figure \ref{fig:mquine_loss}, but the first few training epochs reduce the self-replicating loss too.)There are parallels to be drawn between self-replication in the case of a neural network quine and biological reproduction in nature, as well as specialization at the auxiliary task ... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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4ddbbccf269e5a76fee0f971007a9363de61dafd | subsection | 11 | 14 | Introduction | 8:1798--1828.\bibitem{breivik2001self}Breivik, J. (2001).\newblock Self-organization of template-replicating polymers and thespontaneous rise of genetic information.\newblock {\em Entropy}, 3(4):273--279.\bibitem{wiki:neumann}contributors, W. (2017).\newblock Self-replication --- wikipedia{,} the free encyclopedia.\new... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
] | [
"cs.AI",
"cs.NE"
] | 2,018 | en | Computer Science | [
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5cf5b8ece0f07b151d70c8c05990c3bf4b651e9b | subsection | 12 | 14 | Introduction | (1980).\newblock {\em G{\"o}del, Escher, Bach: an Eternal Golden Braid}.\newblock New York: Vintage Books.\bibitem{hornik1991approximation}Hornik, K. (1991).\newblock Approximation capabilities of multilayer feedforward networks.\newblock {\em Neural networks}, 4(2):251--257.\bibitem{huang2006extreme}Huang, G.-B., Zhu,... | {
"cite_spans": []
} | 1803.05859 | Neural Network Quine | [
"Oscar Chang",
"Hod Lipson"
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"cs.AI",
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d2f2e8a16a68278dae8881438e673f851b0906da | subsection | 13 | 14 | Introduction | Springer.\bibitem{shen2017natural}Shen, J., Pang, R., Weiss, R.~J., Schuster, M., Jaitly, N., Yang, Z., Chen, Z.,Zhang, Y., Wang, Y., Skerry-Ryan, R., et~al. (2017).\newblock Natural tts synthesis by conditioning wavenet on mel spectrogrampredictions.\newblock {\em arXiv preprint arXiv:1712.05884}.\bibitem{hyperneat}St... | {
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6e5b3a24bed4d9648510f0d678968f6ae19a5303 | abstract | 0 | 34 | Abstract | In this paper we use the framework of algebraic effects from programming
language theory to analyze the Beta-Bernoulli process, a standard building
block in Bayesian models. Our analysis reveals the importance of abstract data
types, and two types of program equations, called commutativity and
discardability. We develo... | {
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f10bf79d6c6f0f9852319a04acd84b479137d9f7 | subsection | 1 | 34 | Introduction | From the perspective of programming, a family of Boolean random processes is implemented by a module that supports the following interface:
module type ProcessFactory = sig type process
val new : H -> process
val get : process -> bool end
where H is some type of hyperparameters.
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b2b30324919ff1c4677452d9b15a11ade2419458 | subsection | 2 | 34 | Introduction | Discardability is the requirement that when |x| is not free in |u|,
\Big (\text{|let x = t in u|}\Big ) =
\Big (\text{|u|}\Big ).Together, these properties say that data flow, rather than the control flow, is what matters. For example, in a standard programming language, the purely functional total expressions are comm... | {
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a6b228a4f368973d18e7aa06ae46ee28d87c0b67 | subsection | 3 | 34 | Introduction | REF ).We argue that these results open up a new method for analyzing Bayesian models, based on algebraic effects (see § and This paper formalizes and proves a conjecture from , which is an unpublished abstract.). | {
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8900f27158c2c1f7af607465ea6043e325dcd619 | subsection | 4 | 34 | An algebraic presentation of the Beta-Bernoulli process | In this section, we present syntactic rules for
well-formed client programs of the Beta-Bernoulli module, and
axioms for deriving equations on those programs. | {
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51dc8d781fa3805c75dcd6541c3326b142d6bc50 | subsection | 5 | 34 | An algebraic presentation of finite probability | Recall the module |Bernoulli| from the introduction
which provides a method of sampling with odds (i:j).
We will axiomatize its equational properties. Algebraic effects provide a way to
axiomatize the specific features of this module while putting aside the general properties of programming languages, such as \beta /\e... | {
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59a1249cafb046bcbbd94094e5f7454ce059c5bc | subsection | 6 | 34 | An algebraic presentation of finite probability | The theory of rational convexity is the first-order algebraic theory with binary operations \mathop {{}_{i}\!{?}\!_{j}} for all i,j\in \mathbb {N} such that i+j>0, subject to the axiom schemesw,x,y,z\vdash &(w \mathop {{}_{i}\!{?}\!_{j}} x) \mathop {{}_{i+j}\!{?}\!_{k+l}}(y \mathop {{}_{k}\!{?}\!_{l}} z)
=
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2b2ff3de467df12e93c5fce5a89335a326d170ab | subsection | 7 | 34 | A parameterized algebraic signature for Beta-Bernoulli | In the theory of convex sets, the parameters i,j for |get|
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05fc0b84a23ba6cf15a7c71318eea81b01449546 | subsection | 8 | 34 | Axioms for Beta-Bernoulli | The axioms for the Beta-Bernoulli theory comprise the axioms for rational convexity
(Def. REF ) together with the following axiom schemes.Commutativity.
All the operations commute with each other:
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ccf6a750845be90bbe469c1af2f00a3ec05b8ee3 | subsection | 9 | 34 | Axioms for Beta-Bernoulli | It immediately follows from conjugacy and discardability that x \mathop {{}_{i}\!{?}\!_{j}} y is definable as \nu _{i,j}p.(x\mathop {?_{p}} y) for i,j>0.As an example, consider t(r) = (r \mathop {?_{p}} x) \mathop {?_{p}} (y \mathop {?_{p}} r) that represents tossing a coin with bias p twice, continuing with x or y if ... | {
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a2454a94ffa01d6b45a504fe0b8500ac796f96c4 | subsection | 10 | 34 | A complete interpretation in measure theory | In this section we give an interpretation of terms using measures and integration operators, the standard formalism for probability theory (e.g. , ), and we show that this interpretation is complete (Thm. REF ).
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2d658d15231c63414fc2eb9b2cac064d29706eee | subsection | 11 | 34 | Programs as probability kernels. | Forgetting about abstract types for a moment, terms in the |BetaBern| module are first-order probabilistic programs. So we have a standard denotational semantics due to where terms are interpreted as probability kernels and \nu as integration. Let I= [0,1] denote the unit interval. We write \beta _{i,j} for the \textr... | {
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"raw": "Dexter Kozen. Semantics of probabilistic programs. J. Comput. System Sci., 22(3):328–350, 1981.",
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1f25043c452691db308270127138ba858d45d287 | subsection | 12 | 34 | Programs as probability kernels. | We interpret terms \Gamma \mathop |\Delta \vdash t as probability kernels \llbracket t\rrbracket : I^\ell \times \Sigma ( \llbracket \Delta \rrbracket )\rightarrow [0,1] inductively,
for \vec{p}\in I^\ell and U\in \Sigma ( \llbracket \Delta \rrbracket ) :& \llbracket x_i(p_{j_1},\ldots ,p_{j_m})\rrbracket (\vec{p}, U) ... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
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"Nathanael L. Ackerman",
"Cameron E. Freer",
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48b98de7639d72220b6f2425bde449776f75b9d6 | subsection | 13 | 34 | Interpretation as functionals | We write \mathbb {R}^{I^m} for the vector space of continuous
functions I^m \rightarrow \mathbb {R}, endowed with the supremum norm.
Given a probability kernel \kappa :I^\ell \times \Sigma \big (\sum _{j=1}^kI^{m_j}\big )\rightarrow [0,1]
and \vec{p}\in I^\ell , we define a linear map
\phi _{\vec{p}}: \mathbb {R}^{I^{m... | {
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f3c9d358fd7bb33db9939d0ff04c074f79f1360a | subsection | 14 | 34 | Interpretation as functionals | It is informative to spell out the interpretation of terms {p_1,\ldots ,p_\ell \mathop |x_1 : m_1, \ldots , x_k : m_k\vdash t} as maps
\llbracket t\rrbracket : \mathbb {R}^{I^{m_1}} \times \ldots \times \mathbb {R}^{I^{m_k}} \rightarrow \mathbb {R}^{I^\ell }
since it fits the algebraic notation: we may think of the va... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
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805b806c0ce193fffbca9dc569d5ca4aef2a954d | subsection | 15 | 34 | Technical background on Bernstein polynomials | [Bernstein polynomials]
For i=0,\ldots ,k, we define the i-th basis Bernstein polynomial b_{i,k} of degree k as
b_{i,k}(p) = \binom{k}{i} p^{k-i}(1-p)^i.
For a multi-index I = (i_1,\ldots ,i_\ell ) with 0 \le i_j \le k, we let b_{I,k}(\vec{p}) = b_{i_1,k}(p_1)\cdots b_{i_\ell ,k}(p_\ell ). A Bernstein polynomial is a l... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
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b5f5b3590e7047edb27189b5017afadce03c1172 | subsection | 16 | 34 | Normal forms and completeness | For the completeness proof of the measure-theoretic model, we proceed as follows: To decide \Gamma \mathop |\Delta \vdash t = u for two terms t,u, we transform them into a common normal form whose interpretations can be given explicitly. We then use a series of linear independence results to show that if the interpreta... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
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"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
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5f90014500a6afaa9c9fb30d4ca0bcb26324d2d0 | subsection | 17 | 34 | Stone's normal forms for rational convex sets | Normal forms for the theory of rational convex sets have been described by Stone .
We note that if -\mathop |x_1\ldots x_k : 0\vdash t is a term in the theory of rational convex sets (Def. REF ) then
\llbracket t\rrbracket :\mathbb {R}^k\rightarrow \mathbb {R} is a unital positive linear map that takes rationals to rat... | {
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52f874ec93727cc9a62f8ef9ddf0fc94498951f9 | subsection | 18 | 34 | Characterization and completeness for | This section concerns the normalization of terms using free parameters but no \nu . Consider a single parameter p. If we think of a term t as a syntactic tree, commutativity and discardability can be used to move all occurrences of \mathop {?_{p}} to the root of the tree, making it a tree diagram of some depth k. Let u... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
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5d2376237163184ca59973a849756a7f62102f73 | subsection | 19 | 34 | Characterization and completeness for | \then \then
For example, normalizing (v\mathop {?_{p}} x){\mathop {?_{p}}}(y\mathop {?_{p}} v) gives (v\mathop {?_{p}}(x \mathop {{}_{1}\!{?}\!_{1}} y)){\mathop {?_{p}}} ((x\mathop {{}_{1}\!{?}\!_{1}} y)\mathop {?_{p}} v)=C_2(v,x{\mathop {{}_{1}\!{?}\!_{1}}}y,v).From this we obtain the following completeness result:P... | {
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38b922d477aee0f58cd7665c86c2029267749e10 | subsection | 20 | 34 | Normalization of Beta-Bernoulli | For arbitrary terms {p_1 \ldots p_\ell \mathop |x_1\colon m_1, \ldots , x_s\colon m_s\vdash t}, we employ the following normalization procedure. Using conjugacy and the commutativity axioms (–), we can push all uses of \nu towards the leaves of the tree, until we end up with a tree of ratios and free parameter choices ... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
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"Hongseok Yang",
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c996ee7520186567a9ac6ddbe98220610e5261d7 | subsection | 21 | 34 | Proof of completeness | Consider a chain c = \nu _{i_1,j_1} p_{\ell +1}.\ldots \nu _{i_d,j_d} p_{\ell + d}.\,x(p_{\tau (1)},\ldots ,p_{\tau (m)}). Its measure-theoretic interpretation \llbracket c\rrbracket (p_1,\ldots ,p_\ell ) is a pushforward of a product of d beta distributions, supported on a hyperplane segment that is parameterized by t... | {
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
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24ead67b749c685a4b07df1da02e780635d67c3d | subsection | 22 | 34 | Proof of completeness | Let \sum a_i h_{i*}(\mu _i) = 0 as a signed measure. We show by induction over the dimension of the chains that all a_i vanish. Assume that a_i = 0 whenever the dimension of c_i is less than d, and consider an arbitrary subspace \tau _j of dimension d. We can define a signed Borel measure on I^d by restriction
nullfon... | {
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0.0989990234375,
0.1436767578125,
0.10089111328125,
0.025421142578125,
0.031341552734375,
0.150390625,
0.033447265625,
0.03131103515625,
0.053436279296875,
0.1240234375,
0.0556640625,
0.254638671875,
0.251708984375,
0.01... |
b63b8cdb29310046fa33c498b3f8133a9037d754 | subsection | 23 | 34 | Proof of completeness | The interpretations of these normal forms are given explicitly by
nullfont\displaystyle
\llbracket t\rrbracket (\vec{f})(\vec{p}) = \sum _{j} \frac{w_{Ij}}{w_I} \cdot b_{I,k}(\vec{p}) \cdot \llbracket c_j\rrbracket (\vec{f})(\vec{p}) \text{ where } w_I = \sum _j w_{Ij}
\then \then
and analogously for t^{\prime }. T... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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f90d1d103c1b4f47e26b9719bff9eab8732a55d5 | subsection | 24 | 34 | Extensionality and syntactical completeness | In this section we use the model completeness of the previous section to establish some syntactical results about
the theory of Beta-Bernoulli. Although the model is helpful in informing the proofs, the statements of the results in
this section are purely syntactical.The ultimate result of this section is equational sy... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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c944cdaf8b64a8d27b9b7f8e261a57b4133d77fe | subsection | 25 | 34 | Extensionality | Proposition (Extensionality for closed terms)
Suppose
\Gamma ,q\mathop |\Delta \vdash t and \Gamma ,q\mathop |\Delta \vdash u.
If \Gamma \mathop |\Delta \vdash \nu _{i,j}q.t=\nu _{i,j}q.u for all i,j,
then also \Gamma \mathop |\Delta \vdash t=u.We show the contrapositive. By the model completeness theorem (Thm. REF )... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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180b9f59b1856833caddb411cddf5c1a97dc896d | subsection | 26 | 34 | Extensionality | So, \llbracket \nu _{i_n,j_n}q.t\rrbracket \ne \llbracket \nu _{i_n,j_n}q.u\rrbracket .Proposition (Extensionality for ground terms)
In brief: If {t[^{v_1 \dots v_k}\!/\!_{x_1\dots x_k}]
=
u[^{v_1 \dots v_k}\!/\!_{x_1\dots x_k}]}
for all suitable ground v_1\dots v_k, then t =u.In detail: Consider t and u with
-\matho... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.0125... |
68484d6a98fcf9ab450c4d1de6d941b7b4ae3def | subsection | 27 | 34 | Extensionality | Then
nullfont\displaystyle
\llbracket t[^{v_1\dots v_k}\!/\!_{x_1\dots x_k}]\rrbracket (1,0) =
\llbracket t\rrbracket (0, {\dots },b_{I,k},{\dots }, 0)
\ne \llbracket u\rrbracket (0,{\dots }, b_{I,k},{\dots }, 0)
=
\llbracket u[^{v_1\dots v_k}\!/\!_{x_1\dots x_k}]\rrbracket (1,0).
\then \then
The required
{\lnot \b... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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1cfffad08246efbc5392883ecfa6acff0af8b6bf | subsection | 28 | 34 | Relative syntactical completeness | Proposition (Neumann, )
If t,u are terms in the theory of rational convexity (Def. REF ), then either t=u is derivable or
it implies x\mathop {{}_{i}\!{?}\!_{j}} y=x\mathop {{}_{i^{\prime }}\!{?}\!_{j^{\prime }}} y for all nonzero i,i^{\prime },j,j^{\prime }.The theory of Beta-Bernoulli is syntactically complete rela... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01220869",
"end": 85,
"openalex_id": "https://openalex.org/W2315523708",
"raw": "Walter D. Neumann. On the quasivariety of convex subsets of affine space. Arch. Math., 21:11–16, 1970.",
"source_ref_id": "a988d7521c0da83d7425... | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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b855084cae7a3fe6e061c349a7081d0b70f8b08d | subsection | 29 | 34 | Remark about stateful implementations | In the introduction we recalled the idea of using Pólya's urn to implement a Beta-Bernoulli process
using local (hidden) state.Our equational presentation gives a recipe for understanding the correctness of the stateful
implementation. First, one would give an operational semantics,
and then a basic notion of observati... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-662-46678-0_18",
"end": 531,
"openalex_id": "https://openalex.org/W3106280144",
"raw": "Ales Bizjak and Lars Birkedal. Step-indexed logical relations for probability. In Proc. FOSSACS 2015, pages 279–294, 2015.",
"source... | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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a6a5fcd10ab865df620f4451fccfac219642ccf2 | subsection | 30 | 34 | Conclusion | Exchangeable random processes are central to many Bayesian models.
The general message of this paper is that the analysis of exchangeable random processes, based on basic concepts from programming language theory, depends on three crucial ingredients: commutativity, discardability, and abstract types.
We have illustrat... | {
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{
"arxiv_id": "",
"doi": "10.1198/016214506000000302",
"end": 1163,
"openalex_id": "https://openalex.org/W2158266063",
"raw": "Yee Whye Teh, Michael I. Jordan, Matthew J. Beal, and David M. Blei. Hierarchical Dirichlet processes. J. Amer. Statist. Assoc., 101(476):1... | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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65a21fcffa03376c3940e4b3306f48e8d92bb5fd | subsection | 31 | 34 | Example derivations | In this appendix, we derive equations mentioned in the main text of the paper. The first equation, as found in §REF , is
nullfont\displaystyle x \mathop {{}_{1}\!{?}\!_{1}} y = ((x \mathop {{}_{1}\!{?}\!_{1}} y) \mathop {?_{p}} x) \mathop {?_{p}} (y \mathop {?_{p}} (x \mathop {{}_{1}\!{?}\!_{1}} y)) \then \then
whic... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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