Correction: Nogales, A.G. Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution. Mathematics 2022, 10, 1213

Text Correction

There was some calculation errors in Examples 2 and 3 of the original publication [1]. In Section 7, Example 2, it is written:

$$\begin{array}{c}\begin{array}{cc}\hfill f_{n,k^{\prime }}^{*}(k_1,k_2)& ={\int }_0^1{f}_{\theta }(k_1,k_2)r_{n,k^{\prime }}^{*}\left(\theta \right)dQ\left(\theta \right)\hfill \\ & =K\left(k^{\prime }\right){\int }_0^1{\theta }^{a_n\left(k^{\prime }\right)+k_1}{(1-\theta )}^{b_n\left(k^{\prime }\right)+k_2}d\theta \hfill \\ & =K\left(k^{\prime }\right)B(a_n\left(k^{\prime }\right)+k_1+1,b_n\left(k^{\prime }\right)+k_2+1),\hfill \end{array}\end{array}$$

and it must be written as follows:

$$\begin{array}{c}\begin{array}{cc}\hfill f_{n,k^{\prime }}^{*}(k_1,k_2)& ={\int }_0^1{f}_{\theta }(k_1,k_2)r_{n,k^{\prime }}^{*}\left(\theta \right)dQ\left(\theta \right)\hfill \\ & =\left\{\begin{array}{cc}K\left(k^{\prime }\right)B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)\hfill & \mathrm{if}{k}_2=0\hfill \\ K\left(k^{\prime }\right)B(a_n\left(k^{\prime }\right)+1,b_n\left(k^{\prime }\right)+3)\hfill & \mathrm{if}{k}_1=0,k_2=1\hfill \\ K\left(k^{\prime }\right)B(a_n\left(k^{\prime }\right)+3,b_n\left(k^{\prime }\right)+1)\hfill & \mathrm{if}{k}_1=k_2=1\hfill \end{array}\right..\hfill \end{array}\end{array}$$

This leads to some errors in subsequent expressions of Example 2. Thus, in Section 7 it is written:

$$\begin{array}{c}{f}_{\theta }^{X_2|X_1=k_1}\left(k_2\right)={[k_1+(1-2k_1)(1-\theta )]}^{k_2}{[1-k_1-(1-2k_1)(1-\theta )]}^{1-k_2}\end{array}$$

and it must be written:

$$\begin{array}{c}{f}_{\theta }^{X_2|X_1=k_1}\left(k_2\right)={[k_1+(1-2k_1)(1-\theta )]}^{k_2}{[1-k_1-(1-2k_1)(1-\theta )]}^{1-k_2}\\ =\left\{\begin{array}{cc}1-\theta \hfill & \mathrm{if}{k}_1=k_2\in \{0,1\}\hfill \\ \theta \hfill & \mathrm{if}{k}_1\ne k_2\hfill \end{array}\right.\end{array}$$

and, just below, it is written

$$\begin{array}{c}{f}_{n,k^{\prime },1}^{*}\left(k_1\right)=K\left(k^{\prime }\right)[B(a_n\left(k^{\prime }\right)+k_1+1,b_n\left(k^{\prime }\right)+1)+B(a_n\left(k^{\prime }\right)+k_1+1,b_n\left(k^{\prime }\right)+2)],\end{array}$$

and it must be written as follows:

$$\begin{array}{c}{f}_{n,k^{\prime },1}^{*}\left(k_1\right)=\left\{\begin{array}{cc}K\left(k^{\prime }\right)[B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)+B(a_n\left(k^{\prime }\right)+1,b_n\left(k^{\prime }\right)+3)]\hfill & \mathrm{if}{k}_1=0\hfill \\ K\left(k^{\prime }\right)[B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)+B(a_n\left(k^{\prime }\right)+3,b_n\left(k^{\prime }\right)+1)]\hfill & \mathrm{if}{k}_1=1\hfill \end{array}\right..\end{array}$$

Also, on in Section 7 it is written:

$$\begin{array}{c}{f_{n,x^{\prime }}^{*}}^{X_2|X_1=k_1}\left(k_2\right):={\displaystyle \frac{f_{n,k^{\prime }}^{*}(k_1,k_2)}{f_{n,k^{\prime },1}^{*}\left(k_1\right)}}=\left\{\begin{array}{cc}{\displaystyle \frac{2n+2}{2n+n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+3}}\hfill & \mathrm{if}{k}_1=k_2=0,\hfill \\ {\displaystyle \frac{n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+1}{2n+n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+3}}\hfill & \mathrm{if}{k}_1=0,k_2=1,\hfill \\ {\displaystyle \frac{2n+3}{2n+n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+4}}\hfill & \mathrm{if}{k}_1=1,k_2=0,\hfill \\ {\displaystyle \frac{n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+1}{2n+n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+4}}\hfill & \mathrm{if}{k}_1=k_2=1,\hfill \end{array}\right.\end{array}$$

and it must be written as follows:

$$\begin{array}{c}\begin{array}{cc}\hfill {f_{n,x^{\prime }}^{*}}^{X_2|X_1=k_1}\left(k_2\right)& :={\displaystyle \frac{f_{n,k^{\prime }}^{*}(k_1,k_2)}{f_{n,k^{\prime },1}^{*}\left(k_1\right)}}\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle \frac{B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)}{B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)+B(a_n\left(k^{\prime }\right)+1,b_n\left(k^{\prime }\right)+3)}}={\displaystyle \frac{a_n\left(k^{\prime }\right)+1}{2n+3}}\hfill & \mathrm{if}{k}_1=k_2=0,\hfill \\ {\displaystyle \frac{B(a_n\left(k^{\prime }\right)+1,b_n\left(k^{\prime }\right)+3)}{B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)+B(a_n\left(k^{\prime }\right)+1,b_n\left(k^{\prime }\right)+3)}}={\displaystyle \frac{b_n\left(k^{\prime }\right)+2}{2n+3}}\hfill & \mathrm{if}{k}_1=0,k_2=1,\hfill \\ {\displaystyle \frac{B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)}{B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)+B(a_n\left(k^{\prime }\right)+3,b_n\left(k^{\prime }\right)+1)}}={\displaystyle \frac{b_n\left(k^{\prime }\right)+1}{2n+3}}\hfill & \mathrm{if}{k}_1=1,k_2=0,\hfill \\ {\displaystyle \frac{B(a_n\left(k^{\prime }\right)+3,b_n\left(k^{\prime }\right)+1)}{B(a_n\left(k^{\prime }\right)+2,b_n\left(k^{\prime }\right)+2)+B(a_n\left(k^{\prime }\right)+3,b_n\left(k^{\prime }\right)+1)}}={\displaystyle \frac{a_n\left(k^{\prime }\right)+2}{2n+3}}\hfill & \mathrm{if}{k}_1,k_2=1.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

Finally, on Example 2, in Section 7 it is written:

$$m_n^{*}(k^{\prime },k_1)={f_{n,k^{\prime }}^{*}}^{X_2|X_1=k_1}\left(1\right)=\left\{\begin{array}{cc}{\displaystyle \frac{n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+1}{2n+n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+3}}\hfill & \mathrm{if}{k}_1=0,\hfill \\ {\displaystyle \frac{n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+1}{2n+n_{+0}\left(k^{\prime }\right)+2n_{01}\left(k^{\prime }\right)+4}}\hfill & \mathrm{if}{k}_1=1,\hfill \end{array}\right.$$

and it must be written as follows:

$$m_n^{*}(k^{\prime },k_1)={f_{n,k^{\prime }}^{*}}^{X_2|X_1=k_1}\left(1\right)=\left\{\begin{array}{cc}{\displaystyle \frac{b_n\left(k^{\prime }\right)+2}{2n+3}}\hfill & \mathrm{if}{k}_1=0,\hfill \\ {\displaystyle \frac{a_n\left(k^{\prime }\right)+2}{2n+3}}\hfill & \mathrm{if}{k}_1=1.\hfill \end{array}\right.$$

In Example 3, the calculation errors are also limited to the determination of the posterior predictive density, which obviously has consequences for the expression of the Bayes estimator of the regression curve. To correct these errors, I believe it is advisable to completely rewrite Example 3, leaving some calculation details to the reader. In short, we propose the following changes to this example:

In Example 3 of the original article [1], the text in Section 7, Example 3 should be replaced with the following:

Example 3.

Let $(X_1,X_2)$ have bivariate normal distribution with density:

$$\begin{array}{c}\begin{array}{cc}\hfill f_{\theta }\left(x\right)& :=D{(\sigma ,\rho )}^{{\displaystyle \frac{1}{n}}}exp\left\{-{\displaystyle \frac{1}{2{\sigma }^2(1-{\rho }^2)}}[{(x_1-\theta )}^2\right.\hfill \\ & \left.-2\rho (x_1-\theta )(x_2-\theta )+{(x_2-\theta )}^2]\right\}\hfill \\ & =D{(\sigma ,\rho )}^{{\displaystyle \frac{1}{n}}}exp\left\{-{\displaystyle \frac{1}{2{\sigma }^2(1-{\rho }^2)}}[x_1^2+x_2^2-2\rho x_1{x}_2\right.\hfill \\ & \left.-2(1-\rho )(x_1+x_2)\theta +2(1-\rho ){\theta }^2]\right\},\hfill \end{array}\end{array}$$

(7)

where $\sigma >0$ and $\rho \in [-1,1]$ are assumed to be known and $D(\sigma ,\rho )={\left[2\pi {\sigma }^2\sqrt{1-{\rho }^2}\right]}^{-n}$. Thus:

$$\begin{array}{c}{R}_{\theta }=N_2\left(\left(\begin{array}{c}\theta \\ \theta \end{array}\right),{\sigma }^2\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)\right),X_1,X_2{\sim }_{\theta }N(\theta ,{\sigma }^2)\\ P_{\theta }^{X_2|X_1=x_1}=N((1-\rho )\theta +\rho x_1,{\sigma }^2\sqrt{1-{\rho }^2}),\\ E_{\theta }\left(X_2\right|X_1=x_1)=(1-\rho )\theta +\rho x_1.\end{array}$$

Hence, for a sample

$$x^{\prime }=(x_1^{\prime },\dots ,x_n^{\prime })=(x_{11}^{\prime },x_{12}^{\prime },\dots ,x_{n1}^{\prime },x_{n2}^{\prime })\in {\left({\mathbb{R}}^2\right)}^n,$$

we have that:

$$\begin{array}{c}\begin{array}{cc}\hfill f_{n,\theta }\left(x^{\prime }\right)& =\prod _{i=1}^n{f}_{\theta }\left(x_i^{\prime }\right)\hfill \\ & =D(\sigma ,\rho )exp\left\{-\frac{1}{2{\sigma }^2(1-{\rho }^2)}{\sum }_{i=1}^n\left[{(x_{i1}^{\prime }-\theta )}^2\right.\right.\hfill \\ & \left.\left.-2\rho (x_{i1}^{\prime }-\theta )(x_{i2}^{\prime }-\theta )+{(x_{i2}^{\prime }-\theta )}^2\right]\right\}\hfill \\ & =D(\sigma ,\rho )exp\left\{-{\displaystyle \frac{1}{2{\sigma }^2(1-{\rho }^2)}}\left[2n(1-\rho ){\theta }^2\right.\right.\hfill \\ & \left.\left.-2(1-\rho )s_1\left(x^{\prime }\right)\theta +s_2\left(x^{\prime }\right)-2\rho p\left(x^{\prime }\right)\right]\right\},\hfill \end{array}\end{array}$$

where $s_1\left(x^{\prime }\right):={\sum }_i(x_{i1}^{\prime }+x_{i2}^{\prime })$, $s_2\left(x^{\prime }\right):={\sum }_i({x_{i1}^{\prime }}^2+{x_{i2}^{\prime }}^2)$, $p\left(x^{\prime }\right)={\sum }_i{x}_{i1}^{\prime }{x}_{i2}^{\prime }$.

Let us consider the prior distribution $Q=N(\mu ,{\tau }^2)$ whose density is:

$$g\left(\theta \right)={\displaystyle \frac{1}{\tau \sqrt{2\pi }}}exp\left\{-{\displaystyle \frac{1}{2{\tau }^2}}{(\theta -\mu )}^2\right\}.$$

The posterior density (with respect to the Lebesgue measure) is:

$$\begin{array}{c}\begin{array}{cc}\hfill r_{n,x^{\prime }}^{*}\left(\theta \right)& :={\displaystyle \frac{d{R}_{n,x^{\prime }}^{*}}{d\theta }}\left(\theta \right)=K\left(x^{\prime }\right)f_{n,\theta }\left(x^{\prime }\right)g\left(\theta \right)\hfill \\ & =K^{\prime }\left(x^{\prime }\right)exp\left\{-{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{2n{\tau }^2+{\sigma }^2(1+\rho )}{{\sigma }^2{\tau }^2(1+\rho )}}{\theta }^2\right.\right.\hfill \\ & \left.\left.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2{\displaystyle \frac{s_1\left(x^{\prime }\right){\tau }^2+\mu {\sigma }^2(1+\rho )}{{\sigma }^2{\tau }^2(1+\rho )}}\theta +C\left(x^{\prime }\right)\right]\right\}\hfill \\ & ={\displaystyle \frac{1}{B_1\sqrt{2\pi }}}exp\left\{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\theta -A_1\left(x^{\prime }\right)}{B_1}}\right)}^2\right\},\hfill \end{array}\end{array}$$

where $K^{\prime }\left(x^{\prime }\right)$ and $C\left(x^{\prime }\right)$ are appropriate constants and

$$B_1^2={\displaystyle \frac{{\sigma }^2{\tau }^2(1+\rho )}{2n{\tau }^2+{\sigma }^2(1+\rho )}}\mathrm{and}{A}_1\left(x^{\prime }\right)={\displaystyle \frac{{\tau }^2{s}_1\left(x^{\prime }\right)+{\sigma }^2(1+\rho )\mu }{2n{\tau }^2+{\sigma }^2(1+\rho )}}.$$

So, the posterior distribution given $x^{\prime }$ is $R_{n,x^{\prime }}^{*}=N(A_1\left(x^{\prime }\right),B_1^2)$. The posterior predictive density given $x^{\prime }\in {\left({\mathbb{R}}^2\right)}^n$ is:

$$\begin{array}{c}\begin{array}{cc}\hfill f_{n,x^{\prime }}^{*}\left(x\right)& :={\int }_{\mathbb{R}}{f}_{\theta }\left(x\right)r_{n,x^{\prime }}^{*}\left(\theta \right)d\theta \hfill \\ & ={\displaystyle \frac{1}{B_1{\sqrt{2\pi }}^3{\sigma }^2\sqrt{1-{\rho }^2}}}{\int }_{\mathbb{R}}exp\left\{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\theta -A_2(x,x^{\prime })}{B_2}}\right)}^2\right\}d\theta \hfill \\ & ·exp\left\{{\displaystyle \frac{A_2{(x,x^{\prime })}^2}{2B_2^2}}-{\displaystyle \frac{A_1\left(x^{\prime }\right)}{2B_1^2}}-{\displaystyle \frac{x_1^2+x_2^2-2\rho x_1{x}_2}{2{\sigma }^2(1-{\rho }^2)}}\right\},\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}{B}_2^2:={\displaystyle \frac{B_1^2{\sigma }^2(1+\rho )}{2B_1^2+{\sigma }^2(1+\rho )}},A_2(x,x^{\prime }):={\displaystyle \frac{{\sigma }^2(1+\rho )A_1\left(x^{\prime }\right)+B_1^2(x_1+x_2)}{2B_1^2+{\sigma }^2(1+\rho )}}.\end{array}$$

Some algebraic manipulations allow us to prove that

$$\begin{array}{c}{f}_{n,x^{\prime }}^{*}\left(x\right)=D{({\sigma }_1,{\rho }_1)}^{{\displaystyle \frac{1}{n}}}exp\{-{\displaystyle \frac{1}{2{\sigma }_1^2(1-{\rho }_1^2)}}\\ ·[x_1^2+x_2^2-2{\rho }_1{x}_1{x}_2-2(1-{\rho }_1)(x_1+x_2)m\left(x^{\prime }\right)+2(1-{\rho }_1)m{\left(x^{\prime }\right)}^2]\},\end{array}$$

(8)

where

$$\begin{array}{c}{\rho }_1={\displaystyle \frac{(2\rho n+\rho +1){\tau }^2+\rho (1+\rho ){\sigma }^2}{(2n+\rho +1){\tau }^2+(1+\rho ){\sigma }^2}}\in [-1,1],\\ {\sigma }_1^2={\sigma }^2{\displaystyle \frac{(2n+\rho +1){\tau }^2+(1+\rho ){\sigma }^2}{2n{\tau }^2+(1+\rho ){\sigma }^2}}>0\mathrm{and}\\ m\left(x^{\prime }\right)={\displaystyle \frac{(2n+1){\tau }^2+(1+\rho ){\sigma }^2}{(2n+\rho +1){\tau }^2+(1+\rho ){\sigma }^2}}{\displaystyle \frac{{\tau }^2{s}_1\left(x^{\prime }\right)+(1+\rho ){\sigma }^2\mu }{2n{\tau }^2+(1+\rho ){\sigma }^2}}.\end{array}$$

Comparing the expressions (7) and (8), it is shown that the posterior predictive distribution ${R_{n,x^{\prime }}^{*}}^R$ on ${\mathbb{R}}^2$ given $x^{\prime }\in {\left({\mathbb{R}}^2\right)}^n$ is the bivariate normal distribution

$$\begin{array}{c}{N}_2\left(\left(\begin{array}{c}m\left(x^{\prime }\right)\\ m\left(x^{\prime }\right)\end{array}\right),{\sigma }_1^2\left(\begin{array}{cc}1& {\rho }_1\\ {\rho }_1& 1\end{array}\right)\right).\end{array}$$

It follows that the conditional distribution:

$${\left({R_{n,x^{\prime }}^{*}}^R\right)}^{{\pi }_2|{\pi }_1=x_1}:=N\left((1-{\rho }_1)m\left(x^{\prime }\right)+{\rho }_1{x}_1,{\sigma }_1^2(1-{\rho }_1^2)\right)$$

is the Bayes estimator of the conditional distribution:

$$P_{\theta }^{X_2|X_1=x_1}=N\left((1-\rho )\theta +\rho x_1,{\sigma }^2(1-{\rho }^2)\right)$$

for the squared total variation function, and its density ${f_{n,x^{\prime }}^{*}}^{{\pi }_2|{\pi }_1=x_1}$ is the Bayes estimator of the conditional density:

$$f_{\theta }^{X_2|X_1=x_1}\left(x_2\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi (1-{\rho }^2)}}}exp\left\{-{\displaystyle \frac{1}{2{\sigma }^2(1-{\rho }^2)}}{[x_2-(1-\rho )\theta -\rho x_1]}^2\right\}$$

for the $L^1$-squared loss function.

Moreover, its mean:

$$E_{{R_{n,x^{\prime }}^{*}}^R}\left({\pi }_2\right|{\pi }_1=x_1)=(1-{\rho }_1)m\left(x^{\prime }\right)+{\rho }_1{x}_1$$

is the Bayes estimator of the regression curve:

$$E_{\theta }\left(X_2\right|X_1=x_1)=(1-\rho )\theta +\rho x_1$$

for the squared error loss function.

The original equations numbered (7)–(11) have been renumbered accordingly as (9)–(13) due to the inclusion of Equations (7) and (8).

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.