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

Nogales, Agustín G.

doi:10.3390/math14030468

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{matrix} \begin{matrix} f_{n, k^{'}}^{*} (k_{1}, k_{2}) & = \int_{0}^{1} f_{θ} (k_{1}, k_{2}) r_{n, k^{'}}^{*} (θ) d Q (θ) \\ = K (k^{'}) \int_{0}^{1} θ^{a_{n} (k^{'}) + k_{1}} {(1 - θ)}^{b_{n} (k^{'}) + k_{2}} d θ \\ = K (k^{'}) B (a_{n} (k^{'}) + k_{1} + 1, b_{n} (k^{'}) + k_{2} + 1), \end{matrix} \end{matrix}

and it must be written as follows:

\begin{matrix} \begin{matrix} f_{n, k^{'}}^{*} (k_{1}, k_{2}) & = \int_{0}^{1} f_{θ} (k_{1}, k_{2}) r_{n, k^{'}}^{*} (θ) d Q (θ) \\ = \{\begin{matrix} K (k^{'}) B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) & if k_{2} = 0 \\ K (k^{'}) B (a_{n} (k^{'}) + 1, b_{n} (k^{'}) + 3) & if k_{1} = 0, k_{2} = 1 \\ K (k^{'}) B (a_{n} (k^{'}) + 3, b_{n} (k^{'}) + 1) & if k_{1} = k_{2} = 1 \end{matrix} . \end{matrix} \end{matrix}

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

\begin{matrix} f_{θ}^{X_{2} | X_{1} = k_{1}} (k_{2}) = {[k_{1} + (1 - 2 k_{1}) (1 - θ)]}^{k_{2}} {[1 - k_{1} - (1 - 2 k_{1}) (1 - θ)]}^{1 - k_{2}} \end{matrix}

and it must be written:

\begin{matrix} f_{θ}^{X_{2} | X_{1} = k_{1}} (k_{2}) = {[k_{1} + (1 - 2 k_{1}) (1 - θ)]}^{k_{2}} {[1 - k_{1} - (1 - 2 k_{1}) (1 - θ)]}^{1 - k_{2}} \\ = \{\begin{matrix} 1 - θ & if k_{1} = k_{2} \in {0, 1} \\ θ & if k_{1} \neq k_{2} \end{matrix} \end{matrix}

and, just below, it is written

\begin{matrix} f_{n, k^{'}, 1}^{*} (k_{1}) = K (k^{'}) [B (a_{n} (k^{'}) + k_{1} + 1, b_{n} (k^{'}) + 1) + B (a_{n} (k^{'}) + k_{1} + 1, b_{n} (k^{'}) + 2)], \end{matrix}

and it must be written as follows:

\begin{matrix} f_{n, k^{'}, 1}^{*} (k_{1}) = \{\begin{matrix} K (k^{'}) [B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) + B (a_{n} (k^{'}) + 1, b_{n} (k^{'}) + 3)] & if k_{1} = 0 \\ K (k^{'}) [B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) + B (a_{n} (k^{'}) + 3, b_{n} (k^{'}) + 1)] & if k_{1} = 1 \end{matrix} . \end{matrix}

Also, on in Section 7 it is written:

\begin{matrix} {f_{n, x^{'}}^{*}}^{X_{2} | X_{1} = k_{1}} (k_{2}) : = \frac{f_{n, k^{'}}^{*} (k_{1}, k_{2})}{f_{n, k^{'}, 1}^{*} (k_{1})} = \{\begin{matrix} \frac{2 n + 2}{2 n + n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 3} & if k_{1} = k_{2} = 0, \\ \frac{n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 1}{2 n + n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 3} & if k_{1} = 0, k_{2} = 1, \\ \frac{2 n + 3}{2 n + n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 4} & if k_{1} = 1, k_{2} = 0, \\ \frac{n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 1}{2 n + n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 4} & if k_{1} = k_{2} = 1, \end{matrix} \end{matrix}

and it must be written as follows:

\begin{matrix} \begin{matrix} {f_{n, x^{'}}^{*}}^{X_{2} | X_{1} = k_{1}} (k_{2}) & : = \frac{f_{n, k^{'}}^{*} (k_{1}, k_{2})}{f_{n, k^{'}, 1}^{*} (k_{1})} \\ = \{\begin{matrix} \frac{B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2)}{B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) + B (a_{n} (k^{'}) + 1, b_{n} (k^{'}) + 3)} = \frac{a_{n} (k^{'}) + 1}{2 n + 3} & if k_{1} = k_{2} = 0, \\ \frac{B (a_{n} (k^{'}) + 1, b_{n} (k^{'}) + 3)}{B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) + B (a_{n} (k^{'}) + 1, b_{n} (k^{'}) + 3)} = \frac{b_{n} (k^{'}) + 2}{2 n + 3} & if k_{1} = 0, k_{2} = 1, \\ \frac{B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2)}{B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) + B (a_{n} (k^{'}) + 3, b_{n} (k^{'}) + 1)} = \frac{b_{n} (k^{'}) + 1}{2 n + 3} & if k_{1} = 1, k_{2} = 0, \\ \frac{B (a_{n} (k^{'}) + 3, b_{n} (k^{'}) + 1)}{B (a_{n} (k^{'}) + 2, b_{n} (k^{'}) + 2) + B (a_{n} (k^{'}) + 3, b_{n} (k^{'}) + 1)} = \frac{a_{n} (k^{'}) + 2}{2 n + 3} & if k_{1}, k_{2} = 1 . \end{matrix} \end{matrix} \end{matrix}

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

m_{n}^{*} (k^{'}, k_{1}) = {f_{n, k^{'}}^{*}}^{X_{2} | X_{1} = k_{1}} (1) = \{\begin{matrix} \frac{n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 1}{2 n + n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 3} & if k_{1} = 0, \\ \frac{n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 1}{2 n + n_{+ 0} (k^{'}) + 2 n_{01} (k^{'}) + 4} & if k_{1} = 1, \end{matrix}

and it must be written as follows:

m_{n}^{*} (k^{'}, k_{1}) = {f_{n, k^{'}}^{*}}^{X_{2} | X_{1} = k_{1}} (1) = \{\begin{matrix} \frac{b_{n} (k^{'}) + 2}{2 n + 3} & if k_{1} = 0, \\ \frac{a_{n} (k^{'}) + 2}{2 n + 3} & if k_{1} = 1 . \end{matrix}

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{matrix} \begin{matrix} f_{θ} (x) & : = D {(σ, ρ)}^{\frac{1}{n}} exp \{- \frac{1}{2 σ^{2} (1 - ρ^{2})} [{(x_{1} - θ)}^{2} \\ - 2 ρ (x_{1} - θ) (x_{2} - θ) + {(x_{2} - θ)}^{2}]\} \\ = D {(σ, ρ)}^{\frac{1}{n}} exp \{- \frac{1}{2 σ^{2} (1 - ρ^{2})} [x_{1}^{2} + x_{2}^{2} - 2 ρ x_{1} x_{2} \\ - 2 (1 - ρ) (x_{1} + x_{2}) θ + 2 (1 - ρ) θ^{2}]\}, \end{matrix} \end{matrix}

(7)

where

σ > 0

and

ρ \in [- 1, 1]

are assumed to be known and

D (σ, ρ) = {[2 π σ^{2} \sqrt{1 - ρ^{2}}]}^{- n}

. Thus:

\begin{matrix} R_{θ} = N_{2} ((\begin{matrix} θ \\ θ \end{matrix}), σ^{2} (\begin{matrix} 1 & ρ \\ ρ & 1 \end{matrix})), X_{1}, X_{2} \sim_{θ} N (θ, σ^{2}) \\ P_{θ}^{X_{2} | X_{1} = x_{1}} = N ((1 - ρ) θ + ρ x_{1}, σ^{2} \sqrt{1 - ρ^{2}}), \\ E_{θ} (X_{2} | X_{1} = x_{1}) = (1 - ρ) θ + ρ x_{1} . \end{matrix}

Hence, for a sample

x^{'} = (x_{1}^{'}, \dots, x_{n}^{'}) = (x_{11}^{'}, x_{12}^{'}, \dots, x_{n 1}^{'}, x_{n 2}^{'}) \in {(R^{2})}^{n},

we have that:

\begin{matrix} \begin{matrix} f_{n, θ} (x^{'}) & = \prod_{i = 1}^{n} f_{θ} (x_{i}^{'}) \\ = D (σ, ρ) exp \{- \frac{1}{2 σ^{2} (1 - ρ^{2})} \sum_{i = 1}^{n} [{(x_{i 1}^{'} - θ)}^{2} \\ - 2 ρ (x_{i 1}^{'} - θ) (x_{i 2}^{'} - θ) + {(x_{i 2}^{'} - θ)}^{2}]\} \\ = D (σ, ρ) exp \{- \frac{1}{2 σ^{2} (1 - ρ^{2})} [2 n (1 - ρ) θ^{2} \\ - 2 (1 - ρ) s_{1} (x^{'}) θ + s_{2} (x^{'}) - 2 ρ p (x^{'})]\}, \end{matrix} \end{matrix}

where

s_{1} (x^{'}) : = \sum_{i} (x_{i 1}^{'} + x_{i 2}^{'})

,

s_{2} (x^{'}) : = \sum_{i} ({x_{i 1}^{'}}^{2} + {x_{i 2}^{'}}^{2})

,

p (x^{'}) = \sum_{i} x_{i 1}^{'} x_{i 2}^{'}

.

Let us consider the prior distribution

Q = N (μ, τ^{2})

whose density is:

g (θ) = \frac{1}{τ \sqrt{2 π}} exp \{- \frac{1}{2 τ^{2}} {(θ - μ)}^{2}\} .

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

\begin{matrix} \begin{matrix} r_{n, x^{'}}^{*} (θ) & : = \frac{d R_{n, x^{'}}^{*}}{d θ} (θ) = K (x^{'}) f_{n, θ} (x^{'}) g (θ) \\ = K^{'} (x^{'}) exp \{- \frac{1}{2} [\frac{2 n τ^{2} + σ^{2} (1 + ρ)}{σ^{2} τ^{2} (1 + ρ)} θ^{2} \\ - 2 \frac{s_{1} (x^{'}) τ^{2} + μ σ^{2} (1 + ρ)}{σ^{2} τ^{2} (1 + ρ)} θ + C (x^{'})]\} \\ = \frac{1}{B_{1} \sqrt{2 π}} exp \{- \frac{1}{2} {(\frac{θ - A_{1} (x^{'})}{B_{1}})}^{2}\}, \end{matrix} \end{matrix}

where

K^{'} (x^{'})

and

C (x^{'})

are appropriate constants and

B_{1}^{2} = \frac{σ^{2} τ^{2} (1 + ρ)}{2 n τ^{2} + σ^{2} (1 + ρ)} and A_{1} (x^{'}) = \frac{τ^{2} s_{1} (x^{'}) + σ^{2} (1 + ρ) μ}{2 n τ^{2} + σ^{2} (1 + ρ)} .

So, the posterior distribution given

x^{'}

is

R_{n, x^{'}}^{*} = N (A_{1} (x^{'}), B_{1}^{2})

. The posterior predictive density given

x^{'} \in {(R^{2})}^{n}

is:

\begin{matrix} \begin{matrix} f_{n, x^{'}}^{*} (x) & : = \int_{R} f_{θ} (x) r_{n, x^{'}}^{*} (θ) d θ \\ = \frac{1}{B_{1} {\sqrt{2 π}}^{3} σ^{2} \sqrt{1 - ρ^{2}}} \int_{R} exp \{- \frac{1}{2} {(\frac{θ - A_{2} (x, x^{'})}{B_{2}})}^{2}\} d θ \\ \cdot exp \{\frac{A_{2} {(x, x^{'})}^{2}}{2 B_{2}^{2}} - \frac{A_{1} (x^{'})}{2 B_{1}^{2}} - \frac{x_{1}^{2} + x_{2}^{2} - 2 ρ x_{1} x_{2}}{2 σ^{2} (1 - ρ^{2})}\}, \end{matrix} \end{matrix}

where

\begin{matrix} B_{2}^{2} : = \frac{B_{1}^{2} σ^{2} (1 + ρ)}{2 B_{1}^{2} + σ^{2} (1 + ρ)}, A_{2} (x, x^{'}) : = \frac{σ^{2} (1 + ρ) A_{1} (x^{'}) + B_{1}^{2} (x_{1} + x_{2})}{2 B_{1}^{2} + σ^{2} (1 + ρ)} . \end{matrix}

Some algebraic manipulations allow us to prove that

\begin{matrix} f_{n, x^{'}}^{*} (x) = D {(σ_{1}, ρ_{1})}^{\frac{1}{n}} exp {- \frac{1}{2 σ_{1}^{2} (1 - ρ_{1}^{2})} \\ \cdot [x_{1}^{2} + x_{2}^{2} - 2 ρ_{1} x_{1} x_{2} - 2 (1 - ρ_{1}) (x_{1} + x_{2}) m (x^{'}) + 2 (1 - ρ_{1}) m {(x^{'})}^{2}]}, \end{matrix}

(8)

where

\begin{matrix} ρ_{1} = \frac{(2 ρ n + ρ + 1) τ^{2} + ρ (1 + ρ) σ^{2}}{(2 n + ρ + 1) τ^{2} + (1 + ρ) σ^{2}} \in [- 1, 1], \\ σ_{1}^{2} = σ^{2} \frac{(2 n + ρ + 1) τ^{2} + (1 + ρ) σ^{2}}{2 n τ^{2} + (1 + ρ) σ^{2}} > 0 and \\ m (x^{'}) = \frac{(2 n + 1) τ^{2} + (1 + ρ) σ^{2}}{(2 n + ρ + 1) τ^{2} + (1 + ρ) σ^{2}} \frac{τ^{2} s_{1} (x^{'}) + (1 + ρ) σ^{2} μ}{2 n τ^{2} + (1 + ρ) σ^{2}} . \end{matrix}

Comparing the expressions (7) and (8), it is shown that the posterior predictive distribution

{R_{n, x^{'}}^{*}}^{R}

on

R^{2}

given

x^{'} \in {(R^{2})}^{n}

is the bivariate normal distribution

\begin{matrix} N_{2} ((\begin{matrix} m (x^{'}) \\ m (x^{'}) \end{matrix}), σ_{1}^{2} (\begin{matrix} 1 & ρ_{1} \\ ρ_{1} & 1 \end{matrix})) . \end{matrix}

It follows that the conditional distribution:

{({R_{n, x^{'}}^{*}}^{R})}^{π_{2} | π_{1} = x_{1}} : = N ((1 - ρ_{1}) m (x^{'}) + ρ_{1} x_{1}, σ_{1}^{2} (1 - ρ_{1}^{2}))

is the Bayes estimator of the conditional distribution:

P_{θ}^{X_{2} | X_{1} = x_{1}} = N ((1 - ρ) θ + ρ x_{1}, σ^{2} (1 - ρ^{2}))

for the squared total variation function, and its density

{f_{n, x^{'}}^{*}}^{π_{2} | π_{1} = x_{1}}

is the Bayes estimator of the conditional density:

f_{θ}^{X_{2} | X_{1} = x_{1}} (x_{2}) = \frac{1}{σ \sqrt{2 π (1 - ρ^{2})}} exp \{- \frac{1}{2 σ^{2} (1 - ρ^{2})} {[x_{2} - (1 - ρ) θ - ρ x_{1}]}^{2}\}

for the

L^{1}

-squared loss function.

Moreover, its mean:

E_{{R_{n, x^{'}}^{*}}^{R}} (π_{2} | π_{1} = x_{1}) = (1 - ρ_{1}) m (x^{'}) + ρ_{1} x_{1}

is the Bayes estimator of the regression curve:

E_{θ} (X_{2} | X_{1} = x_{1}) = (1 - ρ) θ + ρ 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.

Reference

Nogales, A.G. Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution. Mathematics 2022, 10, 1213. [Google Scholar] [CrossRef] [Scilit]

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