SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left[ \mu_ {0} ^ {2} \left(\mathbf {W} _ {1}, \mathbf {r} _ {1}\right) \mid \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] (B.46) \\ = \sum_ {\overline {{\mathbf {w}}} ^ {\prime} ^ {m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}) \prod_ {j = 2} ^ {m} P _ {\sigma (\mathbf {Z} _ {j})} (w _ {j} ^ {\prime} | w _ {1} ^ {\prime}, \overline {{\mathbf {w} ^ {\prime}}} ^ {2: j - 1}, \overline {{\mathbf {r}}} ^ {j - 1}, \mathbf {z} _ {j}) P _ {\sigma (\mathbf {Z} _ {1})} (w _ {1} ^ {\prime} | \mathbf {z} _ {1}, \mathbf {r} _ {0}) \qquad (\mathrm {B}. 4 7) \\ = \sum_ {\overline {{\mathbf {w}}} ^ {m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = 1} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left(w _ {j} ^ {\prime} \mid \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1}, \mathbf {z} _ {j}\right) (B.48) \\ = A _ {0}. (B.49) \\ \end{array}
$$
Lemma S.3 (Bias Analysis of g-mSBD Estimators (1)). Let  $\overline{A}$  be the quantity defined as
$$
\bar {A} := \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \breve {\mu} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. \tag {B.50}
$$
For  $i = 2,\dots ,m$
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right)}} \left[ \bar {\pi} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu^ {i + 1} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma \left(\mathbf {Z} _ {i}\right)}} \left[ \bar {\pi} ^ {i - 1} \left\{\check {\mu} ^ {i + 1} - \mu^ {i} \right\} \mid \mathbf {z} _ {i}, \mathbf {r} _ {0} \right] \tag {B.51} \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i - 1} \{\mu_ {0} ^ {i + 1} - \mu^ {i + 1} \} \{\pi^ {i} - \pi_ {0} ^ {i} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i})}} \left[ \overline {{\pi}} ^ {i - 1} \{\breve {\mu} _ {0} ^ {i + 1} - \mu^ {i} \} | \mathbf {z} _ {i}, \mathbf {r} _ {0} \right]. \\ \end{array}
$$
Proof of Lemma S.3. We first rewrite Eq. (B.51) as follows:
$$
\begin{array}{l} \operatorname {E q}. \left(\mathrm {B}. 5 1\right) = \mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {i + 1}\right)}} \left[ \bar {\pi} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu^ {i + 1} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] (B.52) \\ + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i})}} \left[ \bar {\pi} ^ {i - 1} \left\{\check {\mu} ^ {i + 1} - \check {\mu} _ {0} ^ {i + 1} \right\} \mid \mathbf {z} _ {i}, \mathbf {r} _ {0} \right] (B.53) \\ + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i})}} \left[ \bar {\pi} ^ {i - 1} \left\{\check {\mu} _ {0} ^ {i + 1} - \mu^ {i} \right\} | \mathbf {z} _ {i}, \mathbf {r} _ {0} \right]. (B.54) \\ \end{array}
$$
Eq. (B.53)
$$
\begin{array}{l} = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i})}} \left[ \overline {{\pi}} ^ {i - 1} (\overline {{\mathbf {W}}} ^ {i - 1}, \overline {{\mathbf {R}}} ^ {1: i - 1}) \{\check {\mu} ^ {i + 1} (\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i - 1}) - \check {\mu} _ {0} ^ {i + 1} (\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i - 1}) \} | \mathbf {z} _ {i}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i - 1} (\overline {{\mathbf {W}}} ^ {i - 1}, \overline {{\mathbf {R}}} ^ {1: i - 1}) \frac {P _ {\sigma (\mathbf {Z} _ {i})} (\overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} | \mathbf {z} _ {i} , \mathbf {r} _ {0})}{P _ {\sigma (\mathbf {Z} _ {i + 1})} (\overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} | \mathbf {z} _ {i + 1} , \mathbf {r} _ {0})} \{\breve {\mu} ^ {i + 1} (\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i - 1}) - \breve {\mu} _ {0} ^ {i + 1} (\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i - 1}) \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i - 1} \frac {P _ {\sigma (\mathbf {Z} _ {i})} (\overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} | \mathbf {z} _ {i} , \mathbf {r} _ {0})}{P _ {\sigma (\mathbf {Z} _ {i + 1})} (\overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} | \mathbf {z} _ {i + 1} , \mathbf {r} _ {0})} \frac {\mathbb {1} _ {\mathbf {r} _ {i}} (\mathbf {R} _ {i})}{P _ {\sigma (\mathbf {Z} _ {i + 1})} (\mathbf {R} _ {i} | \overline {{\mathbf {W}}} ^ {i - 1} , \overline {{\mathbf {R}}} ^ {1 : i - 1} , \mathbf {r} _ {0} , \mathbf {z} _ {i + 1})} \{\mu^ {i + 1} - \mu_ {0} ^ {i + 1} \} \Bigg | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \bar {\pi} ^ {i - 1} \left(\overline {{\mathbf {W}}} ^ {i - 1}, \overline {{\mathbf {R}}} ^ {1: i - 1}\right) \pi_ {0} ^ {i} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {i}\right) \left\{\mu^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {i}\right) - \mu_ {0} ^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {i}\right) \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right]. \tag {B.55} \\ \end{array}
$$
$$
\begin{array}{l} \text {E q .} (\mathrm {B}. 5 2) + \text {E q .} (\mathrm {B}. 5 3) \\ = \text {E q .} (\mathrm {B}. 5 2) + \text {E q .} (\mathrm {B}. 5 5) \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu^ {i + 1} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \text {E q .} (B. 5 5) \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\mu_ {0} ^ {i + 1} - \mu^ {i + 1} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \operatorname {E q}. (B. 5 5) \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i} \{\mu_ {0} ^ {i + 1} - \mu^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i - 1} \pi_ {0} ^ {i} \{\mu^ {i + 1} - \mu_ {0} ^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \bar {\pi} ^ {i - 1} \left\{\mu_ {0} ^ {i + 1} - \mu^ {i + 1} \right\} \left\{\pi^ {i} - \pi_ {0} ^ {i} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right]. \tag {B.56} \\ \end{array}
$$
$$
\text {E q .} (\mathrm {B}. 5 1) = \text {E q .} (\mathrm {B}. 5 6) + \text {E q .} (\mathrm {B}. 5 4).
$$
Lemma S.4 (Bias Analysis of g-mSBD Estimators (2)). Let  $\overline{A}$  be the quantity defined as
$$
\bar {A} := \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \breve {\mu} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. \tag {B.57}
$$
Then, for  $k = 3,\dots ,m - 1$
$$
\begin{array}{l} \overline {{A}} - A _ {0} = \sum_ {r = k} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {r + 1})}} \left[ \overline {{\pi}} ^ {r - 1} \{\mu_ {0} ^ {r + 1} - \mu^ {r + 1} \} \{\pi^ {r} - \pi_ {0} ^ {r} \} | \mathbf {z} _ {r + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {k})}} \left[ \overline {{\pi}} ^ {k - 1} \{\breve {\mu} _ {0} ^ {k + 1} - \mu^ {k} \} | \mathbf {z} _ {k}, \mathbf {r} _ {0} \right] \\ + \sum_ {i = 1} ^ {k - 2} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \breve {\mu} ^ {2} - \breve {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. \\ \end{array}
$$
Proof of Lemma S.4. The equation holds for  $k = m - 1$ . It can be shown as follows:
$$
\begin{array}{l} \overline {{A}} - A _ {0} \\ = \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {1})}} \left[ \breve {\mu} ^ {2} - \breve {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {m + 1})}} \left[ \bar {\pi} ^ {m} \left\{\check {\mu} _ {0} ^ {m + 2} - \mu^ {m + 1} \right\} \mid \mathbf {z} _ {m + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {m})}} \left[ \bar {\pi} ^ {m - 1} \left\{\check {\mu} ^ {m + 1} - \mu^ {m} \right\} \mid \mathbf {z} _ {m}, \mathbf {r} _ {0} \right] (B.58) \\ + \sum_ {i = 1} ^ {m - 2} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \breve {\mu} ^ {2} - \breve {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. (B.59) \\ \end{array}
$$
Eq. (B.58)
$$
\stackrel {\text {L e m m a S . 3}} {=} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {m + 1})}} \left[ \bar {\pi} ^ {m - 1} \left\{\mu_ {0} ^ {m + 1} - \mu^ {m + 1} \right\} \left\{\pi^ {m} - \pi_ {0} ^ {m} \right\} | \mathbf {z} _ {m + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {m})}} \left[ \bar {\pi} ^ {m - 1} \left\{\check {\mu} _ {0} ^ {m} - \mu^ {m - 1} \right\} | \mathbf {z} _ {m}, \mathbf {r} _ {0} \right]. \tag {B.60}
$$
$$
\begin{array}{l} \overline {{A}} - A _ {0} \\ = \operatorname {E q}. (\mathrm {B}. 6 0) + \sum_ {i = 1} ^ {m - 2} \mathbb {E} _ {P _ {\sigma} (\mathbf {z} _ {i + 1})} \left[ \bar {\pi} ^ {i} \{\check {\mu} ^ {i + 2} - \mu^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma} (\mathbf {z} _ {1})} \left[ \check {\mu} ^ {2} - \check {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. (B. 6 1) \\ \end{array}
$$
For any fixed  $k + 1\in \{m - 1,\dots ,4\}$  , suppose the following holds:
$$
\begin{array}{l} \overline {{A}} - A _ {0} \\ = \sum_ {r = k + 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {r + 1})}} \left[ \bar {\pi} ^ {r - 1} \left\{\mu_ {0} ^ {r} - \mu^ {r} \right\} \left\{\pi^ {r + 1} - \pi_ {0} ^ {r + 1} \right\} \mid \mathbf {z} _ {r + 1}, \mathbf {r} _ {0} \right] (B.62) \\ + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k + 1})}} \left[ \bar {\pi} ^ {k} \left\{\check {\mu} _ {0} ^ {k + 2} - \mu^ {k + 1} \right\} \mid \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] (B.63) \\ + \sum_ {i = 1} ^ {k - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] (B.64) \\ + \mathbb {E} _ {P _ {\sigma} (\mathbf {z} _ {1})} \left[ \breve {\mu} ^ {2} - \breve {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. (B.65) \\ \end{array}
$$
We note that this holds when  $k = m - 2$ , as shown in Eq. (B.61). We will now show that it will hold for  $k$ , too. First,
$$
\text {E q .} (\mathrm {B}. 6 3) + \text {E q .} (\mathrm {B}. 6 4) \tag {B.66}
$$
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {k + 1})}} \left[ \overline {{\pi}} ^ {k} \{\breve {\mu} _ {0} ^ {k + 2} - \mu^ {k + 1} \} | \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] + \sum_ {i = 1} ^ {k - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i} \{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k + 1})}} \left[ \overline {{\pi}} ^ {k} \{\check {\mu} _ {0} ^ {k + 2} - \mu^ {k + 1} \} | \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k})}} \left[ \overline {{\pi}} ^ {k - 1} \{\check {\mu} ^ {k + 2} - \mu^ {k + 1} \} | \mathbf {z} _ {k}, \mathbf {r} _ {0} \right] \\ + \sum_ {i = 1} ^ {k - 2} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} ^ {i} \{\tilde {\mu} ^ {i + 2} - \mu^ {i + 1} \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right], \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {\text {L e m m a S . 3}} {=} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k + 1})}} \left[ \overline {{\pi}} ^ {k - 1} \left\{\mu_ {0} ^ {k + 1} - \mu^ {k + 1} \right\} \left\{\pi^ {k} - \pi_ {0} ^ {k} \right\} | \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k})}} \left[ \overline {{\pi}} ^ {k - 1} \left\{\tilde {\mu} _ {0} ^ {k + 1} - \mu^ {k} \right\} | \mathbf {z} _ {k}, \mathbf {r} _ {0} \right] \\ + \sum_ {i = 1} ^ {k - 2} \mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {i + 1}\right)}} \left[ \bar {\pi} ^ {i} \left\{\check {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right]. \tag {B.67} \\ \end{array}
$$
$$
\begin{array}{l} \overline {{A}} - A _ {0} \\ = \text {E q .} (\mathrm {B}. 6 2) + \text {E q .} (\mathrm {B}. 6 3) + \text {E q .} (\mathrm {B}. 6 4) + \text {E q .} (\mathrm {B}. 6 5) \\ = \text {E q .} (\mathrm {B}. 6 2) + \text {E q .} (\mathrm {B}. 6 7) + \text {E q .} (\mathrm {B}. 6 5) \\ = \sum_ {r = k + 1} ^ {m} \mathbb {E} _ {P _ {\sigma} (\mathbf {z} _ {r + 1})} \left[ \overline {{\pi}} ^ {r - 1} \{\mu_ {0} ^ {r + 1} - \mu^ {r + 1} \} \{\pi^ {r} - \pi_ {0} ^ {r} \} | \mathbf {z} _ {r + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma} (\mathbf {z} _ {k + 1})} \left[ \overline {{\pi}} ^ {k - 1} \{\mu_ {0} ^ {k + 1} - \mu^ {k _ {1}} \} \{\pi^ {k} - \pi_ {0} ^ {k} \} | \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] \\ + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k})}} \left[ \bar {\pi} ^ {k - 1} \left\{\breve {\mu} _ {0} ^ {k + 1} - \mu^ {k} \right\} \mid \mathbf {z} _ {k}, \mathbf {r} _ {0} \right] \\ + \sum_ {i = 1} ^ {k - 2} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\check {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] \\ + \text {E q .} (B. 6 5) \\ \end{array}
$$
Therefore, the equation holds for  $k$ , too. This completes the proof.
Lemma S.5 (Bias Analysis of g-mSBD Estimators (3)).
$$
\mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {2})}} \left[ \pi^ {1} \{\check {\mu} _ {0} ^ {3} - \mu^ {2} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] - \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \check {\mu} ^ {2} - \check {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] = \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {2})}} \left[ \{\mu_ {0} ^ {2} - \mu^ {2} \} \{\pi^ {1} - \pi_ {0} ^ {1} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right].
$$
Proof of Lemma S.5. Note that
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \breve {\mu} ^ {2} (\mathbf {W} _ {1}, \mathbf {R} _ {0}) - \breve {\mu} _ {0} ^ {2} (\mathbf {W} _ {1}, \mathbf {R} _ {0}) | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \frac {P _ {\sigma (\mathbf {Z} _ {1})} (\mathbf {W} _ {1} , \mathbf {R} _ {0} | \mathbf {z} _ {1} , \mathbf {r} _ {0})}{P _ {\sigma (\mathbf {Z} _ {2})} (\mathbf {W} _ {1} , \mathbf {R} _ {0} | \mathbf {z} _ {2} , \mathbf {r} _ {0})} \{\check {\mu} ^ {2} (\mathbf {W} _ {1}, \mathbf {R} _ {0}) - \check {\mu} _ {0} ^ {2} (\mathbf {W} _ {1}, \mathbf {R} _ {0}) \} \Bigg | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \frac {P _ {\sigma (\mathbf {Z} _ {1})} (\mathbf {W} _ {1} , \mathbf {R} _ {0} | \mathbf {z} _ {1} , \mathbf {r} _ {0})}{P _ {\sigma (\mathbf {Z} _ {2})} (\mathbf {W} _ {1} , \mathbf {R} _ {0} | \mathbf {z} _ {2} , \mathbf {r} _ {0})} \frac {\mathbb {1} _ {\mathbf {r} _ {1}} (\mathbf {R} _ {1})}{P _ {\sigma (\mathbf {Z} _ {2})} (\mathbf {R} _ {1} | \mathbf {W} _ {0} , \mathbf {z} _ {2} , \mathbf {r} _ {0})} \left\{\mu^ {2} (\mathbf {W} _ {1}, \mathbf {R} _ {1}) - \mu_ {0} ^ {2} (\mathbf {W} _ {1}, \mathbf {R} _ {1}) \right\} \middle | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma} (\mathbf {z} _ {2})} \left[ \pi_ {0} ^ {1} \left(\mathbf {W} _ {1}, X _ {1}\right) \left\{\mu^ {2} \left(\mathbf {W} _ {1}, X _ {1}\right) - \mu_ {0} ^ {2} \left(\mathbf {W} _ {1}, X _ {1}\right) \right\} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right]. \\ \end{array}
$$
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {2})}} \left[ \pi^ {1} \{\check {\mu} _ {0} ^ {3} - \mu^ {2} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] - \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \check {\mu} ^ {2} - \check {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \pi^ {1} \{\mu_ {0} ^ {2} - \mu^ {2} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] - \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {1})}} \left[ \tilde {\mu} ^ {2} - \tilde {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \pi^ {1} \{\mu_ {0} ^ {2} - \mu^ {2} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] - \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \pi_ {0} ^ {1} \{\mu_ {0} ^ {2} - \mu^ {2} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \{\mu_ {0} ^ {2} - \mu^ {2} \} \{\pi^ {1} - \pi_ {0} ^ {1} \} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right]. \\ \end{array}
$$
Lemma S.6 (Bias Analysis of g-mSBD Estimators). Let  $\overline{A}$  be the quantity defined as
$$
\bar {A} := \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} ^ {i} \left\{\breve {\mu} ^ {i + 2} - \mu^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \breve {\mu} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right], \tag {B.68}
$$
where  $\pi^i, \mu^i$  are arbitrary nuisances for true nuisances  $\pi_0^i$  and  $\mu_0^i$  defined in Def. 2. Let  $A_0$  denote the  $g$ -mSBD in Def. 1. Then,
$$
\bar {A} - A _ {0} = \sum_ {r = 1} ^ {m - 1} O _ {P _ {\sigma \left(\mathbf {z} _ {r + 1}\right)}} \left(\| \pi^ {r} - \pi_ {0} ^ {r} \| \| \mu^ {r + 1} - \mu_ {0} ^ {r + 1} \|\right). \tag {B.69}
$$
Proof of Lemma S.6. By Lemmas (S.3,S.4,S.5).
We also use the following results which are used by Kennedy et al. [2020].
Lemma S.7 (Decomposition). Let  $\mathcal{D} \sim P$  denote a finite sample set following a distribution  $P$ . Let  $h(V; \eta)$  denote an arbitrary random function taking  $\eta$  as a nuisance. For any  $\eta, \eta_0$ ,
$$
\begin{array}{l} \mathbb {E} _ {\mathcal {D}} [ h (\mathbf {V}; \eta) ] - \mathbb {E} _ {P} [ h (\mathbf {V}; \eta_ {0}) ] (B.70) \\ = \mathbb {E} _ {\mathcal {D} - P} [ h (V; \eta_ {0}) ] + \mathbb {E} _ {\mathcal {D} - P} [ h (V; \eta) - h (V; \eta_ {0}) ] + \mathbb {E} _ {P} [ h (V; \eta) - h (V; \eta_ {0}) ]. (B.71) \\ \end{array}
$$
Proof of Lemma S.7.
$$
\begin{array}{l} \mathbb {E} _ {\mathcal {D}} [ h (\mathbf {V}; \eta) ] - \mathbb {E} _ {P} [ h (\mathbf {V}; \eta_ {0}) ] (B.72) \\ = \mathbb {E} _ {\mathcal {D} - P} [ h (\mathbf {V}; \eta_ {0}) ] + \mathbb {E} _ {\mathcal {D}} [ h (\mathbf {V}; \eta) - h (\mathbf {V}; \eta_ {0}) ] (B.73) \\ = \mathbb {E} _ {\mathcal {D} - P} [ h (\mathbf {V}; \eta_ {0}) ] + \mathbb {E} _ {\mathcal {D} - P} [ h (\mathbf {V}; \eta) - h (\mathbf {V}; \eta_ {0}) ] + \mathbb {E} _ {P} [ h (\mathbf {V}; \eta) - h (\mathbf {V}; \eta_ {0}) ]. (B.74) \\ \end{array}
$$
Lemma S.8 (Continuous Mapping Theorem for  $L_{2}(P)$ ). Let  $X_{n}$ ,  $X$  denote a random sequence defined on a metric space  $S$ . Suppose a function  $g: S \to S'$  (where  $S'$  is another metric space) is bounded and continuous almost everywhere. Then,
$$
X _ {n} \stackrel {L _ {2} (P)} {\rightarrow} X \Longrightarrow g \left(X _ {n}\right) \stackrel {L _ {2} (P)} {\rightarrow} g (X). \tag {B.75}
$$
Proof of Lemma S.8. We first note that  $X_{n} \stackrel{L_{2}(P)}{\rightarrow} X$  implies  $X_{n} \stackrel{p}{\rightarrow} X$ . Then, by continuous mapping theorem,  $g(X_{n}) \stackrel{p}{\rightarrow} g(X)$ . Then,
$$
\lim  _ {n \rightarrow \infty} \| g (X _ {n}) - g (X) \| ^ {2} = \lim  _ {n \rightarrow \infty} \int_ {\mathcal {X}} | g (X _ {n}) - g (X) | ^ {2} d [ P ] \stackrel {*} {=} \int_ {\mathcal {X}} \lim  _ {n \rightarrow \infty} | g (X _ {n}) - g (X) | ^ {2} d [ P ] = 0, \tag {B.76}
$$
where the equation  $\stackrel{*}{=}^*$  holds by the dominated convergence theorem, which is applicable since  $g(X_{n}), g(X)$  are bounded functions (from the given condition) and  $X_{n} \xrightarrow{p} X$ .
Proposition 1 (Asymptotic Analysis of g-mSBD Estimators). Assume that the nuisance estimates  $\mu_{\ell}^{i}$  and  $\pi_{\ell}^{i}$  are  $L_{2}$ -consistent; i.e.,  $\| \mu_{\ell}^{i + 1} - \mu_0^{i + 1}\|_{P_{\sigma (\mathbf{Z}_{i + 1})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1})}}(1)$ ,  $\| \breve{\mu}_{\ell}^{i + 2} - \breve{\mu}_0^{i + 2}\|_{P_{\sigma (\mathbf{Z}_{i + 1})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1})}}(1)$  and  $\| \pi_{\ell}^{i} - \pi_{0}^{i}\|_{P_{\sigma (\mathbf{Z}_{i + 1})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1})}}(1)$  for  $i = 1,\dots ,m - 1$ , and  $\| \breve{\mu}_{\ell}^{2} - \breve{\mu}_{0}^{2}\|_{P_{\sigma (\mathbf{Z}_{1})}} = o_{P_{\sigma (\mathbf{Z}_{1})}}(1)$ . Let  $n_i\coloneqq |\overline{D}_{\mathbf{z}_i}|$  for  $i\in \{1,\dots ,m\}$ . Then,
$$
\hat {A} - A _ {0} = \sum_ {i = 1} ^ {m} R _ {i} + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m - 1} O _ {P _ {\sigma \left(\mathbf {z} _ {i + 1}\right)}} \left(\| \mu_ {\ell} ^ {i + 1} - \mu_ {0} ^ {i + 1} \| \| \pi_ {\ell} ^ {i} - \pi_ {0} ^ {i} \|\right), \tag {10}
$$
Proof of Proposition 1. We start the proof by noting that
$$
\hat {A} - A _ {0} = \frac {1}{L} \sum_ {\ell = 1} ^ {L} \left\{\hat {A} _ {\ell} - A _ {0} \right\}, \tag {B.77}
$$
where  $\hat{A}_{\ell}$  is defined in Def. 3. This proof focuses on analyzing  $\hat{A}_{\ell} - A_{0}$ .
Also, we recall that  $\overline{D}_{\mathbf{z}_i,\ell}$  for all  $\mathbf{z}_i$  follows the distribution  $P_{\sigma (\mathbf{Z}_i)}(\mathbf{V}|\mathbf{r}_0,\mathbf{z}_i)$ . Throughout the proof, we will denote
$$
P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right) \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0}} (\mathbf {V}) := P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right)} \left(\mathbf {V} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0}\right). \tag {B.78}
$$
Then, each  $\hat{A}_{\ell} - A_{0}$  in Eq. (B.77) is given as follow: