\section{Introduction}

\IEEEPARstart{T}{he} relationship between conditional entropy (equivocation) or mutual information, and best possible quality of decoding is an important concept in information theory. The best possible quality of a decoding scheme, when quantified by the minimal probability of error $\epsilon$, does not uniquely determine the value of equivocation or mutual information, but various upper and lower bounds have been proved, see Sec. \ref{sec_existing_theory}.

Here we discuss a scenario when not only $\epsilon$, but the complete joint probability distribution $p(x,\hat{x})$ of signals $x$ and maximum a posteriori
decodes $\hat{x}$ is available. We refer to $p(x,\hat{x})$ as the confusion matrix. To our knowledge, such a scenario has not been extensively studied in the literature, despite having practical relevance for estimation of mutual information, as we point out in Sec. \ref{sec_motivation}. In this article, we derive an upper bound on mutual information (and a corresponding lower bound on equivocation) that is based on the confusion matrix and is tighter than the known similar bound by Kovalevsky and others \cite{Kovalevsky1968,Tebbe1968,Feder1994} based on probability of error alone. The inequality in our bound can be proved quickly using the bound by Kovalevsky, as we show in Sec. \ref{sec_quick_proof}. However, we also include a self-contained derivation in Sec. \ref{sec_our_proof}, where we construct the distribution of channel outputs that minimizes equivocation $H(X|Y)$ under our constraints.

\subsection{Equivocation, mutual information and the minimal probability of error}
\label{sec_existing_theory}

We consider a signal variable (message) $X$ that is communicated through a channel with output $Y$ and then decoded, obtaining a ``decode'' $\hat{X}$ -- forming a Markov chain $X \leftrightarrow Y \leftrightarrow \hat{X}$. The equivocation $H(X|Y)$ quantifies the uncertainty in $X$ if the value of $Y$ is given. Conversely, the mutual information $I(X;Y)$ measures how much information about $X$ is contained in $Y$. It is not surprising that both $H(X|Y)$ and $I(X;Y)$ can be related to the minimal probability of error while decoding, $\epsilon = \Pr(X \neq \hat{X})$.

Accurate decoding, i.e., low $\epsilon$, requires sufficiently low equivocation $H(X|Y)$. This is quantified by Fano's inequality \cite{Cover2006}. The mutual information between the true signal and the channel output,  $I(X;Y) = H(X)-H(X|Y)$, needs to be sufficiently high, and this is described by rate-distortion theory \cite{Shannon1959}.

Here we focus on the opposite bounds. If the minimal probability of error $\epsilon$ is specified, there is also a minimal possible equivocation. The following lower bound was derived for discrete $X$ with finite support by Kovalevsky \cite{Kovalevsky1968} and later Tebbe and Dwyer \cite{Tebbe1968} and Feder and Merhav \cite{Feder1994}. It reads
\begin{equation}
H(X|Y) \geq \phi^\ast (\epsilon), \label{eq_feder_merhav}
\end{equation}
where $\phi^\ast (\epsilon)$ is a piecewise linear function that coincides with $-\log{(1-\epsilon)}$ at points $\epsilon=0$, $1/2$, $2/3$, $\dots$, $(|\mathcal{X}|-1)/|\mathcal{X}|$ (we use $\log = \log_2$ throughout the paper, and $\mathcal{X}$ is the support of $X$), and it can be written using the floor and ceiling functions,
\begin{align}
\phi^\ast (\epsilon) &= \alpha(\epsilon) \log{ \left\lfloor \frac{1}{1- \epsilon} \right\rfloor} + \left(1-\alpha(\epsilon) \right) \log{ \left\lceil \frac{1}{1- \epsilon} \right\rceil}, \label{eq_phi} \\
\alpha(\epsilon) &= \left\lfloor \frac{1}{1- \epsilon} \right\rfloor \left( (1-\epsilon) \left\lceil \frac{1}{1- \epsilon} \right\rceil -1  \right). \label{eq_alpha}
\end{align}
The function $\phi^\ast (\epsilon)$ is plotted in Fig. \ref{fig_phi}.

\begin{figure}[!t]
\centering
\includegraphics[width=2.7in]{phi-eps-converted-to.pdf}
\caption{Plot of the functions $\phi^\ast(\epsilon)$ and $-\log{(1-\epsilon)}$. The two functions intersect at $\epsilon=0$, $1/2$, $2/3$, $\dots$, $(|\mathcal{X}|-1)/|\mathcal{X}|$ (black dots), and in between $\phi^\ast(\epsilon)$ is piecewise linear.}
\label{fig_phi}
\end{figure}

The bound \eqref{eq_feder_merhav} has been generalized to countably infinite support of $X$ by Ho and Verd\'u \cite{Ho2010}. Sason and Verd\'u \cite{Sason2018} proved a generalisation of \eqref{eq_feder_merhav} for Arimoto-R\'enyi conditional entropy of arbitrary order.

The bound \eqref{eq_feder_merhav} is tight when only the overall probability of error $\epsilon$ is available. However, when more constraints on the the joint distribution of $X$ and $Y$ are given, tighter bounds can be obtained. Prasad \cite{Prasad2015} introduced two series of lower bounds on $H(X|Y)$ based on partial knowledge of the posterior distribution $p(x|y)$. The first is in terms of the $k$ largest posterior probabilities $p(x|y)$ for each $y$, that we could label $p_1(y), p_2(y), \dots, p_k(y)$ in descending order (where $1 \leq k \leq |\mathcal{X}|$). The second series of bounds by Prasad is in terms of the averages of $p_1(y), p_2(y), \dots, p_k(y)$ across all $y$.

Hu and Xing \cite{Hu2016} focused on binary signal $X$ and derived a bound tighter than \eqref{eq_feder_merhav} by taking into account the prior distribution of signals $p(x)$. Hu and Xing also discuss suboptimal (other than maximum a posteriori) decoding, which is otherwise rare in the related literature.

\subsection{Motivation: estimation of mutual information} \label{sec_motivation}

Here we extend the bound \eqref{eq_feder_merhav} to account for the situation when the complete confusion matrix -- the joint distribution $p(x,\hat{x})$ is known. We are motivated by the following scenario: suppose that the goal is to estimate the mutual information $I(X;Y)$ from a finite set of $(x,y)$ samples. Moreover, assume that the space of possible channel outputs $\mathcal{Y}$ is large (much larger than the space of signals, $|\mathcal{Y}|\gg|\mathcal{X}|$), making a direct calculation of $I(X;Y)$ by means of their joint distribution $p(x,y)$ infeasible due to insufficient sampling. In such a case, one approach (used e.g. in neuroscience \cite{Borst1999}) is to construct a decoder, map each $y$ into a decode $\hat{x}$ and estimate the confusion matrix $p(x,\hat{x})$. Then the post-decoding mutual information $I(X;\hat{X})$ can be calculated and used as a lower bound on $I(X;Y)$ due to the data processing inequality \cite{Cover2006}. However, the gap between $I(X;\hat{X})$ and $I(X;Y)$ is not known (but see a discussion of this gap in \cite{Samengo2002}), and an upper bound on $I(X;Y)$ based on $p(x,\hat{x})$ is desirable. Our result is such a bound, for the specific case of maximum a posteriori decoder.

While mutual information $I(X;Y)$ has this practical importance, we formulate our result as an equivalent lower bound on equivocation $H(X|Y) = H(X)-I(X;Y)$ first. This is simpler to state and prove.

\section{Statement of the bound}

Given the joint distribution $p(X,\hat{X})$ of signals $X$ (discrete with finite support) and maximum a posteriori decodes $\hat{X}$ based on the channel output $Y$, the equivocation $H(X|Y)$ is bounded from below by
\begin{equation}
H(X|Y) \geq \sum_{\hat{x}} p(\hat{x}) \, \phi^\ast (\epsilon_{\hat{x}} ), \label{eq_bound_equivocation}
\end{equation}
where $\epsilon_{\hat{x}} = p(X \neq \hat{X} | \hat{x}) = 1 - p(X = \hat{x} | \hat{X} = \hat{x})$ is the probability of error for the decode $\hat{x}$ and the function $\phi^\ast$ is defined in \eqref{eq_phi}, \eqref{eq_alpha}.

Equivalently, we can bound the mutual information $I(X;Y)$ from above:
\begin{align}
I(X;Y) &= H(X) - H(X|Y)  \nonumber \\
 &\leq H(X) - \sum_{\hat{x}} p(\hat{x}) \, \phi^\ast (\epsilon_{\hat{x}} ).
\end{align}

These bounds are tight, and we construct the distributions $p(y|\hat{x})$ and $p(x|y)$ that achieve equality in Sec. \ref{sec_our_proof}.

\subsection{Comments on the bound}

We note that since the function $\phi^\ast (\epsilon_{\hat{x}})$ is convex, we can apply Jensen's inequality to the right hand side of \eqref{eq_bound_equivocation} and recover the bound \eqref{eq_feder_merhav} by Kovalevsky \cite{Kovalevsky1968},
\begin{equation}
H(X|Y) \geq \phi^\ast \left( \sum_{\hat{x}} p(\hat{x}) \, \epsilon_{\hat{x}} \right)
= \phi^\ast (\epsilon).
\end{equation}

Both bounds coincide in case of binary signal $|\mathcal{X}| = 2$, or any other case when the probability of error is less than $1/2$, $\epsilon_{\hat{x}} < 1/2$ for all $\hat{x}$. On this range, $\phi^\ast (\epsilon_{\hat{x}}) = 2 \epsilon_{\hat{x}}$ and the bound simplifies to
\begin{equation}
H(X|Y) \geq 2 \sum_{\hat{x}} p(\hat{x}) \, \epsilon_{\hat{x}} = 2 \epsilon,
\end{equation}
as has been noted in \cite{Feder1994} and before.

\subsection{Example calculation}

As an illustration, we apply our bound \eqref{eq_bound_equivocation} to an example confusion matrix and compare it to the bound \eqref{eq_feder_merhav} that is in terms of error probability $\epsilon$ only.

The confusion matrix considered is depicted in Fig. \ref{fig_simpleCalc} (A) for the case $|\mathcal{X}|=5$. We vary the size $|\mathcal{X}|$ of the space of signals $\mathcal{X}=\{ 1,2,\dots,|\mathcal{X}|\}$, and the confusion matrix always takes the form
\begin{equation}
    p(x,\hat{x}) =
    \begin{cases}
    \frac{1}{2|\mathcal{X}|}; \qquad  &  x=\hat{x}<|\mathcal{X}|, \\
    \frac{1}{2|\mathcal{X}|}; \qquad  &  x<|\mathcal{X}|,\, \hat{x}=|\mathcal{X}|, \\
    \frac{1}{|\mathcal{X}|};  \qquad  &  x=\hat{x}=|\mathcal{X}|, \\
    0;                        \qquad  &  x \neq \hat{x},\, \hat{x}<|\mathcal{X}|.
  \end{cases} \label{eq_example_calculation}
\end{equation}
This distribution has the property that while most of the decodes have zero probability of being incorrect ($\epsilon_{\hat{x}}=0$ for $\hat{x}<|\mathcal{X}|$), the last one has a high probability of being incorrect, $\epsilon_{\hat{x}}=(|\mathcal{X}|-1)/(|\mathcal{X}|+1)$ for $\hat{x}=|\mathcal{X}|$. Our bound \eqref{eq_bound_equivocation} takes this into account -- which makes it substantially tighter than the bound \eqref{eq_feder_merhav} based only on the overall probability of error $\epsilon$. This can be seen in Fig. \ref{fig_simpleCalc} (B), where both lower bounds are plotted. We also plot the post-decoding conditional entropy $H(X|\hat{X})$ which serves as the upper bound on the true value of $H(X|Y)$.

\begin{figure}[!t]
\centering
\begin{tabular}{m{1.3 in} m{1.8 in}}

    \includegraphics[width=\linewidth]{simple_calc_conf_matrix-eps-converted-to.pdf} &
    \includegraphics[width=\linewidth]{simple_calc-eps-converted-to.pdf}
\end{tabular}
\caption{Example application of the bound. (A) The joint distribution of signals and decodes $p(x,\hat{x})$ for which we compute the bound, defined in Eq. \eqref{eq_example_calculation}. Here for the case $|\mathcal{X}|=5$. (B) Bounds on conditional entropy (equivocation) $H(X|Y)$ plotted for different sizes of signal space $|\mathcal{X}|$. $H(X|Y)$ is bounded from above by $H(X|\hat{X})$ (blue points). Our novel lower bound (Eq. \eqref{eq_bound_equivocation}) is in orange and the bound by Kovalevsky (Eq. \eqref{eq_feder_merhav}) in green. Our bound \eqref{eq_bound_equivocation} is the tightest possible given the confusion matrix.}
\label{fig_simpleCalc}
\end{figure}

\section{Proof of the bound}

We offer two alternative proofs of the bound here. The first proves it as a simple consequence of the bound \eqref{eq_feder_merhav} by Kovalevsky. It is short, but it leaves open the question of tightness. We therefore focus on the second proof, which is self-contained, implies tightness and perhaps offers additional insights, since it includes a derivation of the distribution of channel outputs $p(y|\hat{x})$, $p(x|y)$ that minimizes $H(X|Y)$.

Throughout the proofs, the spaces of possible values of $X$ and $Y$ are written as $\mathcal{X}$ and $\mathcal{Y}$ respectively. The decoding function is denoted $g : \mathcal{Y} \rightarrow \mathcal{X}$ and is based on the maximum a posteriori rule, $g(y) \in \underset{x}{\operatorname{argmax}} \, p(x|y)$. Finally, $\mathcal{Y}_{\hat{x}} = \{ y \in \mathcal{Y} \, | \, g(y) = \hat{x} \}$ is the set of all $y$ that decode into $\hat{x}$.

\subsection{A quick proof of inequality following Kovalevsky's bound} \label{sec_quick_proof}

The left hand side of \eqref{eq_bound_equivocation}, the equivocation $H(X|Y)$ can be written as
\begin{equation}
H(X|Y) = \sum_{\hat{x}} p(\hat{x}) \int_{\mathcal{Y}_{\hat{x}}} H(X|Y=y) \,dp(y|\hat{x}),
\end{equation}
where the term $\int_{\mathcal{Y}_{\hat{x}}} H(X|Y=y) \,dp(y|\hat{x})$ is the entropy of $X$ conditional on $Y$, but with the values of $Y$ only limited to $\mathcal{Y}_{\hat{x}}$. Since it has the form of conditional entropy, we can use the Kovalevsky bound \eqref{eq_feder_merhav} and obtain our result \eqref{eq_bound_equivocation}.

This establishes the inequality in our bound, but it does not tell us if equality can be achieved -- and if it can, for what distribution of $Y$ does it happen. We address this in the following section.

\section{Proof by minimization of equivocation}
\label{sec_our_proof}

For simplicity, we formulate the derivation for discrete $Y$. However, as we comment in Sec. \ref{sec_discussion}, the derivation applies to continuous $Y$ with only minor modifications.

For clarity, let us state the minimization problem we are solving. We minimize
\begin{align}
H(X|Y) = \sum_{\hat{x}} p(\hat{x}) \sum_{y \in \mathcal{Y}_{\hat{x}}} p(y|\hat{x}) H(X|Y=y) \label{eq_objective_function_full}
\end{align}
with respect to $p(y|\hat{x})$ and $p(x|y)$, with the constraints given by the confusion matrix and maximum a posteriori decoding:
\begin{align}
\forall x, \hat{x}: \qquad & \sum_y p(x|y) p(y|\hat{x}) = p(x|\hat{x}), \label{eq_constraints_conf} \\
\forall \hat{x}, \forall y \in \mathcal{Y}_{\hat{x}}: \qquad & \hat{x} \in \underset{x}{\operatorname{argmax}} \, p(x|y). \label{eq_constraints_dec}
\end{align}

Note in \eqref{eq_objective_function_full} that the minimization can be done separately for each $\hat{x}$, since the corresponding $\mathcal{Y}_{\hat{x}}$ are disjoint. Hence we have $|\mathcal{X}|$ independent minimization problems with the objective function
\begin{equation}
\sum_{\mathcal{Y}_{\hat{x}}} p(y|\hat{x}) H(X|Y=y). \label{eq_objective_function}
\end{equation}

Note also that we do not have any constraint on $|\mathcal{Y}|$, the number of elements of $\mathcal{Y}$. We actually exploit this flexibility in the proof. However, it turns out (see Propositions 1 and 2) that when the minimum is achieved, there can be only a limited number of $y$ values with different distribution $p(x|y)$.

Our approach is based on update rules for $p(y|\hat{x})$ and $p(x|y)$ that decrease the objective function \eqref{eq_objective_function} while respecting the constraints \eqref{eq_constraints_conf}, \eqref{eq_constraints_dec}. In fact, the updates also change $|Y|$. The minimum of $H(X|Y)$ is achieved when the update rules can no longer be used to decrease it -- and such situations can be characterized and the corresponding $H(X|Y)$ can be calculated.

It is instructive to have in mind the following visualization of our minimization problem, which we use to illustrate the update rules in Fig. \ref{fig_updateRules}. The distribution $p(x,y|\hat{x})$ for some $\hat{x}$, with $y$ restricted to $y \in \mathcal{Y}_{\hat{x}}$ can be represented as a matrix, with a row for each $x$ and a column for each $y$. Normalized columns correspond to $p(x|y)$ and the sum of each column is $p(y|\hat{x})$. The constraint \eqref{eq_constraints_conf} means that each row has a fixed sum, $p(x|\hat{x})$, and the constraint \eqref{eq_constraints_dec} means that one row (e.g. the first) contains the dominant elements of all columns. The objective function \eqref{eq_objective_function} is a weighted sum of entropies of all columns. Our minimization will consist of adding and removing columns, and moving probability mass within rows.

In the following, a probability distribution is called \emph{flat} if all non-zero elements are equal, i.e. there are $n$ non-zero elements and all have probabilities $1/n$. The number $n$ is called its \emph{length}.

\subsection*{Proposition 1: equivocation minimized by flat $p(x|y)$}
\label{subsec_proposition1}

The minimum of the objective function \eqref{eq_objective_function}, given constraints \eqref{eq_constraints_conf}, \eqref{eq_constraints_dec} can only be achieved when the distributions $p(x|y)$ are flat for all $y$.

\begin{IEEEproof}
Suppose that there is a channel output $y'$ with a non-flat distribution $p(x|y')$. Then, the following update rule, illustrated in Fig. \ref{fig_updateRules} (A), will decrease the objective function \eqref{eq_objective_function}.

We label the elements of $\mathcal{X}$ as $x_1, x_2, \dots, x_{|\mathcal{X}|}$ such that
\begin{equation}
p(x_1|y') \geq p(x_2|y') \geq \dots \geq p(x_{|\mathcal{X}|}|y') \geq 0,
\end{equation}
where at least two of the inequalities are sharp (otherwise $p(x|y')$ would be flat). Note that $x_1$ must be the decode of $y'$, i.e. $g(y') = \hat{x} = x_1$. The proposed update is to replace $y'$ by $y'_1, y'_2, \dots, y'_{|\mathcal{X}|}$ with flat distributions $p(x|y'_i)$,
\begin{align}
p(x_j|y'_i) &= \begin{cases}
1/i; & j \leq i \\
0; & j > i,
\end{cases} \label{eq_prop1_xy} \\
p(y'_i|\hat{x}) &= \begin{cases}
i p(y'|\hat{x})  \left( p(x_i|y') - p(x_{i+1}|y') \right); & i < |\mathcal{X}| \\
i p(y'|\hat{x}) \, p(x_i|y'); & i = |\mathcal{X}|.
\end{cases}  \label{eq_prop1_yx}
\end{align}
Intuitively, this replaces $y'$ by multiple elements $y'_i$ with flat distributions $p(x|y'_i)$ covering the first $1, 2, \dots, |\mathcal{X}|$ elements of the ordered $x_1, x_2, \dots, x_{|\mathcal{X}|}$. It can be confirmed that this replacement respects the constraints \eqref{eq_constraints_conf}. All $y'_i$ still decode into $\hat{x} = x_1$, and the probability associated with $y'$ is merely divided among the elements $y'_i$,
\begin{align}
\sum_i p(y'_i|\hat{x}) &= p(y'|\hat{x}), \label{eq_prop1_argument1} \\
\sum_i p(x_j|y'_i) p(y'_i|\hat{x}) &= p(x_j|y') p(y'|\hat{x}). \label{eq_prop1_argument2}
\end{align}
See Fig. \ref{fig_updateRules} for an example.

Now we look at the change in the objective function \eqref{eq_objective_function} induced by this replacement. Before the replacement, $y'$ contributes the amount
\begin{equation}
p(y'|\hat{x}) H(X|Y=y'), \label{eq_prop1_oldH}
\end{equation}
where $H(X|Y=y')$ is the entropy of a single random variable with distribution $p(x|y')$. After the replacement, the total contribution of all $y'_1, y'_2, \dots, y'_{|\mathcal{X}|}$ is
\begin{multline}
\sum_i p(y'_i|\hat{x}) H(X|Y=y'_i) = \\
= p(y'|\hat{x}) \sum_i \frac{p(y'_i|\hat{x})}{p(y'|\hat{x})} H(X|Y=y'_i), \label{eq_prop1_newH}
\end{multline}
where the latter sum has the form of a conditional entropy of a variable with a marginal distribution $p(x|y')$, conditioned on the value of $y'_i$ distributed according to $p(y'_i|\hat{x})/p(y'|\hat{x})$ (this follows from Eq. \eqref{eq_prop1_argument1}, \eqref{eq_prop1_argument2}). Since conditioning decreases entropy, our replacement decreases the objective function \eqref{eq_objective_function}.

The only case when our the proposed replacement cannot be used to decrease the objective function is when $p(x|y)$ is flat for all $y$. Therefore flat $p(x|y)$ must be a characteristic of any solution to our minimization problem.
\end{IEEEproof}

Note that there are only $2^{|\mathcal{X}|-1}$ different possible flat distributions $p(x|y)$ with nonzero $p(X=\hat{x}|y)$, which means that we need at most $2^{|\mathcal{X}|-1}$ elements in $\mathcal{Y}_{\hat{x}}$ to achieve the minimum equivocation. However, as the following proposition will show, there are further restrictions on $p(x|y)$ at the minimum.

Reflecting that only flat $p(x|y)$ are of further interest in the minimization, we say that the channel output $y$ has length $l$ if $p(x|y)$ has length $l$.

\begin{figure}[!t]
\centering
\includegraphics[width=2.8in]{updateRule_1-eps-converted-to.pdf}\\
\includegraphics[width=2.8in]{updateRule_2v2-eps-converted-to.pdf}
\caption{Illustrations of the update rules used to prove (A) Proposition 1 and (B) Proposition 2. Displayed is the joint distribution $p(x,y|\hat{x})$.
(A) A channel output $y'$ with a non-flat distribution $p(x|y')$ is replaced by $y'_1, y'_2, \dots, y'_{4}$ with flat distributions $p(x|y'_i)$, such that $y'_1, y'_2, \dots, y'_{4}$ still decode into $x_1$ and the confusion matrix is not affected. This replacement decreases $H(X|Y)$, our objective function. The elements of $\mathcal{X}$ are labeled in decreasing order of $p(x,y|\hat{x})$.
(B) Two channel outputs, $y_1$ and $y_2$, have flat distributions $p(x|y_{1,2})$ with $3$ and $1$ nonzero elements respectively. We replace $y_1$ by $\bar{y}_1$ and $\bar{\bar{y}}_1$, and then transfer probability $p(x_2,\bar{y}_1|\hat{x})$ to $p(x_2,y_2|\hat{x})$ (dotted red arrow). The distributions $p(x|\bar{y}_1)$, $p(x|\bar{\bar{y}}_1)$ and $p(x|y_{2})$ remain flat, and the objective function $H(X|Y)$ is decreased.
}
\label{fig_updateRules}
\end{figure}

\subsection*{Proposition 2: minimization restricts lengths of $p(x|y)$ }
\label{subsec_proposition2}

Building on Proposition 1, we further claim that if equivocation is minimized, no two channel outputs $y_1,y_2 \in \mathcal{Y}_{\hat{x}}$ can have lengths differing by more than $1$.

\begin{IEEEproof}
As before, we introduce an update rule. Recalling the visualization with a column for each $y$, this update rule will move a nonzero element from a longer column to a shorter column, as shown in Fig. \ref{fig_updateRules} (B).

Take two elements $y_1,y_2 \in \mathcal{Y}_{\hat{x}}$ that have flat distributions $p(x|y_1)$ and $p(x|y_2)$ with lengths $a$ and $b$ respectively where $a>b$. Assume that $a$ and $b$ differ by more than one, $a - b > 1$. This means that we can choose an element $x' \in \mathcal{X}$ such that $p(x'|y_1) = 1/a$ and $p(x'|y_2) = 0$. Assume momentarily that $p(y_1|\hat{x})/a = p(y_2|\hat{x})/b$ (we will relax this assumption later). Then we can replace $y_1,y_2$ by $y'_1$ and $y'_2$, such that
\begin{itemize}
\item $p(x|y'_1)$ is flat with length $a-1$. It is nonzero for the same $x$ as $p(x|y_1)$, except for $x'$ where it is zero.
\item $p(x|y'_2)$ is flat with length $b+1$. It is nonzero for the same $x$ as $p(x|y_2)$, and also for $x'$.
\end{itemize}
Given that $p(y_1|\hat{x})/a = p(y_2|\hat{x})/b$, we can also choose the probabilities $p(y'_1|\hat{x})$ and $p(y'_2|\hat{x})$ such that $y'_1$, $y'_2$ contribute the same amount to $p(x|\hat{x}) = \sum_{y} p(x|y) p(y|\hat{x})$ as $y_1$ and $y_2$ did, ensuring that constraints \eqref{eq_constraints_conf} are respected:
\begin{align}
p(y'_1|\hat{x}) &= \frac{a-1}{a} p(y_1|\hat{x}), \\
p(y'_2|\hat{x}) &= \frac{b+1}{b} p(y_2|\hat{x}).
\end{align}

Now we show that the proposed replacement reduces the objective function. Before the replacement, the contribution to the objective function \eqref{eq_objective_function} by $y_1$ and $y_2$ was
\begin{equation}
p(y_1|\hat{x}) \log{a} + p(y_2|\hat{x}) \log{b}.
\end{equation}
After the replacement, $y'_1$ and $y'_2$ contribute by
\begin{equation}
\frac{a-1}{a} p(y_1|\hat{x}) \log{(a-1)} + \frac{b+1}{b} p(y_2|\hat{x}) \log{(b+1)}.
\end{equation}
The difference of these contributions $\Delta$, after minus before replacement, has the form
\begin{equation}
\Delta = \frac{p(y_1|\hat{x})} {a} \left(   f(b+1) - f(a) \right),
\end{equation}
where $f(t) = t \log{t} - (t-1) \log{(t-1)}$ is an increasing function for $t\geq 1$. Since $b+1 < a$, we have $\Delta < 0$, meaning that the objective function is reduced.

This update rule is applicable to any $y_1,y_2 \in \mathcal{Y}_{\hat{x}}$ with lengths $a$ and $b$ such that $a-b>1$ respectively. We have, however, further required that $p(y_1|\hat{x})/a = p(y_2|\hat{x})/b$. This requirement can be avoided. If $p(y_1|\hat{x})/a > p(y_2|\hat{x})/b$, we first split $y_1$ into $\bar{y}_1$ and $\bar{\bar{y}}_1$ with
\begin{align}
p(\bar{y}_1|\hat{x}) &= a \, p(y_2|\hat{x})/b, \\
p(\bar{\bar{y}}_1|\hat{x}) &= p(y_1|\hat{x}) - a \, p(y_2|\hat{x})/b, \\
p(x|\bar{y}_1) &= p(x|\bar{\bar{y}}_1) = p(x|y_1),
\end{align}
such that the above mentioned update rule can be applied to $\bar{y}_1$ and $y_2$ while $\bar{\bar{y}}_1$ is left unchanged, see Fig. \ref{fig_updateRules} (B). If $p(y_1|\hat{x})/a < p(y_2|\hat{x})/b$, we can proceed analogously by splitting $y_2$.

We can decrease the objective function by repeatedly applying this generalized update rule. Therefore, the minimum can only be achieved when the lengths of $p(x|y)$ for $y \in \mathcal{Y}_{\hat{x}}$ vary by no more than 1.
\end{IEEEproof}

Note that by repeated application of this update rule, in a finite number of steps we reach a state with only up to two lengths (per $\hat{x}$) that differ by at most 1. As shown in the next section, such a state implies a specific value of $H(X|Y)$. Together with the update rule in the proof of Proposition 1, this gives us an algorithm to find the distributions $p(y|\hat{x})$ and $p(x|y)$ that achieves the minimum $H(X|Y)$. The algorithm can start from an arbitrary initialization of $p(y|\hat{x})$ and $p(x|y)$ that follows the constraints \eqref{eq_constraints_conf}, \eqref{eq_constraints_dec} and finishes in a finite number of steps.

It remains to be determined what are the (at most two) allowed lengths of $y \in \mathcal{Y}_{\hat{x}}$ and how the elements $y$ with these lengths contribute to the equivocation $H(X|Y)$.

\subsection*{Admissible lengths of $p(x|y)$}

Let us call the two admissible lengths $l_{\hat{x}}$ and $l_{\hat{x}}+1$. Given $\hat{x}$, the total probability of all $y \in \mathcal{Y}_{\hat{x}}$ with length $l_{\hat{x}}$ is $\alpha_{\hat{x}}$, and those of length $l_{\hat{x}}+1$ have probability $1-\alpha_{\hat{x}}$. Then from the constraint \eqref{eq_constraints_conf}, we can write the probability that $\hat{x}$ is the correct decode
\begin{equation}
1-\epsilon_{\hat{x}} = \frac{\alpha_{\hat{x}}}{ l_{\hat{x}} }+ \frac{1-\alpha_{\hat{x}}} {l_{\hat{x}}+1}, \label{eq_alpha_length}
\end{equation}
from which we can deduce that $\frac{1} {l_{\hat{x}}+1} \leq 1-\epsilon_{\hat{x}} \leq \frac{1}{ l_{\hat{x}} }$, and that the two admissible lengths must be
\begin{equation}
l_{\hat{x}} = \left\lfloor \frac{1}{1- \epsilon_{\hat{x}}} \right\rfloor
\text{ and } l_{\hat{x}} + 1 = \left\lceil \frac{1}{1- \epsilon_{\hat{x}}} \right\rceil, \label{eq_result_lengths}
\end{equation}
unless $\frac{1}{1- \epsilon_{\hat{x}}}$ is an integer -- in that case the floor and ceiling coincide into a single admissible length.

Now, from equations \eqref{eq_alpha_length} and \eqref{eq_result_lengths} we can determine that
\begin{equation}
\alpha_{\hat{x}} = \left\lfloor \frac{1}{1- \epsilon_{\hat{x}}} \right\rfloor \left( (1-\epsilon_{\hat{x}}) \left\lceil \frac{1}{1- \epsilon_{\hat{x}}} \right\rceil -1  \right) = \alpha(\epsilon_{\hat{x}}) \label{eq_result_alpha}
\end{equation}
is the total probability (given $\hat{x}$) of $y \in \mathcal{Y}_{\hat{x}}$ with length $\lfloor \frac{1}{1- \epsilon_{\hat{x}}} \rfloor$.

Finally, the minimal value of equivocation is simply
\begin{equation}
H(X|Y) \geq \sum_{\hat{x}} p(\hat{x}) \left( \alpha_{\hat{x}} \log{l_{\hat{x}}} + (1-\alpha_{\hat{x}}) \log{(l_{\hat{x}} + 1)} \right),
\end{equation}
which together with equations \eqref{eq_result_lengths} and \eqref{eq_result_alpha} constitutes our main bound, as stated in \eqref{eq_bound_equivocation}.

\section{Discussion} \label{sec_discussion}

We have introduced a tight lower bound on equivocation in terms of the maximum a posteriori confusion matrix, and proved it in two ways. The first is a proof of the inequality, starting from a similar bound by Kovalevsky \cite{Kovalevsky1968}, but it does not prove that the bound is tight. Therefore, we developed a second proof, in which we construct the distribution of channel outputs that minimizes the equivocation and achieves equality in our bound.

Central to the latter approach are two update rules for the distribution of the channel outputs. These update rules exploit the fact that equivocation can be, under our constraints, minimized by (1) making the posterior distributions $p(x|y)$ flat and (2) making sure that these flat distributions contain similar numbers of nonzero elements.

We formulated the proof for discrete random variables $X$ and $Y$, but it can be extended. If $X$ is discrete but $Y$ continuous, application of a modified version of the first update rule would result in $2^{|\mathcal{X}|}$ regions in the $\mathcal{Y}_{\hat{x}}$ space corresponding to each of the $2^{|\mathcal{X}|}$ possible flat distributions $p(x|y')$. For example, the region associated with a flat distribution of length $|\mathcal{X}|$, that is $p(x|y') = 1/|\mathcal{X}|$, would have a total probability $\int_{\mathcal{Y}_{\hat{x}}} |\mathcal{X}| \min_x{p(x|y)} dp(y|\hat{x})$. These subsets of $\mathcal{Y}$ where $p(x|y)$ is constant can then be treated like discrete values, and the rest of our derivation applies.

Bounds on equivocation (or mutual information) in terms of the confusion matrix are, to our knowledge, not common -- despite their relevance for estimation of mutual information. We hope that our result can be useful for these purposes, and that it sheds some light on the gap between mutual information before and after decoding. However, its applicability is restricted by the assumption of maximum a posteriori decoding, and relaxing this assumption remains an interesting challenge.

\section*{Acknowledgment}

The authors would like to thank Sarah A. Cepeda-Humerez for helpful discussions.


\section{Introduction}
Markov Chain Monte Carlo (MCMC) methods have been widely applied  in many areas for more than 40 years \citep{hastings70monte}. In particular, they are successful when the target distribution $\pi$ is available only up to a normalizing constant. To sample from such a distribution, various MCMC algorithms have been developed. A typical MCMC algorithm is designed using a transition kernel $P$  that  has $\pi$ as stationary distribution.
See for example \citet{meyn93markov,robert04monte,roberts04general}, and the references therein.

Constructing a transition kernel to sample efficiently from a given distribution, although conceptually easy, is rather difficult in practice. The difficulty is that generic algorithms such as the Metropolis--Hastings algorithm requires a careful choice and tuning of the proposal kernel.
The development of adaptive MCMC (AMCMC) methods stems partly from these difficulties. Instead of having a fixed Markov kernel $P$, at each round $n$ an AMCMC algorithm selects a kernel $P_{\what\theta_n}$ from a family of Markov kernels $\{P_\theta\}_{\theta\in\Theta}$, where the value (parameter) $\what\theta_n$ is computed based on possibly all the samples generated up to time $n$, so that the transition kernel is automatically self-adapted. This approach is very appealing in practice, as it frees the users from parameter tuning, and provides a better exploration-exploitation balance in the performance.
As a consequence, AMCMC algorithms often yield smaller asymptotic errors in Monte Carlo estimations. The theoretical and methodological studies of AMCMC have drawn attentions of many researchers lately. See for example the survey by \citet{atchade11adaptive} and the references therein.

In this paper, we investigate the convergence rates of two AMCMC algorithms: the {\it Importance Resampling MCMC (IRMCMC)}  algorithm introduced by \citet{atchade09resampling}, and the {\it Equi-Energy (EE) sampler} by \citet{kou06equi}. The IRMCMC algorithm is also referred to {\it interacting annealing} algorithm \citep{bercu12fluctuations}. For the EE sampler, we actually focus on a simplified version, which is sometimes referred to as {\it interacting tempering} algorithm \citep{fort11central}.

Throughout the paper we denote by $\{X_n\}_{n\in{\mathbb N}}$ the random process generated by either of these algorithms. A common feature is that in either case,  the dynamics of $\{X_n\}_{n\in{\mathbb N}}$ is driven by a sequence of random measures $\what\theta_n$ computed from an auxiliary chain $\{Y_n\}_{n\in{\mathbb N}}$. Most of the theoretical results available so far focused on  the convergence of marginal distributions
\[
{\cal L}_{X_n}\Rightarrow\pi,
\] and on the law of large numbers:
\[
\lim_{n\to\infty}\frac1n \summ i1n f(X_i) = \pi(f) \mbox{ almost surely.}
\]
See for example \citet{andrieu08note,andrieu11nonlinear}, \citet{atchade09resampling,atchade10cautionary} and \citet{fort11convergence} (there is a mistake in the proof of \citep{atchade10cautionary}, pointed out in \citep{fort11convergence}).
Central limit theorems for such AMCMC algorithms have only been considered recently by \citet{fort11central} and \citet{bercu09functional,bercu12fluctuations}. In short, introducing the auxiliary chain makes the stochastic process no longer Markov, which raises considerable technical difficulties. We point out that there are other classes of AMCMC algorithms, for which the parameters take values in finite dimensional spaces (e.g.~the adaptive Metropolis algorithm introduced by~\citet{haario99adaptive}). The analysis of such algorithms is out of the scope of this paper.

In this paper, we study the {\it convergence rate} (or mixing time) of the IRMCMC and EE algorithms. That is, we provide upper bounds on the distances between ${\cal L}_{X_n}$ (the distribution of $X_n$) and the target distribution. This type of rate differs and complements rate of convergence obtained in central limit theorems. Mixing time results provide information on the burn-in time of the algorithm, whereas central limit theorems (such as those mentioned above)  deal with the rate of convergence and fluctuations of Monte Carlo averages.
Beside \citet{andrieu07efficiency} who considered AMCMC with a finite-dimensional parameter, and to the best of our knowledge, the mixing time of  AMCMC has not been investigated so far.

We show that the IRMCMC algorithm has convergence rate of order $O(n^{-1})$. In particular, we also provide a simple example, for which the convergence rate has lower bound $1/n$. That is, one should not expect a rate  superior to $O(n^{-1})$ in general. We also show that for $m$-tuple IRMCMC (to be defined in section~\ref{sec:mIRMCMC}), the convergence rate is also within $O(n^{-1})$.
For the EE sampler, under some regularity conditions, we show that the rate of convergence is $O(n^{-1/2})$ in terms of a slightly weaker norm than the total variation distance.  Our results are qualitative, in that the constants in the rates are not explicit. But they clarify what is known about these algorithms, and suggest in particular that longer  burn-in should be selected for the EE sampler.

The rest of the paper is organized as follows. The remaining of the introduction gives a general description of the algorithms considered in the paper and introduces some notation. Section~\ref{sec:IRMCMC} is devoted to IRMCMC algorithm. The convergence rate is established in Section~\ref{sec:IRMCMCrate}, and for multiple IRMCMC in Section~\ref{sec:mIRMCMC}. Section~\ref{sec:EE} is devoted to EE sampler. The convergence rate is established in Section~\ref{sec:EErate}. A remark on the connection to parallel tempering is provided in Section~\ref{sec:mEE}.

\subsection{A class of AMCMC algorithms}
We describe the general framework of AMCMC considered in this paper.
Let ${\cal X}$ denote a general state space. An AMCMC algorithm  is a stochastic process $\{(X_n,Y_n)\}_{n\geq 0}$ in ${\cal X}\times{\cal X}$, designed such that the main chain $X_n$ converges to the target distribution $\pi$ in a certain sense to be described precisely later. Let ${\cal P}$ denotes the set of all probability measures on ${\cal X}$, and $\{K_\theta,\;\theta\in{\cal P}\}$ a set of transition kernels on ${\cal X}$.  Let $P$ be a transition kernel on ${\cal X}$ with invariant probability measure $\pi$. Starting from $P$ and $\{K_\theta,\theta\in{\cal P}\}$, we consider the family of transition kernel $P_\theta$ given by
\[
P_\theta(x,\cdot) = (1-\epsilon)P(x,\cdot) + \epsilon K_{\theta}(x,\cdot),\;\;\theta\in{\cal P},\;x\in{\cal X}.
\]

The dynamics of the AMCMC algorithms considered in this paper can be unified as follows: given ${\cal F}_n = \sigma(X_0,\dots,X_n,Y_0,\dots,Y_n)$, $X_{n+1}$ and $Y_{n+1}$ are conditionally independent, and for all bounded measurable functions $h:{\cal X}\to{\mathbb R}$,
\[{\mathbb E}_x \left(h(X_{n+1})\mid {\cal F}_n,\,\{Y_k\}_{k\geq n+1}\right)={\mathbb E}_x \left(h(X_{n+1})\mid {\cal F}_n\right)= P_{\wt\theta_n}h(X_n),  \mbox{ almost surely}
\]
where $\wt \theta_n(\cdot)\ensuremath{\stackrel{\mathrm{def}}{=}} n^{-1}\sum_{j=1}^n \delta_{Y_j}(\cdot)$, denotes the empirical measure associated to the auxiliary chain $\{Y_n\}_{n\geq 0}$. Each algorithm is determined by the choice of the kernels $K_\theta$. For the IRMCMC,
\[
K_\theta(x,A)=\frac{\int_A w(z)\theta({\rm d} z)}{\int_{{\cal X}}w(z){\rm d} z},
\]
where $w(z)={{\rm d}\pi}/{{\rm d}\pi_Y}(z)$ (see Section~\ref{sec:notation} for our convention on $\pi$ and ${\rm d}\pi$), while for the EE, the following choice is made
\[
K_\theta(x,A)=\textbf{1}_A(x) + \int_{{\cal X}}\left(1\wedge\frac{\pi(z)\pi_Y(x)}{\pi(x)\pi_Y(z)}\right)\left(\textbf{1}_A(z)-\textbf{1}_A(x)\right)\theta({\rm d} z).
\]
In both cases,  $\pi_Y$ is an auxiliary distribution chosen such that it is relatively close to $\pi$, and  easy to sample from (at least easier than $\pi$). We assume that the evolution of the auxiliary train is independent of the main chain in the sense that for bounded measurable function $h:{\cal X}\to{\mathbb R}$,
\[
{\mathbb E}(h(Y_{n+1})\mid {\cal F}_n) = {\mathbb E}(h(Y_{n+1})\mid Y_0,\dots,Y_n), \mbox{ almost surely.}
\]
The description of the dynamics of the joint process $\{(X_n,Y_n)\}_{n\geq 0}$ is completed by specifying the dynamics of $Y_{n+1}\mid \F_n$, which is either a Markov chain with target distribution $\pi_Y$, or the main chain of another adaptive MCMC with target distribution $\pi_Y$, not necessarily Markov.

The rationale of these algorithms can be viewed as follows. For $\theta=\theta_\star=\pi_Y$, the Markov chain with transition kernel $P_{\theta_\star}$ have nice mixing properties, due to the choice of $\pi_Y$. Unfortunately, however, it is not possible to implement directly the kernel $P_{\theta_\star}$. The idea here therefore is (i) to run an auxiliary chain $\{Y_n\}_{n\geq 0}$ with limiting distribution $\pi_Y$, so that the  empirical measure $\wt \theta_n$  converges to $\theta_\star$, and (ii) to sample $X_{n+1}$ from
\[
P_{\wt\theta_n}(X_n,\cdot) = (1-\epsilon)P(X_n,\cdot) + \epsilon K_{\wt\theta_n}(X_n,\cdot)\,,
\]
which approximates $P_{\theta_\star}(X_n,\cdot)$ as $n\to\infty$.

\subsection{Notation}\label{sec:notation}
We assume that the state space ${\cal X}$ is a Polish space equipped with a metric $\textsf{d}$, and ${\cal B}$ is the associated Borel $\sigma$-algebra. In addition, $({\cal X},{\mathbb B})$ is a measure space with a reference $\sigma$-finite measure, which we denote for short by ${\rm d} x$. Let $\pi$ and $\pi_Y$ be  probability measures on $({\cal X},{\mathbb B})$.
We assume that $\pi$ and $\pi_Y$ are both absolutely continuous with respect to ${\rm d} x$ and with a little abuse of notation, we also use $\pi$ and $\pi_Y$ to denote the density respectively. That is, we write $\pi({\rm d} x) = \pi(x){\rm d} x$ and similarly for $\pi_Y$. For a transition kernel $Q$, a measure $\nu$ and a function $h$, we shall write $\nu Q(\cdot)\ensuremath{\stackrel{\mathrm{def}}{=}} \int\nu({\rm d} z)Q(z,\cdot)$, and $Qh(\cdot)\ensuremath{\stackrel{\mathrm{def}}{=}} \int Q(\cdot,{\rm d} z)h(z)$.  We denote $\what\pi_{Y,n}$ the empirical measure associated to  the auxiliary chain $\indn Y$ defined by
\[
\what \pi_{Y,n}(\cdot) \ensuremath{\stackrel{\mathrm{def}}{=}} \frac1n\summ i1n\delta_{Y_i}(\cdot)\,.
\]
At times, we also use the notation $\wt \theta_n(\cdot)$ to denote $\what \pi_{Y,n}(\cdot)$. For functions $f:{\cal X}\to{\mathbb R}$, we write
\[
\what\pi_{Y,n}(\wb f) \ensuremath{\stackrel{\mathrm{def}}{=}} \what\pi_{Y,n}(f) - \pi_Y(f).
\]
We let $C$ denote general constants that do not depend on $n$, but may change from line to line.

\newpage
\section{Importance Resampling MCMC}\label{sec:IRMCMC}
We consider the  {importance-resampling Markov Chain Monte Carlo} method described in \citet{atchade09resampling}.

\begin{Algo}[IRMCMC]\label{algo:IRMCMC}
Fix $\epsilon\in(0,1)$. Pick arbitrary $X_0 = x_0$ and $Y_0 = y_0$. Let $P$ be an arbitrary Markov kernel with stationary distribution $\pi$. At each round $n$,  $X_n$ and $Y_n$ are conditionally independent given $\F_{n-1}$, and
\[
X_{n}\mid \F_{n-1} \sim \left\{
\begin{array}{l@{\mbox{ w.p.~}}l}
P(X_{n-1},\cdot) & 1-\epsilon\,,\\
\what\theta_{n-1}(\cdot) &  \epsilon\,,
\end{array}\right.
\]
where $\what\theta_n$ is the (randomly) weighted empirical distribution defined by
\begin{equation}\label{eq:theta}
\what\theta_n(\cdot) = \summ i1{n}\frac{\wt w(Y_i)}{\summ j1{n}\wt w(Y_j)}\delta_{Y_i}(\cdot)=\frac{\int_\cdot \wt w(z)\what\pi_{Y,n}({\rm d} z)}{\int_{{\cal X}} \wt w(z)\what\pi_{Y,n}({\rm d} z)},
\end{equation}
with $\wt w(y) \propto \pi(y)/\pi_Y(y)=: w(y)$, and $\what\theta_0 = \delta_{y_0}$. We assume $|w|_\infty\ensuremath{\stackrel{\mathrm{def}}{=}}\sup_{x\in{\cal X}}|w(x)|<\infty$.
\end{Algo}
\begin{Rem}\label{rem:w}
The assumption on the boundedness of $w$ is not too restrict. Indeed, very often in practice, we have $\wt\pi$, the un-normalized density function of $\pi$ as a bounded function, and set the auxiliary chain with stationary distribution $\wt\pi_Y\propto\pi_Y$ obtained by $\wt\pi_Y = \wt\pi^T$ with $T\in(0,1)$. In this case, $\wt w = \wt \pi/\wt \pi_Y$ is bounded and thus so is $w$.
\end{Rem}

Equivalently, for any bounded function $f:{\cal X}\to{\mathbb R}$,
\[
{\mathbb E} (f(X_{n+1})\mid {\cal F}_n) = P_{\what\theta_n}f(X_n) \mbox{ almost surely, }
\]
where for all probability measures $\theta$ on ${\cal X}$, $P_\theta(x,\cdot)$ is defined by
\begin{equation}\label{eq:Ptheta}
P_\theta(x,\cdot) = (1-\epsilon)P(x,\cdot) + \epsilon \theta(\cdot)\,.
\end{equation}
For the time being, we make no particular assumption on the dynamics of the auxiliary chain $\{Y_n\}_{n\geq 0}$.
\subsection{Convergence rate of IRMCMC}\label{sec:IRMCMCrate}
The following equivalent representation of Algorithm~\ref{algo:IRMCMC} is useful. Let the auxiliary chain be as above. Furthermore, let $\{Z_n\}_{n\geq 0}$ be a sequence of independent and identically distributed random variables with $\mathbb P(Z_1 = 1) = 1-\mathbb P(Z_1 = 0) =  \epsilon$. Furthermore, we assume that $\{Z_n\}_{n\geq 0}$ and $\{Y_n\}_{n\geq 0}$ are independent and for each $n\geq 1$, $Z_n$ and $\F_{n-1}$ are independent.  Then, at round $n$, we can introduce $Z_n$, and write the conditional distribution of $X_n$ given $Z_n,\F_{n-1}$ as
\[
X_n\mid \F_{n-1},Z_n\sim\left\{
\begin{array}{l@{\mbox{ if }}l}
P(X_{n-1},\cdot) & Z_n = 0\\
\what\theta_{n-1}(\cdot) & Z_n = 1\,.
\end{array}
\right.
\]
Define
\begin{equation}\label{eq:tau}
\tau_0 =0, \tau_{i+1} = \min\{k>\tau_i: Z_k = 1\} \mbox{ and } n^* = \max\{k: \tau_k\leq n\}\,.
\end{equation}
Observe that at each time $\tau_k>0$, conditioning on $Y_0, Y_1,\dots,Y_{\tau_k-1}$, $X_{\tau_k}$ is sampled from $\what\theta_{\tau_k-1}$, independent of $X_0,\dots,X_{\tau_k-1}$. Furthermore, $Y_0,\dots,Y_n$ are independent from $\tau_1,\dots,\tau_{n^*}$. Therefore, we first focus on
\begin{equation}\label{eq:eta}
\eta_n\ensuremath{\stackrel{\mathrm{def}}{=}} \mathbb P(X_{n+1}\in\cdot\mid Z_{n+1}=1) = {\mathbb E}\what\theta_{n}(\cdot)\,,n\in{\mathbb N}\,.
\end{equation}
We first obtain a bound on the total variation distance $\nnTV{\eta_n-\pi}$.
Recall that, given two probability distributions $\mu$ and $\nu$, the total variation distance $\nnTV{\mu-\nu}$ is defined by:
\begin{equation}\label{eq:W}
\nnTV{\mu-\nu} = \frac12\sup_{|f|\leq 1}|\mu(f)-\nu(f)|\,.
\end{equation}
For convenience, write
\begin{equation}\label{eq:Bn}
B_n\ensuremath{\stackrel{\mathrm{def}}{=}} |w|_\infty\sup_{|f|\leq 1}{\mathbb E}\what\pi_{Y,n}(\wb f) + 2|w|_\infty^2\sup_{|f|\leq 1}{\mathbb E}^\prime{\what\pi_{Y,n}(\wb f)}^2, n\in{\mathbb N}.
\end{equation}
Recall that throughout we assume $|w|_\infty<\infty$.
\begin{Lem}\label{lem:etak} For all $n\in{\mathbb N}$,
\begin{equation}\label{eq:etanTV}
\nnTV{\eta_n-\pi}
\leq B_n.
\end{equation}
\end{Lem}
\begin{Rem}
A special case of $\nnTV{\eta_n-\pi}$, when $w\equiv 1$ (equal weights), is the so-called {\it cesaro mixing time} of $\{Y_n\}_{n\geq 0}$. See for example~\citet[Chapter 6.6]{levin09markov}.
\end{Rem}

The proof of Lemma~\ref{lem:etak} is postponed to next subsection.
We have no explicit requirement on $\pi_Y$, except that $w= \pi/\pi_Y$ is bounded. However, the convergence of $Y_1,Y_2,\dots$ to $\pi_Y$ is implicitly ensured when we require further $\sup_{|f|\leq 1}{\mathbb E}(\what\pi_{Y,n}(f) - \pi_Y(f)) + \sup_{|f|\leq 1}{\mathbb E}(\what\pi_{Y,n}(f) - \pi_Y(f))^2\to 0$ as $n\to\infty$.
Indeed, these rates yield an upper bound on the convergence rate of ${\cal L}_{X_n}\Rightarrow \pi$, as shown in the following theorem.
We set $B_0 = B_{-1} = 1$.
\begin{Thm}\label{thm:IRMCMC}
Consider $\indn X$ generated from Algorithm~\ref{algo:IRMCMC}.
Then,
\begin{equation}\label{eq:boundXn}
\nnTV{{\cal L}_{X_n}-\pi}
\leq \sum_{\ell=0}^n (1-\epsilon)^{n-\ell}B_{\ell-1}.
\end{equation}
Furthermore, for any bounded measurable function $f$,
\begin{multline}\label{eq:L2fX}
\PE\bb{\frac1{\sqrt n}\summ i1n(f(X_i)-\pi(f))}^2\\
\leq \frac{80\epsilon^{-2}|f|_\infty^2}n + 64\epsilon^{-2}|f|_\infty^2 + |f|_\infty^2^\prime{\frac1{\sqrt n}\summ 01{n-1}\sqrt{B_k}}^2, n\in{\mathbb N}.
\end{multline}

\end{Thm}
The proof of Theorem~\ref{thm:IRMCMC} is postponed to next subsection.
\begin{Rem}
The control of the total variation distance depends on the initial position of the auxiliary chain, as in general, $B_n$ depends on the initial position $Y_0 = y_0$. We omit this dependence throughout this paper. At the same time, it is obvious that the initial position $X_0 = x_0$ is irrelevant.
\end{Rem}
\begin{Rem}
In  Theorem~\ref{thm:IRMCMC}, we do not assume  any ergodicity assumption on the kernel $P$. In the case $P$ is say, geometrically ergodic, one can improve (\ref{eq:boundXn}) quantitatively by bounding the term $\nnTV{\eta_k P^{n-k}-\pi}$ more effectively. For example, if $P$ is uniformly ergodic with rate $\rho$, then (\ref{eq:boundXn}) would become
\[\nnTV{{\cal L}_{X_n}-\pi}
\leq \sum_{\ell=0}^n \left[\rho(1-\epsilon)\right]^{n-\ell}B_{\ell-1}.\]
A similar improvement can be formulated for (\ref{eq:L2fX}). However, these improvements do not change the rate but only the constant in the corollary below. Beside, such improvements will not be easily available if $P$ is sub-geometrically ergodic.
\end{Rem}
Now, as a corollary we obtain an upper bound on the convergence rate of IRMCMC algorithm, under the following assumption.
\assumpH
\item \label{A:Y}
There exist a finite constant $C$ such that for all measurable function $f:\;{\cal X}\to\mathbb R$,  with $|f|_\infty\leq 1$,
\begin{equation}\label{eq:AY}
{\mathbb E} \what\pi_{Y,n}(\wb f)\leq \frac {C}n \quad\mbox{ and }\quad \PE\bbpp{\what\pi_{Y,n}(\wb f)}^2 \leq \frac{C}n .
\end{equation}
\assumpE
\begin{Rem}
For example, if $\indn Y$ is  a Markov chain  with transition kernel $P_Y$ and stationary distribution $\pi_Y$, then the first part of~\eqref{eq:AY} holds if $P$ is geometrically ergodic \citep{roberts04general}. The second part of~\eqref{eq:AY} essentially assumes that the sample variances of $\{f(Y_n)\}_{n\in{\mathbb N}}$ is bounded, which also holds for geometrically ergodic Markov chains under appropriate moment condition on $f$ (e.g. $\pi(|f|^{2+\epsilon})<\infty$ with $\epsilon>0$). Occasionally this condition can fail, as the sample variance might become infinity in the limit. See for example~\citet{haggstrom05central} and~\citet{haggstrom07variance}.
\end{Rem}

\begin{Coro}\label{coro:1}
Consider the importance resampling MCMC (Algorithm~\ref{algo:IRMCMC}). If Assumption~\ref{A:Y} holds, then there exists a finite constant $C$ such that
\[
\nnTV{{\cal L}_{X_n}-\pi} \leq \frac {C}n\,.
\]
Furthermore for any bounded measurable function $f$,
\[\PE\bb{\frac1{\sqrt n}\summ i1n(f(X_i)-\pi(f))}^2 \leq C |f|_\infty^2\,, n\in{\mathbb N}\,.\]
\end{Coro}

\subsection{Proofs of Lemma~\ref{lem:etak} and Theorem~\ref{thm:IRMCMC}}
\begin{proof}[Proof of Lemma~\ref{lem:etak}]
 Fix $n\geq 1$ and write $\what\pi_Y \equiv\what\pi_{Y,n}$.
Rewrite $\eta_n(f)$ as,
\begin{eqnarray*}
\eta_n(f) & = & {\mathbb E}\bbpp{\summ j1{n}\frac{w(Y_j)}{\summ l1{n}w(Y_l)}f(Y_j)} \nonumber\\
& = & {\mathbb E}\bb{\frac1n\summ j1{n}w(Y_j)f(Y_j)+ \bbpp{1-\frac1n{\summ j1{n}w(Y_j)}}\summ j1{n}\frac{w(Y_j)f(Y_j)}{\summ l1{n}w(Y_l)}}\nonumber\\
& = & {\mathbb E}\bb{\what\pi_{Y,n}(wf) + (\pi_{Y}(w)-\what\pi_{Y,n}(w))\what\theta_n(f)}\,,
\end{eqnarray*}
where in the second term above we used the fact that $\pi_Y(w)=1$.
Since $\pi(f) = \pi_Y(wf)$,
\begin{multline}
 \nnTV{\eta_n-\pi} = \sup_{|f|\leq 1}\frac12\bbpp{\eta_n(f)-\pi(f)}\\
\leq \frac12\sup_{|f|\leq 1}{{\mathbb E}\what\pi_{Y,n}(\wb{wf})} + \frac12\sup_{|f|\leq 1}{{\mathbb E}^\prime{\what\pi_{Y,n}(\wb w)\what\theta_n(f)}}\\
\leq |w|_\infty{\sup_{|f|\leq 1}{{\mathbb E}\what\pi_{Y,n}(\wb f)} + \frac12\sup_{|f|\leq 1}{{\mathbb E}\bb{\what\pi_{Y,n}(\wb w)\bbpp{\what\theta_n(f)-\pi_Y(wf)}}}}.\label{eq:etan}
\end{multline}
By Cauchy--Schwarz inequality,
\begin{multline}
\sup_{|f|\leq 1}{\mathbb E}\bb{{\what \pi_{Y,n}(\wb w)}\bbpp{\what\theta_n(f)-\pi_Y(wf)}}\\
\leq \bb{{\mathbb E}\bbpp{\what \pi_{Y,n}(\wb w)}^2}^{1/2}\times \sup_{|f|\leq 1}\bb{{\mathbb E}\bbpp{\what\theta_n(f)-\pi_Y(wf)}^2}^{1/2}\,.\label{eq:etan2}
\end{multline}
The first term is bounded by $|w|_\infty\sup_{|f|\leq 1}\sbb{{\mathbb E}\spp{\what \pi_{Y,n}(\wb f)}^2}^{1/2}$. For the second term, observe that
\begin{multline}\label{eq:etan3}
{\mathbb E}\bbpp{\what\theta_n(f)-\pi_Y(wf)}^2 \\
\leq 2{\mathbb E} \bbpp{\what\theta_n(f)-\what\pi_{Y,n}(wf)}^2 + 2 {\mathbb E} \bbpp{\what\pi_{Y,n}(wf) - \pi_Y(wf)}^2,
\end{multline}
and
\begin{multline}
{\mathbb E} \bbpp{\what\theta_n(f)-\what\pi_{Y,n}(wf)}^2 = {\mathbb E} \bbpp{\summ j1{n}\frac{w(Y_j)f(Y_j)}{\summ l1{n}w(Y_l)} - \frac1n\summ j1{n}w(Y_j)f(Y_j)}^2\\
= {\mathbb E}\bb{\bbpp{1-\what\pi_{Y,n}(w)}^2\what\theta_n^2(f)}
\leq {\mathbb E}\bbpp{\pi_Y(w)-\what\pi_{Y,n}(w)}^2\\
\leq |w|_\infty^2\sup_{|g|\leq 1}{\mathbb E}\bbpp{\what \pi_{Y,n}(\wb g)}^2,\label{eq:etan4}
\end{multline}
and the above calculation holds for all $f: |f|\leq 1$. Combining~\eqref{eq:etan},~\eqref{eq:etan2},~\eqref{eq:etan3} and~\eqref{eq:etan4} yields the desired result.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:IRMCMC}]
 We recall that $\tau_{n^*}$ is the last time $k$ before $n$ that the main chain is sampled from $\what\theta_{k-1}$. Now, we can write
\begin{eqnarray}
\nnTV{{\cal L}_{X_n}-\pi} & = & \sup_{|f|\leq 1}\frac12|{\mathbb E} f(X_n) - \pi (f)|\nonumber\\
& = & \sup_{|f|\leq 1}\frac12\abs{\summ k0n{\mathbb E}(f(X_n), \tau_{n^*} = k) - \pi(f)}\nonumber\\
& = & \sup_{|f|\leq 1}\frac12\abs{\summ k0n\mathbb P(\tau_{n^*} = k)\sbb{{\mathbb E}(f(X_n)\mid \tau_{n^*} = k) - \pi(f)}}\nonumber\\
& \leq & \summ k0n\mathbb P(\tau_{n^*} = k)\sup_{|f|\leq 1}\frac12|{\mathbb E}(f(X_n)\mid \tau_{n^*} = k) - \pi(f)|.\label{eq:bound}
\end{eqnarray}
Observe that the conditional distribution of $X_n$ given that $\tau_{n^*} = k\geq 1$, is $\eta_{k-1}P^{n-k}$ (set $\eta_0=\delta_{Y_0}$). Then,
\begin{eqnarray*}
\sup_{|f|\leq 1}\frac12|{\mathbb E}(f(X_n)\mid\tau_{n^*} = k) - \pi(f)| & = &
\sup_{|f|\leq 1}\frac12|\eta_{k-1}P^{n-k}(f) - \pi(f)| \\
& = & \nnTV{\eta_{k-1}P^{n-k}-\pi}\,.
\end{eqnarray*}
By  the fact that $\pi P = \pi$, we have $\nnTV{\eta_{k-1} P^{n-k}-\pi}  \leq \nnTV{\eta_{k-1}-\pi}\leq B_{k-1}$, by Lemma \ref{lem:etak}. Also $\mathbb P(\tau_{n^*} = k)=\epsilon(1-\epsilon)^{n-k}$ for $k=1,\dots,n$ and $\mathbb P(\tau_{n^*} = 0) = (1-\epsilon)^n$. Thus,~\eqref{eq:bound} becomes \eqref{eq:boundXn}.

To establish (\ref{eq:L2fX}), we show that the partial sum $\sum_{k=1}^n \left(f(X_k)-\pi(f)\right)$ admits a well behaved martingale approximation. For a probability measure $\theta$ on ${\cal X}$, define
\[\pi_\theta(A)=\epsilon\sum_{j=0}^\infty(1-\epsilon)^j(\theta P^j)(A),\;\;A\in{\cal B}.\]
Clearly, $\pi_\theta$ is a  probability measure on $({\cal X},{\cal B})$, and in fact we have for all $A\in{\cal B}$,
\begin{multline*}
\pi_\theta P_\theta(A)=\epsilon\sum_{j=0}^\infty (1-\epsilon)^j \int(\theta P^j)({\rm d} z)\left((1-\epsilon)P(z,A)+\epsilon\theta(A)\right)\\
=\epsilon\sum_{j=0}^\infty (1-\epsilon)^{j+1}(\theta P^{j+1})(A)+\epsilon\theta(A)=\pi_\theta(A).\end{multline*}
This means that the kernel $P_\theta$ is invariant by $\pi_\theta$. It is also easy to check by induction that for any bounded measurable function $f$, and $n\geq 1$,
\begin{equation}\label{eq:ratePtheta1}
P_\theta^nf(x)-\pi_\theta(f)=(1-\epsilon)^nP^nf(x)-\epsilon\sum_{j=n}^\infty(1-\epsilon)^j(\theta P^j)f.\end{equation}
It then follows that
\begin{equation}\label{eq:ratePtheta2}
\nnTV{P_\theta^n(x,\cdot)-\pi_\theta}\leq 2(1-\epsilon)^n.
\end{equation}
As a result of (\ref{eq:ratePtheta2}), the function
\[g_\theta(x)=\sum_{j=0}^\infty \left(P_\theta^jf(x)-\pi_\theta(f)\right),\]
is well-defined with $|g_\theta|_\infty\leq 2\epsilon^{-1}|f|_\infty$, and satisfies Poisson's equation
\begin{equation}\label{eq:PoissonPtheta}
g_\theta(x)-P_\theta g_\theta(x)=f(x)-\pi_\theta(f),\;\;\;x\in{\cal X}.\end{equation}
In particular, we have $f(X_k)-\pi_{\what\theta_{k-1}}(f)=g_{\what\theta_{k-1}}(X_k)-P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_k)$, almost surely. Using this, we write:
\begin{multline*}
\sum_{k=1}^n \left(f(X_k)-\pi(f)\right)=\sum_{k=1}^n \left(\pi_{\what\theta_{k-1}}(f)-\pi(f)\right)+ \sum_{k=1}^n \left(f(X_k)-\pi_{\what\theta_{k-1}}(f)\right)\\
=\sum_{k=1}^n \left(\pi_{\what\theta_{k-1}}(f)-\pi(f)\right)+ \sum_{k=1}^n \left(g_{\what\theta_{k-1}}(X_k)-P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k-1})\right)\\
+\sum_{k=1}^n \left(P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k-1})-P_{\what\theta_{k}}g_{\what\theta_{k}}(X_{k})\right) + \sum_{k=1}^n \left(P_{\what\theta_{k}}g_{\what\theta_{k}}(X_{k})-P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k})\right).
\end{multline*}
Using (\ref{eq:PoissonPtheta}), the definition of $g_\theta$, and (\ref{eq:ratePtheta1}), it is easy to check that for any probability measures $\theta,\theta'$ and $x\in{\cal X}$,
\[P_\theta g_\theta(x)-P_{\theta'}g_{\theta'}(x)=\int(\theta'-\theta)({\rm d} z)\left(\epsilon\sum_{j=0}^\infty j(1-\epsilon)^j P^jf(z)\right).\]
this implies that the term $\sum_{k=1}^n \left(P_{\what\theta_{k}}g_{\what\theta_{k}}(X_{k})-P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k})\right)$ is a telescoping sum and we have
\begin{multline*}
\left|\sum_{k=1}^n \left(P_{\what\theta_{k}}g_{\what\theta_{k}}(X_{k})-P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k})\right)\right|\\
\leq
\abs{^\prime{\what\theta_n-\what\theta_0}^\prime{\epsilon\sif j0j(1-\epsilon)^jP^jf}}
\leq \frac{2(1-\epsilon)}\epsilon|f|_\infty.
\end{multline*}
The term $\sum_{k=1}^n \left(P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k-1})-P_{\what\theta_{k}}g_{\what\theta_{k}}(X_{k})\right)$ is also a telescoping sum and we have
\[
\left|\sum_{k=1}^n \left(P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k-1})-P_{\what\theta_{k}}g_{\what\theta_{k}}(X_{k})\right)\right|\leq 4\epsilon^{-1}|f|_\infty.
\]
From the definition of $\pi_\theta$, notice that we can write
\[\sum_{k=1}^n \left(\pi_{\what\theta_{k-1}}(f)-\pi(f)\right)=\sum_{k=1}^n \what\theta_{k-1}(f_\epsilon-\pi(f_\epsilon)),\]
where $f_\epsilon(x)=\epsilon\sum_{j=0}^\infty(1-\epsilon)^j P^j f(x)$.
Thus,
\begin{multline*}
{\mathbb E}\bb{\summ k1n^\prime{\pi_{\what\theta_{k-1}}(f)-\pi(f)}}^2 \\
\leq
^\prime{\summ k1n{\mathbb E}^{1/2}\what\theta_{k-1}^2(f_\epsilon - \pi(f_\epsilon))}^2
\leq |f|_\infty^2^\prime{\summ k0{n-1}\sqrt{B_k}}^2,
\end{multline*}
where in the last equality, we use the fact that  $\sup_{|f|_\infty\leq 1}\PE\what\theta_k^2(f-\pi(f))\leq B_k$ which is (\ref{eq:etan3}) and is proved as part of  Lemma~\ref{lem:etak}.

Finally we also notice that $\sum_{k=1}^n \left(g_{\what\theta_{k-1}}(X_k)-P_{\what\theta_{k-1}}g_{\what\theta_{k-1}}(X_{k-1})\right) =:\summ k1n D_k$ is a martingale with respect to $\{\F_n\}$, whence
\[
{\mathbb E}^\prime{\summ k1nD_k}^2  = \summ k1n{\mathbb E} D_k^2\leq 4n\sup_{\theta}|g_\theta|_\infty^2\leq 16\epsilon^{-2}|f|_\infty^2n .
\]
 Using all the above, we obtain~\eqref{eq:L2fX}.
\end{proof}

\subsection{An example on the lower bound}
We provide an example where $O(n^{-1})$ is also the lower bound of the rate for both $\nnTV{\eta_n-\pi}$ and $\nnTV{{\cal L}_{X_n}-\pi}$. This shows that the rate in our upper bound in Corollary~\ref{coro:1} is optimal.
\begin{Example}\label{rem:2state}
Consider the simple case when ${\cal X} = \{\pm1\}$, and $\pi = \pi_Y$. In this case, the weight function is uniform ($w \equiv 1$). Suppose the auxiliary chain $\{Y_n\}_{n\geq 0}$ has transition matrix
\[
P_Y = \bbpp{
\begin{array}{cc}
1-a & a\\
b & 1-b
\end{array}
}, \mbox{ with } a,b\in(0,1)\,.
\]
The corresponding Markov chain has stationary distribution $\pi_Y = (a+b)^{-1}(b,a)$ and eigenvalues $\lambda_1 = 1,\lambda_2 = 1-a-b$. Suppose $a+b\neq 1$ and the chain starts at $Y_0 = -1$. By straight-forward calculation, $\mathbb P(Y_n = -1) = a/(a+b) + b/(a+b)\lambda_2^n$. Then,
\begin{multline*}
{\mathbb E}\what\pi_{Y,n}(\{-1\}) - \pi_Y(\{-1\}) \\
\equiv \frac1n\summ i1{n}(\mathbb P(Y_i = -1) - \pi_Y(\{-1\})) = \frac a{a+b}\frac1n\frac{\lambda_2-\lambda_2^{n+1}}{1-\lambda_2}\,.
\end{multline*}
It then follows from the definition that $\nnTV{\eta_n-\pi}\geq C/n$.

Furthermore, in~\eqref{eq:Ptheta} set $P(x,\cdot) = \pi(\cdot)$. That is, $P$ is the {\it best} kernel we can put into the algorithm, in the sense that it takes one step to arrive at the stationary distribution (although this is too ideal to be practical). Now,
\begin{eqnarray*}
\mathbb P(X_n = -1) - \pi(\{-1\}) & = & (1-\epsilon)\pi(\{-1\}) + \epsilon{\mathbb E}\what\pi_{Y,n}(\{-1\}) - \pi(\{-1\})\\
& = & \epsilon\bbpp{{\mathbb E}\what\pi_{Y,n}(\{-1\}) - \pi_Y(\{-1\})}\,.
\end{eqnarray*}
It then follows that $\nnTV{{\cal L}_{X_n}-\pi}\geq C/n$.

\end{Example}

\subsection{Multiple IRMCMC}\label{sec:mIRMCMC}
We discuss importance-resampling MCMC algorithm in forms of multiple chains and establish a similar convergence rate as in Section~\ref{sec:IRMCMCrate}.
\begin{Algo}[Multiple IRMCMC]\label{algo:mIRMCMC}
We construct iteratively $m$ discrete-time stochastic processes $X\topp \ell \equiv \{X_n\topp \ell\}_{n\geq 0}, \ell=0,\dots, m$ as follows. Fix $\epsilon\in(0,1)$.
Let $X\topp 0$ be a Markov chain with target distribution $\pi_0$ starting at $x_0$. Then iteratively, for each $\ell=1,\dots,m$ with $X\topp {\ell-1}$ constructed, design $X\topp{\ell}$ starting from $x_\ell$, so that $X\topp\ell$ and $X\topp{\ell-1}$ interact as the main chain and the auxiliary chain respectively in Algorithm~\ref{algo:IRMCMC}. Namely, let $P_\ell$ be a Markov kernel with stationary distribution $\pi_\ell$, and sample $X_{n+1}\topp\ell$ from $P_{\ell,\what\theta\topp{\ell-1}_{n}}(X_n\topp \ell,\cdot)$ with
\[
P_{\ell,\theta}(x,\cdot) = (1-\epsilon) P_\ell(x,\cdot) + \epsilon\theta(\cdot)
\]
and
\[
\what\theta\topp{\ell-1}_n(\cdot) = \summ i1{n}\frac{w_\ell(X_i\topp{\ell-1})}{\summ j1{n}w_\ell(X_j\topp{\ell-1})}\delta_{X_i\topp{\ell-1}}(\cdot),
\]
with $w_\ell(x) = {\pi_\ell(x)}/{\pi_{\ell-1}(x)}, x\in{\cal X}$.
Note that the $\ell$-th chain $X\topp \ell$ at time $n$, depends on $\{X_k\topp \ell\}_{k=0,\dots,n-1, \ell=1,\dots m-1}$.
We assume that $\max_{\ell=1,\dots,m}|w_\ell|_\infty<\infty$.
\end{Algo}
In view of Theorem~\ref{thm:IRMCMC}, it suffices to control
\begin{equation}\label{eq:Bnl}
B_n\topp\ell \ensuremath{\stackrel{\mathrm{def}}{=}} \sup_{|f|\leq 1}{\mathbb E}\what\pi_{X\topp{\ell},n}(\wb f) + \sup_{|f|\leq 1}{\mathbb E}^\prime{\what\pi_{X\topp{\ell},n}(\wb f)}^2, n\in{\mathbb N},
\end{equation}
where this time $\what\pi_{X\topp\ell,n}(\wb f) \ensuremath{\stackrel{\mathrm{def}}{=}} \what\pi_{X\topp\ell,n}(f) - \pi_{\ell-1}(f)$.
In fact, it suffices to control $B_n^{(0)}$, which is the purpose of the following assumption.

\assumpH
\item \label{A:mIRMCMC}
As $n\to\infty$, the initial Markov chain $\{X_n\topp 0\}_{n\geq 0}$ satisfies $B_n^{(0)}\leq C/n$.
\assumpE

\begin{Thm}\label{thm:mIRMCMC}
Consider the multiple IRMCMC (Algorithm~\ref{algo:mIRMCMC}) for which Assumption~\ref{A:mIRMCMC} holds and $\max_{\ell=1,\dots,m}|w_\ell|_\infty<\infty$. Then for $\ell = 1,\dots,m$, there exists a finite constant $C$ such that
\begin{equation}\label{eq:lognm}
\nnTV{{\cal L}_{X_n\topp \ell} - \pi_\ell} \leq \frac{C}n\,,
\end{equation}
and for any bounded measurable function $f$,
\begin{equation}\label{eq:varm}
{\mathbb E}\bb{\frac1{\sqrt n}\summ i1n^\prime{f(X_i\topp{\ell})-\pi(f)}}^2 \leq C.\end{equation}

\end{Thm}
\begin{proof} Simply observe that  (\ref{eq:lognm}) and (\ref{eq:varm}) imply that $B_n^{(\ell)}\leq C/n$, as $n\to\infty$. By Theorem \ref{thm:IRMCMC}, this implies in turn that (\ref{eq:lognm}) and (\ref{eq:varm}) hold with $\ell$ replaced by $\ell+1$. Given \ref{A:mIRMCMC}, the result follows by induction.
\end{proof}

\newpage
\section{Equi-Energy Sampler}\label{sec:EE}
In this section, we consider the simplified EE sampler as follows. Recall that the auxiliary chain $\{Y_n\}_{n\geq 0}$ evolves independently from the main chain $\{X_n\}_{n\geq 0}$.
\begin{Algo}[Equi-Energy sampler]\label{algo:EE}
Fix $\epsilon\in(0,1)$. Start $X_0 = x_0$ and $Y_0 = y_0$. At each round $n$,  generate
\[
X_n \sim\left\{
\begin{array}{ll}
P(X_{n-1},\cdot) & \mbox{ w.p.~} 1-\epsilon\\
K_{\what\theta_{n-1}}(X_{n-1},\cdot) & \mbox{ w.p.~} \epsilon
\end{array}
\right.\,,
\]
where $\what\theta_n = \what\pi_{Y,n}$ is the empirical measure associated to $\{Y_n\}_{n\geq 0}$ and $K_\theta$ is defined by
\[
K_\theta(x,A)=\textbf{1}_A(x) + \int_{{\cal X}}\left(1\wedge\frac{\pi(z)\pi_Y(x)}{\pi(x)\pi_Y(z)}\right)\left(\textbf{1}_A(z)-\textbf{1}_A(x)\right)\theta({\rm d} z).
\]
In other words,
for all non-negative functions $h:{\cal X}\to{\mathbb R}$ and $n\in{\mathbb N}$,
\begin{equation}\label{dynEE}
\PE_{x}\left(h(X_{n+1})\mid \F_{n}\right)=P_{\what\theta_{n}}h(X_{n}) \;\;\mbox{ almost surely,}
\end{equation}
where for any probability measure $\theta$ on ${\cal X}$, $P_\theta$ is defined as
\begin{equation}\label{eq:EE}
P_\theta(x,A)=(1-\epsilon)P(x,A)+\epsilon K_\theta(x,A),
\end{equation}
Recall that we write $\pi({\rm d} x) \equiv \pi(x){\rm d} x$ and similarly for $\pi_Y$ with a little abuse of language, and $w(x) = \pi(x)/\pi_Y(x)$. We assume $|w|_\infty<\infty$.
\end{Algo}

The kernel $K_{\theta_\star}$ is the Independent Metropolis kernel with target $\pi$ and proposal $\theta_\star = \pi_Y$. It is well known that under the assumption $|w|_\infty<\infty$ (recall Remark~\ref{rem:w}), the kernel $K_{\theta_\star}$ is uniformly ergodic \citep{mengersen96rates}, and this property is inherited by $P_{\theta_\star}$. That is, there exists $\rho\in (0,1)$, such that
\begin{equation}\label{eq:rateconv}
  \nnTV{P^n_{\theta_\star}(x,\cdot) - \pi(\cdot)}\leq C \rho^n,\;\;\;n\geq 0.
\end{equation}

\subsection{Convergence rate of EE sampler}\label{sec:EErate}
We make the following assumptions.

\assumpH
\item \label{A:subGaussian}
There exist a finite universal  constant $C$ such that for any measurable function $f:\;{\cal X}\to\mathbb R$,  with $|f|_\infty\leq 1$,
\[
\sup_{n}\PP\left(\left|\frac{1}{\sqrt{n}}\summ j1n ^\prime{f(Y_j) - \pi_Y(f)}\right|>x\right)  \leq C\exp\left(-\frac{x^2}{C\sigma^2(f)}\right),
\]
where $\sigma^2(f)\ensuremath{\stackrel{\mathrm{def}}{=}} \int_{\cal X} f^2(x)\pi_Y({\rm d} x)$.
\assumpE

\assumpH
\item \label{A:w} The function $w:\;{\cal X}\to\mathbb R$ is continuous (with respect to the metric on ${\cal X}$),  and
\begin{equation}\label{eq:w}
\sup_{x\in{\cal X}}\frac{\phi(x)}{w^2(x)}<\infty,
\end{equation}
where $\phi(x)\ensuremath{\stackrel{\mathrm{def}}{=}} \pi_Y\left(\{z:\; w(z)\leq w(x)\}\right)$.
\assumpE

\assumpH
\item \label{A:continuous} The kernel $P$ is such that if $f:{\cal X}\to\mathbb R$ is continuous, then $Pf$ is also continuous.
\assumpE
\begin{Rem}Deviation bounds as in~\ref{A:subGaussian} are available for various conditions on transition kernels. See for example \citet[Proposition 1.2]{cattiaux08deviation}.
\end{Rem}
\begin{Rem}
Assumption~\ref{A:w} is not restrictive. For example, consider ${\cal X} = {\mathbb R}$ and $\pi_Y = \pi^T$ with some $T\in(0,1)$. For the sake of simplicity, we focus on $x\in{\mathbb R}_+$ and define $\phi_+(x) \ensuremath{\stackrel{\mathrm{def}}{=}} \pi_Y(\{z>0:w(z)\leq w(x)\})$. Suppose the density $\pi(x)$ decays asymptotically as $x^{-\alpha}$ for $\alpha>1$ as $x\to\infty$. Then, $\pi_Y(x)\sim x^{-T\alpha}$ and $w(x)\sim x^{(T-1)\alpha}$. Here and below, we write $a(x)\sim b(x)$ if $\lim_{x\to\infty}a(x)/b(x) = 1$.  Assume further that $T\alpha>1$. Then, $\phi_+(x)\sim (T\alpha-1)^{-1} x^{1-T\alpha}$ and
\[
\frac{\phi_+(x)}{w^2(x)} \sim\frac1{T\alpha-1}x^{1+2\alpha-3T\alpha}.
\]
Therefore,~\eqref{eq:w} holds, if $T>(1+2\alpha)/(3\alpha)$.
\end{Rem}

\begin{Thm}\label{thm:EE}
Consider the Equi-Energy sampler described as above and suppose that Assumptions \ref{A:subGaussian}--\ref{A:continuous} hold. Then, there exists a constant $C$, such that for all continuous functions $f:{\cal X}\to{\mathbb R}$ and $n\in{\mathbb N}$,
\begin{equation}\label{eq:thm:EE}
\left|{\mathbb E}\left(f(X_n)-\pi(f)\right)\right|\leq \frac{C|f|_\infty}{\sqrt{n}}.
\end{equation}
\end{Thm}
\begin{proof}
Fix $n\geq 2$ and $1\leq q\leq n$. Fix $f:\;{\cal X}\to\mathbb R$ with $|f|_\infty= 1$. Then write
\[
\PE_xf(X_n)-P_{\theta_\star}^nf(x)=\PE_x\left(P_{\theta_\star}^{n-q}f(X_q)-P_{\theta_\star}^nf(x)\right)-\PE_x\left(P_{\theta_\star}^{n-q}f(X_q)-f(X_n)\right).
\]
For the first term we can use \eqref{eq:rateconv} to get:
\[\left|\PE_x\left(P_{\theta_\star}^{n-q}f(X_q)-P_{\theta_\star}^nf(x)\right)\right|\leq C \rho^{n-q},\]
for some finite constant $C$ that does not depend on $f$. For the second term, we write:
\begin{eqnarray}
& & \PE_x\left(P_{\theta_\star}^{n-q}f(X_q)-f(X_n)\right)\nonumber\\
& = & \PE_x\left[\sum_{j=q}^{n-1}\left(P_{\theta_\star}^{n-j}f(X_j)-P_{\theta_\star}^{n-j-1}f(X_{j+1})\right)\right]\nonumber\\
& = & \sum_{j=q}^{n-1}\PE_x\left[P_{\theta_\star}^{n-j}f(X_j)-\PE_x\left(P_{\theta_\star}^{n-j-1}f(X_{j+1})\mid \F_j\right)\right]\nonumber\\
& = & \sum_{j=q}^{n-1}\PE_x\left[P_{\theta_\star}^{n-j}f(X_j)-P_{\what\theta_j}P_{\theta_\star}^{n-j-1}f(X_{j})\right]\nonumber\\
& = &\sum_{j=q}^{n-1}C_0 \rho^{n-j-1}\PE_x\left[\left(P_{\theta_\star}-P_{\what\theta_j}\right)\zeta_{n,j}(X_j)\right],\label{eq:Ex}
\end{eqnarray}
where in the last line we write
\[
\zeta_{n,j}(x)=\frac{P_{\theta_\star}^{n-j-1}(f(x)-\theta_\star(f))}{C_0\rho^{n-j-1}},\;\;x\in{\cal X}\,,
\]
with $C_0$ and $\rho$ chosen as in~\eqref{eq:rateconv}.
As a consequence of~\eqref{eq:rateconv}, $|\zeta_{n,j}|_\infty\leq 1$. It is also continuous by the continuity of $f$ and Assumption~\ref{A:continuous}.

To simplify the notation, for any function $g:\;{\cal X}\to \mathbb R$, define

\begin{equation}\label{eq:Hg}
H_g(x,z)\ensuremath{\stackrel{\mathrm{def}}{=}}\alpha(x,z)\left(g(z)-g(x)\right),\quad x,z\in{\cal X},\end{equation}
where
\begin{equation}\label{eq:alpha}
\alpha(x,z)\ensuremath{\stackrel{\mathrm{def}}{=}} 1\wedge \frac{w(z)}{w(x)}.
\end{equation}
 Thus, we can write
\[
P_\theta g(x)-P_{\theta_\star}g(x)=\epsilon\int H_g(x,z)(\theta({\rm d} z)-\theta_\star({\rm d} z)).
\]
Always based on $g:{\cal X}\to{\mathbb R}$, we introduce the class of functions
\[
{\cal F}_g\ensuremath{\stackrel{\mathrm{def}}{=}}\ccbb{z\mapsto H_g(x,z):\;x\in{\cal X}},
\]
and the empirical process
\[{\mathbb G}_n(h)\ensuremath{\stackrel{\mathrm{def}}{=}} \frac{1}{\sqrt{n}}\sum_{j=1}^n \left(h(Y_j)-\pi_Y(h)\right),\;\;\; h\in{\cal F}_g.
\]
Therefore, the expectation term in~\eqref{eq:Ex} becomes
\begin{multline*}
 {\mathbb E}_x\bb{\bbpp{P_{\theta_\star}-P_{\what\theta_j}}\xi_{n,j}(X_j)}
= \epsilon{\mathbb E}_x\bb{\int H_{\xi_{n,j}}(X_j,z)(\theta_\star({\rm d} z)-\what\theta_j({\rm d} z))}\\
= -\epsilon{\mathbb E}_x\bb{\frac1j\summ \ell1j H_{\xi_{n,j}}(X_j,Y_\ell) - \int_{\cal X} H_{\xi_{n,j}}(X_j,z)\theta_\star({\rm d} z)}\\
=  -\frac{\epsilon}{\sqrt{j}}{\mathbb E}_x\bb{{\mathbb G}_j\bbpp{H_{\xi_{n,j}}(X_j,\cdot)}}\,,
\end{multline*}
whence
\begin{eqnarray*}
\left|\PE_x\left(P_{\theta_\star}^{n-q}f(X_q)-f(X_n)\right)\right| & = & \abs{\epsilon\sum_{j=q}^{n-1}\frac{C_0\rho^{n-j-1}}{\sqrt{j}}{\mathbb E}_x\bb{{\mathbb G}_j\bbpp{H_{\xi_{n,j}}(X_j,\cdot)}}}\\
 & \leq & C_0\sum_{j=q}^{n-1}\frac{\rho^{n-j-1}}{\sqrt{j}}\PE_x\left(\sup_{h\in \F_{\zeta_{n,j}}}\left|{\mathbb G}_j(h)\right|\right).
\end{eqnarray*}
We prove in Lemma \ref{lem2} below that for any continuous function $g:{\cal X}\to\mathbb R$ such that $|g|_\infty\leq 1$,
\[
\PE_x\left(\sup_{h\in \F_{g}}\left|{\mathbb G}_n(h)\right|\right)\leq C,
\]
for some constant $C$ that does not depend on $n$ nor $g$. We conclude that
\[
\left|\PE_x\left(P_{\theta_\star}^{n-q}f(X_q)-f(X_n)\right)\right|\leq C\sum_{j=q}^{n-1}\frac{1}{\sqrt{j}}\rho^{n-j-1}.
\]
Thus for any $1\leq q\leq n$,
\[
\left|\PE_x\left(f(X_n)\right)-\theta_\star(f)\right|\leq C \left\{\rho^n + \rho^{n-q} + \epsilon \sum_{j=q}^{n-1}\frac{\rho^{n-j-1}}{\sqrt{j}}\right\}\leq Cn^{-1/2},
\]
by choosing $q=n-\lfloor\frac{-\log n}{2\log\rho}\rfloor$.

\end{proof}

We rely on the following technical result on the auxiliary chain $\{Y_n\}_{n\geq 0}$.

\begin{Lem}\label{lem2}Suppose that Assumptions \ref{A:subGaussian} and \ref{A:w} hold, and let $g:{\cal X}\to\mathbb R$ be continuous such that $|g|_\infty\leq 1$. Then
\[
\PE_x\left(\sup_{h\in \F_g}\left|{\mathbb G}_n(h)\right|\right)\leq C,\]
for a constant $C$ that does not depend on $n$.
\end{Lem}
\begin{proof}
Throughout the proof $n\geq 1$ is fixed. Assumption \ref{A:subGaussian} suggests the following metric on $\F_g$:
\[\textsf{d}(h_1,h_2)=\sigma(h_1-h_2)=\left(\int_{\cal X}\left(h_1(x)-h_2(x)\right)^2\pi_Y({\rm d} x)\right)^{1/2},\]
which has the following properties.
For  $x_1,x_2\in{\cal X}$, it is easy to check that
\begin{equation}\label{dist1}
\left |H_g(x_1,z)-H_g(x_2,z)\right|\leq 2\left|\alpha(x_1,z)-\alpha(x_2,z)\right| + \left|g(x_1)-g(x_2)\right|.\end{equation}
It follows that
\begin{multline}\label{dist2}
\textsf{d}\left(H_g(x_1,\cdot),H_g(x_2,\cdot)\right)\\
\leq \sqrt 2\left|g(x_1)-g(x_2)\right|
+ 2\sqrt2\sqrt{\int\left|\alpha(x_1,z)-\alpha(x_2,z)\right|^2\pi_Y({\rm d} z)}.\end{multline}
This implies that the diameter of $\F_g$ is bounded by $\delta(\F_g)=4\sqrt{2}$.  It also implies that with respect to $\textsf{d}$, the empirical process $\{{\mathbb G}_n(h),\;h\in \F_g\}$ is separable. Indeed, for $x\in{\cal X}$ arbitrary and $h=H_g(x,\cdot)$, using the Polish assumption, we can find a sequence $x_m\in{\cal X}$ ($x_m$ belongs to a countable subset of ${\cal X}$) such that $x_m\to x$, as $m\to\infty$. Setting $h_m=H_g(x_m,\cdot)$, it follows from (\ref{dist2}) and the continuity of $u$ and $E$ that $h_m\to h$ in $(\F,\textsf{d})$, and (\ref{dist1}) easily show that ${\mathbb G}_n(h_m)-{\mathbb G}_n(h)=n^{-1/2}\sum_{\ell=1}^n\left(H_g(x,Y_\ell)-H_g(x_m,Y_\ell)\right)+\sqrt{n}\pi_Y\left(H_g(x,\cdot)-H_g(x_m,\cdot)\right)\to 0$ as $m\to\infty$ for all realizations of $\{Y_1,\ldots,Y_n\}$.

For any  $h_1,h_2\in\F_{g}$, Assumption \ref{A:subGaussian} implies that for any $x>0$
\[
\PP_x\left(\left|{\mathbb G}_n(h_1)-{\mathbb G}_n(h_2)\right|>x\right)\leq C\exp\left(-\frac{x^2}{c\textsf{d}^2(h_1,h_2)}\right).
\]
Then we apply \citet[Corollary 2.2.8]{vandervaart96weak} to conclude that for $h_0\in\F_g$, there exists a constant $C$ independent of $g$, such that
\[
\PE_x\left(\sup_{h\in \F_{g}}\left|{\mathbb G}_n(h)\right|\right)\leq \PE_x|{\mathbb G}_n(h_0)| + C\int_0^{\delta(\F_g)} \sqrt{1+\log\textsf{D}(\epsilon,\F_g,\textsf{d})}\;{\rm d}\epsilon<\infty,\]
where $\textsf{D}(\epsilon,\F_g,\textsf{d})$ is the packing number of $\F_g$ with respect to $\textsf{d}$. Assumption \ref{A:subGaussian} shows that $\PE_x|{\mathbb G}_n(h_0)|<\infty$. To control the entropy number, we further bound the right hand of (\ref{dist2}).

Without loss of generality, assume $x_1,x_2\in{\cal X}$ and $w(x_1)<w(x_2)$.  If $w(x_1)\vee w(x_2)\leq w(z)$, then $\alpha(x_1,z)-\alpha(x_2,z)=0$.  If $w(z)\leq w(x_1)$, then
\[
\left|\alpha(x_1,z)-\alpha(x_2,z)\right|^2=\left|\frac{w(z)}{w(x_1)} - \frac{w(z)}{w(x_2)}\right|^2\leq \frac{1}{w(x_1)^2}\left(w(x_2)-w(x_1)\right)^2.
\]
If $w(x_1)\leq w(z)\leq w(x_2)$, then
\[\left|\alpha(x_1,z)-\alpha(x_2,z)\right|^2=\left|1- \frac{w(z)}{w(x_2)}\right|^2
\leq \frac{1}{w(x_2)^2}\left(w(x_2)-w(x_1)\right)^2.\]

Thus
\begin{multline*}
\int\left|\alpha(x_1,z)-\alpha(x_2,z)\right|^2\pi_Y({\rm d} z)\\
\leq ^\prime{\frac{\phi(x_1)}{w(x_1)^2}
 + \frac{\phi(x_2)}{w(x_2)^2}}\left(w(x_2)-w(x_1)\right)^2\leq C\left(w(x_2)-w(x_1)\right)^2,
\end{multline*}
where $\phi(x)\ensuremath{\stackrel{\mathrm{def}}{=}} \pi_Y\left(\{z:\; w(z)\leq w(x)\}\right)$, and the last inequality follows from \ref{A:w}.
Together with (\ref{dist2}), we conclude from this bound that
\begin{equation}\label{lip}
\textsf{d}\left(H_g(x_1,\cdot),H_g(x_2,\cdot)\right)
\leq C (\left|g(x_1)-g(x_2)\right|+ \left|w(x_2)-w(x_1)\right|).
\end{equation}
Since $|g|_\infty\leq 1$ and $w(x)\in [0,|w|_\infty]$, this implies that the $\epsilon$-packing number of $(\F_g,\textsf{d})$ is at most of order  $\epsilon^{-2}$, so that $\int_0^{\delta(\F_g)} \sqrt{1+\log\textsf{D}(\epsilon,\F_g,\textsf{d})}\;{\rm d}\epsilon\leq C \int_0^{\delta(\F_g)}\sqrt{1+\log(1/\epsilon)}d\epsilon<\infty$.
This proves the lemma.
\end{proof}

\subsection{Connection with Parallel Tempering}\label{sec:mEE}
Our results suggest that the EE sampler mixes relatively slowly. A plausible reason for this slow mixing is the dependence on the entire sample path $\{Y_k\}_{0\leq k\leq n}$. The EE sampler is closely related to the Parallel Tempering (PT) algorithm of \citep{geyer91markov}, which suggests that it might be  possible to exploit this  connection by deriving versions of the EE sampler with better mixing properties. Like the EE sampler, a 2-chain  PT generates a stochastic process $\{(X_n,Y_n)\}_{n\geq 0}$ where with probability $1-\epsilon$, $X_n$ is generated from $P(X_{n-1},\cdot)$ and with probability $\epsilon$, one proposes to swap the two chains. Thus PT is closely related to a EE-type algorithm where the empirical measure $\what\pi_{Y,n}$ would be replaced by a Dirac measure $\delta_{Y_n}$.  However, we show that in general, this new algorithm does not maintain the correct stationary distribution. We hope that the discussion in this section would be helpful in conceptualizing new adaptive algorithms in the future.

The modified EE sampler is as follows. Let $\{Y_n\}_{n\geq 0}$ be the auxiliary chain with transition kernel $P_Y$ and stationary distribution $\pi_Y$. Let $\{X_n\}_{n\geq 0}$ be a chain satisfying the following assumption: for all continuous and bounded function $f$,
\[
{\mathbb E}[f(X_{n+1})\mid X_0,\dots,X_n,Y_0,\dots,Y_n] = P_{\delta_{Y_n}}f(X_n), n\in{\mathbb N},
\]
where $P_\theta$ is as in~\eqref{eq:EE}, and denote the stationary distribution of $P$ by $\pi_X$.

\begin{Rem}
The difference from the EE sampler is that we replace $\what\pi_{Y,n}$ by $\delta_{Y_n}$. If, when $X_{n+1}$ is moving to $Y_n$, we also make $Y_{n+1}$ move to $X_n$, then we are allowing {\it swaps} between the two chains. Such swaps are in the spirit of parallel tempering algorithms (see e.g.~\citet{geyer91markov}).
\end{Rem}
A nice property of this algorithm is that $\{(X_n,Y_n)\}_{n\geq 0}$ is a Markov chain. Indeed, it has transition kernel
\begin{equation}\label{eq:MEE}
P_{X,Y}(x,y,{\rm d} z,{\rm d} w) = P_{\delta y}(x,{\rm d} z)P_Y(y,{\rm d} w)\,.
\end{equation}

This Markov chain may not have the desired stationary distribution. Let $\pi_{X,Y}$ denote the stationary distribution and let $\pi_{X,Y}\topp i, i=1,2$ denote its two marginal distributions. Naturally, we wish $\pi_{X,Y}\topp 1 = \pi_X$ and $\pi_{X,Y}\topp 2 = \pi_Y$.
By construction, the latter identity is always true.
However, the former does not always hold.
Since $P_\theta = (1-\epsilon)P + \epsilon K_\theta$
and $P(x,{\rm d} z)P_Y(y,{\rm d} w)$ has stationary distribution $\pi_X({\rm d} z)\pi_Y({\rm d} w)$, instead of~\eqref{eq:MEE} it suffices to focus on the transition kernel
\begin{equation}\label{eq:MEE1}
P_{X,Y}(x,y,{\rm d} z,{\rm d} w) = K_{\delta_Y}(x,{\rm d} z)P_Y(y,{\rm d} w)\,.
\end{equation}

Consider the simple case when both chains take values from $\{\pm1\}$.
Let the auxiliary chain has the following transition matrix and stationary distribution:
\[
P_Y = \left(
\begin{array}{cc}
1- a & a\\
b & 1-b
\end{array}
\right)\quad\mbox{ and }\quad \pi_Y = ^\prime{\frac b{a+b}, \frac a{a+b}}\,.
\]
Recall that in this case,
\[
K_{\delta y}(x,z) = \alpha(x,y)\indd{z=y} + (1-\alpha(x,y))\indd {z=x}\,,
\]
with
\[
\alpha(x,y) = 1\wedge \frac{\pi_X(y)}{\pi_X(x)}\frac{\pi_Y(x)}{\pi_Y(y)}\,.
\]
Write $c = \alpha(1,-1)$ and $d= \alpha(-1,1)$. Then, one can write the transition matrix of $P_{X,Y}$ as follows:
\[
\begin{array}{l|llll}
& (1,1) & (1,-1) & (-1,1)& (-1,-1)\\
\hline
(1,1) & 1-a & a & 0 & 0\\
(1,-1) & (1-c)b & (1-c)(1-b) & cb & c(1-b)\\
(-1,1) & d(1-a) & da & (1-d)(1-a) & (1-d)a\\
(-1,-1) & 0 & 0 & b & 1-b
\end{array}
\]
For example, the table reads as
\[
P(1,-1,1,-1) = P_{\delta_{-1}}(1,1)P_Y(-1,-1) = (1-c)(1-b)\,.
\]
We solve $\pi_{X,Y}P_{X,Y} = \pi_{X,Y}$ and obtain
\[
\pi_{XY} = \frac1{(a+b)((1-a-b)cd + ac + bd)}
\left(
\begin{array}{c}
 bd(b+(1-a-b)c)
\\
 abd\\
 abc\\
ac(a+(1-a-b)d)
 \end{array}
 \right)^\top\,.
\]
To see that $\pi_{X,Y}\topp1$ does not always equal $\pi_X$, take for example $a=b=1/3$ and $\pi_X = (2/3,1/3)$. In this case, $c = 1/2$, $d=1$, and $\pi_{X,Y} = (3/8,1/4,1/8,1/4)$, whence $\pi_{X,Y}\topp1 = (5/8,3/8)\neq \pi_X$.

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