text:
\section{Introduction}
Quantum thermometry \cite{Pasquale2018,Mehboudi2019a} is the current prime example of open quantum metrology. It comprises methods to infer the temperature of a thermal reservoir by probing it with small quantum systems and exploiting the quantum nature of the probes and the measurement scheme to enhance the precision of the temperature estimate. The advantages of these methods have found experimental applications in weakly invasive in-vitro thermometry of microscopic organisms~\cite{Fujiwara2020} as well as low-temperature monitoring in ultracold quantum gases \cite{leanhardt2003cooling,Ruostekoski2009,bloch2012quantum,Mehboudi2019,Carcy2021}.

Most theoretical considerations have focused on equilibrium thermometry in the asymptotic limit of large measurement data so that the Cram\'{e}r-Rao bound applies \cite{DeP2016,DeP2017,Hovhannisyan2018,Razavian2019,Bouton2020,Gebbia2020,Zhang2020}, mostly relying on qubits as the paradigmatic quantum probes \cite{Brunelli2012,Jevtic2015b,Cavina2018,Mitchison2020}. Strategies for improved temperature precision, as quantified in terms of the thermal Fisher information \cite{Pasquale2018,Mehboudi2019a}, consist in tailoring the energy spectrum of high-dimensional probes for a desired temperature range \cite{correa2015individual}, optimizing the binning of measurement outcomes \cite{Potts2019,Jorgensen2020,Hovhannisyan2021}, or using a catalyst \cite{Henao2021}.

In practice however, thermal equilibrium between the reservoir and the probes may not always be achievable \cite{Guo2015,Correa2016,Sekatski2021}, or even desirable \cite{Stace2010,Hofer2017b,Pati2020,Henao2021a}. In fact, the transient dynamics induced by repeated finite-time collisions with nonequilibrium quantum probes can result in enhanced precision compared to equilibrium thermometry \cite{Kiilerich2018,Seah2019a,Feyles2019}, once again quantified by the Fisher information.

While the Fisher information is the key figure of merit in most studies, its predictive power is limited to the ideal scenario when measurements can be repeated many times. In the opposite, less explored, regime when only scarce data is available, the Bayesian parameter estimation framework is more appropriate. It encodes any information or bias known beforehand into a \emph{prior} distribution of expected temperatures, which is then updated by the measured outcome into a \emph{posterior} distribution from which to infer the temperature estimate. This allows one to describe both single uninformed measurement runs as well as optimized adaptive protocols. Applied before in Heisenberg-limited phase estimation \cite{Higgins2007,Wiebe2016,Yan2018,Gebhart2021}, the Bayesian formalism was only recently introduced in the context of thermometry \cite{rubio2020global,alves2020bayesian,Jorgensen2021,mehboudi2021fundamental}.

Here we study uninformed Bayesian estimation from a single measurement outcome, assuming a fixed measurement scheme, but no further knowledge about the expected reservoir temperature. Results are evaluated for sequences of identical qubits probing the reservoir. We elaborate on the crucial role of the chosen estimator and prior distribution for the attainable accuracy within the range of detectable temperatures, which can be inferred from the likelihood function. In particular, given that the moments of the posterior temperature distribution typically do not exist, common choices such as the mean estimator necessitate temperatures be restricted to a finite range, which implies that the thermometry protocol cannot be truly uninformed. We propose two new estimators based on relative deviations in the temperature that alleviate the divergence problem and generally give better estimates with lower uncertainty and bias. Moreover, we compare fully thermalized to partially thermalized probes, showing that the latter are sensitive to a wider range of temperatures at the expense of the maximum achievable accuracy.

This paper is organised as follows: Sec.~\ref{sec:bayes} details the Bayesian approach to thermometry, including the prior distribution and all the estimators and measures of uncertainty that we consider. We then focus on our case study of qubit probes in and out of thermal equilibrium in Sec.~\ref{sec:model}, followed by a numerical benchmark assessment of the various estimators in Sec.~\ref{sec:results}. Specifically, for fully thermalised probes, we compare the average estimates and errors as functions of the true temperature in Sec.~\ref{sec:equilResults}, the average performance as a function of the probe number in Sec.~\ref{sec:globalScaling}, and also the influence of the prior in Sec.~\ref{sec:priorCompare}. Finally, we demonstrate the advantage of partially thermalized qubits in Sec.~\ref{sec:noneqResults}, before we conclude in Sec.~\ref{sec:conclusion}.

\section{Bayesian Framework}
\label{sec:bayes}

In the uninformed Bayesian approach to thermometry \cite{Lehmann1998,jaynes2003probability}, there is no prior knowledge about the temperature $T$ of the thermal reservoir in question, apart from its positivity and its influence on the outcomes of the chosen measurement scheme. We assume a single measurement that gives one of $N+1$ different outcomes, labeled by $n=0,\ldots N$. The achievable temperature sensitivity is encapsulated in the \emph{likelihood function}: the conditional probability $P(n|T)$ to observe outcome $n$ given the temperature value $T$, as predicted by the theoretical model of the measurement procedure.

Typically, one can associate a characteristic energy scale $E$ to the measurement scheme and thus express temperature in units of $E/k_B$.
Temperature sensitivity could be optimized by adjusting $E$ if prior information about the expected temperature range were available \cite{correa2015individual}, but we seek to infer $T \geq 0$ without that knowledge at fixed $E$. Bayesian parameter estimation theory requires us to make additional assumptions about (i) the \emph{prior} $P^{(0)} (T)$ that describes the ``expected'' distribution of temperatures prior to any measurement, (ii) a \emph{cost function} $c(\theta_n , T) \geq 0$ quantifying how ``wrong'' the temperature estimate $\theta_n$ based on the observed outcome $n$ is when the true temperature value was $T$, and (iii) a suitable \emph{error measure} that quantifies the uncertainty around the temperature estimate $\theta_n$ without reference to the inaccessible true value $T$.

For the temperature estimator itself, $\boldsymbol{\theta} = (\theta_0,\ldots \theta_N)$, there exists a natural, optimal choice determined by (i) and (ii): the argument $\boldsymbol{\vartheta} = \argmin_{\boldsymbol{\theta}} \mathcal{C}(\boldsymbol{\theta})$ that minimizes the average cost over all possible true temperatures $T$ and outcomes $n$,
\begin{eqnarray}
   \mathcal{C}(\boldsymbol{\theta}) &=&  \int_0^{\infty} \!\! \mathrm{d} T\, P^{(0)}(T) \sum_{n=0}^N P(n|T) c(\theta_n, T) \nonumber \\
    &=& \sum_{n=0}^N P(n) \int_0^{\infty} \!\! \mathrm{d} T\, P(T|n) c(\theta_n, T).  \label{eq:costFunc}
\end{eqnarray}
In the second line, we have applied Bayes' rule and expressed the average in terms of the normalized \emph{posterior}
\begin{eqnarray} \label{eq:posterior}
    P(T|n) &=& \frac{P(n|T)P^{(0)}(T)}{P(n)}, \nonumber \\
    P(n) &=& \int_0^\infty \!\! \mathrm{d} T \, P(n|T)P^{(0)}(T),
\end{eqnarray}
which encodes the updated expectation about the distribution of temperatures after the measurement outcome $n$ was obtained. Here we include only possible outcomes, assuming $P(n)>0$. Moreover, given the infinite range of possible $T$-values, one must choose an appropriate prior $P^{(0)}(T)$ so as to ensure that all outcome probabilities $P(n)$ be finite. Moments of temperature, $\langle T^k\rangle = \int\mathrm{d} T\, P(T|n) T^k$, might however still diverge.

\subsection{The Jeffreys prior}

For the estimation of a single continuous parameter $T$, it is standard practice to assume Jeffreys' prior \cite{Lehmann1998,Ghosh2011}. It represents the least informative starting point, because it maximizes the information gain in terms of the relative entropy between the prior and the posterior and it is invariant under re-parametrizations $T \to f(T)$. Jeffreys' prior is given by the square root of the Fisher information of the likelihood function with respect to the parameter,
\begin{equation}
    P^{(0)}(T) \propto \sqrt{\mathit{I}(T)} = \sqrt{ \sum_n P(n|T) \left[\partial_T \ln P(n|T)\right]^2}.
\end{equation}
The Fisher information $I(T)$ expresses the local temperature resolution of the given measurement scheme. Accordingly, Jeffreys' prior distribution assigns more weight to those temperatures at which the scheme is most sensitive. Other distributions could be chosen for convenience, but they would lead to additional bias by restricting the temperature range or by favoring temperatures that cannot be detected that well. When data is limited, the choice of prior affects the temperature estimator and error measure noticeably, as we will exemplify in Sec.~\ref{sec:priorCompare}.

Moreover, the norm of the prior might not always be finite, which poses no problem in practice if the relevant posteriors \eqref{eq:posterior} and estimates $\vartheta_n$ remain well-defined. However, this is not the case for simple examples like the constant flat prior, or the scale-invariant prior $P^{(0)}(T) \propto 1/T$, and one must therefore restrict them to a finite temperature interval in practice.

\subsection{Cost functions and estimators}\label{sub:estimators}

The cost function $c(\theta,T)$ measures the penalty of wrong temperature estimates by assigning a positive number to any deviation from the true $T$-value. One may choose to penalize deviations differently depending on their relative or absolute size, which then determines the optimal estimator $\boldsymbol{\vartheta}$ as well as an associated error measure $\boldsymbol{\varepsilon}$ \cite{jaynes2003probability}. Note that certain combinations of prior and $c(\theta,T)$ can result in diverging average costs \eqref{eq:costFunc}, estimates $\vartheta_n$, or errors $\varepsilon_n$, and should therefore be avoided. In our case study, we consider six different estimators: (md) the mode of the posterior, (2) the absolute mean, (2r) the relative mean, (1) the median, (1r) the relative median, and (2l) the logarithmic mean.

A widely used choice is the maximum-likelihood estimator with estimates $\vartheta_n^{(\rm{ml})} = \argmax_\theta P(n|\theta) $. It follows by assuming the (improper) flat prior $P^{(0)}(T) \propto 1$ and minimizing the average over the singular cost function $c^{(\rm{ml})}(\theta_n,T) = -\delta(\theta_n - T)$, which penalizes \emph{any} finite deviation by the same amount.
For the Jeffreys prior used here, or for any other prior that restricts the admitted temperature range, the estimator denoted by  $\boldsymbol{\vartheta}^{\rm{(md)}}$ is rather given by the mode of the posterior, $\vartheta_n^{(\rm{md})} = \argmax_\theta P(\theta|n) $.

Another common cost function penalizes square absolute deviations, $c^{(2)}(\theta_n,T) = (\theta_n - T)^2$, which makes the mean of the posterior the optimal estimator, denoted $\boldsymbol{\vartheta}^{(2)}$ with estimates $\vartheta_n^{(2)} = \langle T\rangle$. The error can be measured in multiples of the corresponding standard deviation, provided that the posterior has finite first and second moments, which will however not be the case here.

A viable alternative is obtained by defining the cost function in terms of the \emph{relative} square deviation, $c^{(\rm{2r})}(\theta_n,T) = (\theta_n/T - 1)^2$, which penalizes deviations relative to the absolute temperature scale. This results in the relative mean estimator $\boldsymbol{\vartheta}^{\rm{(2r)}}$, the estimates of which will not diverge due to unrestricted temperatures, $\vartheta_n^{\rm{(2r)}} = \langle T^{-1}\rangle/\langle T^{-2}\rangle$.

The problem of diverging moments is also partly alleviated by using the absolute median estimator $\boldsymbol{\vartheta}^{(1)}$, which merely assumes that the posteriors \eqref{eq:posterior} be normalizable. It optimizes the average over the 1st-moment cost function $c^{(1)}(\theta_n,T) = |\theta_n - T|$, which would still diverge for an unrestricted temperature range. Nevertheless, the $N+1$ estimates $\vartheta_n^{(1)}$ remain finite and are defined through the identity
\begin{equation} \label{eq:median}
    \int_0^{\vartheta_n^{(1)}} \!\! \mathrm{d} T \, P(T|n) = \int_{\vartheta_n^{(1)}}^\infty \!\! \mathrm{d} T \, P(T|n) = \frac{1}{2}.
\end{equation}
Notably, $\boldsymbol{\vartheta}^{(1)}$ is invariant under any re-parametrization $T \to f(T)$, including simple rescaling and typical conversion formulas from temperature to excitation numbers. The median estimator shares this feature with the maximum-likelihood estimator using a flat prior, and with the credibility-based error \eqref{eq:error90} below.

Once again, we could decide to penalize relative deviations instead and divide the cost function by the absolute temperature,  $c^{\rm{(1r)}}(\theta_n,T) = |\theta_n/T - 1|$, circumventing the problem of diverging $T$-moments. This results in the relative median estimator $\boldsymbol{\vartheta}^{\rm{(1r)}}$ with estimates $\vartheta^{\rm{(1r)}}_n$ given by the median of the re-normalized posterior distributions $\propto P(T|n)/T$.

One obtains a different class of estimators if one converts $T$ into a derived, physically motivated parameter $f(T)$ and applies one of the above standard cost functions in this new parameter space. Natural parametrizations of temperature would be the fermionic or bosonic excitation number, or simply $f(T) = \ln T$. The latter was recently proposed in combination with the square deviation $c^{\rm{(2l)}} (\theta_n, T) = \ln^2 (\theta_n/T)$ in log-temperature space, which results in the logarithmic mean estimator $\boldsymbol{\vartheta}^{\rm{(2l)}}$ with estimates
\begin{equation}\label{eq:RubioEstimator}
    \vartheta_n^{\rm{(2l)}} = \frac{E}{k_B} \exp \left[ \int \!\! \mathrm{d} T\, P(T|n) \ln \left( \frac{k_B T}{E}\right)\right],
\end{equation}
given a reference energy scale $E$ \cite{rubio2020global}. The authors employed the $1/T$-prior (i.e.~flat prior in $\ln T$), which necessitated a high- and low-temperature cutoff to guarantee finite estimates; Jeffreys' prior alleviates this issue.

\subsection{Uncertainty and error measures}

In the asymptotic large-data limit $N\to \infty$, the Bernstein-von Mises theorem states that the posterior \eqref{eq:posterior} will be sharply peaked like a Gaussian around the true $T$-value and thus the temperature estimates based on the various mentioned estimators should all eventually converge to that true value \cite{Vaart1998,LeCam}.

At small $N$ however, the posterior distribution remains broad and influenced by the prior, leading to a high degree of uncertainty as well as likely deviations from the true value. It is therefore crucial to associate errors $\varepsilon_n$ to the estimated $\vartheta_n$, which faithfully reproduce the actual uncertainty about the true temperature without knowing it. In fact, the precise (and often biased) $\vartheta_n$-values are not relevant so long as the corresponding uncertainty around them is large.

In Bayesian single-parameter estimation, uncertainty can be universally measured in terms of posterior quantiles: parameter values $\theta_n^{X\%}$ at which the cumulative posterior distribution reaches a certain percentage level,
\begin{equation}\label{eq:quantile}
    \int_0^{\theta_n^{X\%}} \!\!\mathrm{d} T \, P(T|n) \stackrel{!}{=} X\%.
\end{equation}
Quantiles are parametrization-invariant if Jeffreys' prior is used, they are independent of the chosen estimator, and they exist whenever the posterior has finite norm. We can reasonably claim with, say, $90\%$ credibility that the true temperature lies within the $5\%$- and the $95\%$-quantile for measured $n$, $\theta_n^{5\%} \lesssim T \lesssim \theta_n^{95\%}$. This suggests a temperature uncertainty of
\begin{equation}\label{eq:error90}
    \varepsilon_n^{90\%} (\vartheta_n) = \theta_n^{95\%} - \theta_n^{5\%}
\end{equation}
regardless of the choice of estimator $\boldsymbol{\vartheta}$. The so defined credibility region reflects what temperatures the experimenter deems possible with 90\% probability after obtaining a single measurement outcome $n$.

Bayes' rule links the credibility region around $\vartheta_n$ to the corresponding confidence interval $[\vartheta_a,\vartheta_b]$ of estimated temperatures, as determined by the greatest integers $a,b$ such that the cumulative sum of likelihoods yields $\sum_{n=0}^a P(n|T) \leq 5\%$ and $\sum_{n=0}^a P(n|T) \leq 95\%$ at a given true $T$ \cite{Jaynes1976, jaynes2003probability}. The confidence interval depends on the chosen estimator and reflects the range of temperature estimates an experimenter would obtain in 90\% of measurement instances on a reservoir at temperature $T$.
A well-behaved prior and estimator would ensure that the 90\%-credibility and the 90\%-confidence intervals are comparable and cover both the estimate $\vartheta_n$ and the true temperature $T$ for any possible outcome $n$ and detectable $T$-value.

Our numerical case study in Sec.~\ref{sec:results} reveals that the credibility-based error \eqref{eq:error90} tends to overestimate the actual deviation between estimated and true temperature whenever the latter exceeds the reference scale $E/k_B$. Hence, \eqref{eq:error90} is a rather conservative error measure, and errors based on the estimators' underlying cost functions are a convenient and possibly more accurate alternative.

To this end, consider the posterior-averaged cost $\langle c(\vartheta_n, T) \rangle$ given outcome $n$.
It quantifies how much the estimated $\vartheta_n$ could still deviate on average from the unknown true temperature. If the cost function $c(\vartheta_n, T)$ is already in units of temperature, then an appropriate multiple of its posterior average (divided by $\vartheta_n$) can directly serve as an absolute (or relative) error measure. Otherwise, the average must be translated to an associate temperature scale first. For the cost functions considered here, we can define the temperature errors as
\begin{eqnarray} \label{eq:errors}
    \varepsilon_n^{(1)} &=& 4.12\, \langle c^{(1)} (\vartheta_n^{(1)}, T) \rangle, \\
    \varepsilon_n^{\rm{(1r)}} &=& 4.12\, \vartheta_n^{\rm{(1r)}} \langle c^{\rm{(1r)}} (\vartheta_n^{\rm{(1r)}}, T) \rangle , \nonumber \\
    \varepsilon_n^{(2)} &=& 3.29\, \sqrt{ \langle c^{(2)} (\vartheta_n^{(2)}, T) \rangle }, \nonumber \\
    \varepsilon_n^{\rm{(2r)}} &=& 3.29\, \vartheta_n^{\rm{(2r)}} \sqrt{ \langle c^{\rm{(2r)}} (\vartheta_n^{\rm{(2r)}}, T) \rangle }, \nonumber \\
    \varepsilon_n^{\rm{(2l)}} &=& 2 \vartheta_n^{\rm{(2l)}} \sinh \left[ \frac{3.29}{2} \sqrt{ \langle c^{\rm{(2l)}} (\vartheta_n^{\rm{(2l)}}, T) \rangle } \right] \nonumber
\end{eqnarray}
The singular cost function underlying the estimator $\boldsymbol{\vartheta}^{\rm{(md)}}$ does not produce a meaningful error measure. For fair comparison, the prefactors in the first four lines are chosen such that the errors all agree with the $90\%$-credibility measure \eqref{eq:error90} for Gaussian posteriors sharply peaked around the estimated temperatures, as expected in the large-$N$ limit. For the logarithmic estimator $\boldsymbol{\vartheta}^{\rm{(2l)}}$, the posterior-averaged cost function gives the variance $\sigma_n^2 = \langle c^{\rm{(2l)}} (\vartheta_n^{\rm{(2l)}}, T) \rangle$ around the estimated $\ln \vartheta_n^{\rm{(2l)}}$ in $\ln T$-space. We obtain from this a 90\%-matched temperature interval by converting the two boundary values $\ln \vartheta_n^{\rm{2l}} \pm 3.29 \sigma_n/2$ to $T$-space and taking the difference.

\section{Qubit thermometry}
\label{sec:model}

\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{sketch.pdf}
\caption{\label{fig:sketch}Sketch of the qubit thermometer. $N$ ground-state qubits exchange heat with a bosonic reservoir at temperature $T$ and rate $\gamma$, each for a duration $\tau$. This partial thermalization brings the qubit ensemble to a mixed state with excitation probability $q_{\tau}$. The experimenter infers the temperature from the measured number of excitations by means of a Bayesian estimator $\boldsymbol{\vartheta}$.}
\end{figure}

For our case study, we consider the energy-based qubit thermometry setting sketched in Fig.~\ref{fig:sketch}, in which a bosonic thermal reservoir of temperature $T$ is successively probed by $N$ identical qubits with energy gap $E$. Each probe exchanges heat with the reservoir through a weak thermal contact over a fixed duration $\tau$, after which it is decoupled again and replaced with the next qubit probe. Once every probe has undergone its partial equilibration with the reservoir, the experimenter measures the number $n=0,\ldots N$ of excitations in the ensemble.
We assume that the qubit probes are prepared in their ground state,  which corresponds to the most temperature-sensitive full-swap protocol of collisional thermometry studied in \cite{Seah2019a}. Correlations between subsequent probes as studied in \cite{Shu2020} are precluded here.

The thermal coupling is described by the standard master equation for a qubit state $\rho$ in an oscillator bath, with rate parameter $\gamma$. In the rotating frame,
\begin{equation}
    \dot{\rho} = \frac{\gamma}{1-e^{-E/k_B T}} \left( \mathcal{D}[|0\rangle\langle 1|]\rho + e^{-E/k_B T} \mathcal{D}[|1\rangle\langle 0|]\rho \right),
\end{equation}
with $\mathcal{D}[\Op{A}]\rho = \Op{A}\rho\Op{A}^\dagger - \{\Op{A}^\dagger\Op{A},\rho\}/2$, and $|0\rangle$ and $|1\rangle$ the qubit's ground and excited state. Crucially, the master equation predicts an enhanced effective thermalization rate of approximately $\gamma k_B T/E$ in the high-temperature regime $k_B T \gg E$. After the coupling time $\tau$, the initial ground state evolves into a mixture of ground and excited state with excitation probability
\begin{equation}\label{eq:qtau}
q_\tau (T) = \frac{1 - e^{-\gamma \tau \coth (E/2k_B T)} }{1+e^{E/k_B T}}.
\end{equation}
In the limit of long coupling times, $\gamma \tau \gg 1$, the probes equilibrate to the Gibbs state, $q_\infty (T) = 1/(1+e^{E/k_B T})$. Measuring the number of excitations in the probe ensemble then amounts to an equilibrium thermometry scheme with $N$ repetitions.

Measurement sensitivity degrades exponentially in the low-temperature limit $k_B T \ll E$ where $q_\tau (T) \sim e^{-E/k_B T} $ converges to zero. At high temperatures and thermal equilibrium, the excitation probability saturates at $1/2$ and the sensitivity also degrades, $ q_\infty (T) \approx 1/2 - E/4k_B T$ for $k_B T \gg E$. In contrast, a finite time $\tau$ yields
\begin{equation} \label{eq:qtau_highT}
    q_\tau (T) \xrightarrow{k_B T \gg E} \left( \frac{1}{2} - \frac{E}{4k_B T} \right) \left(1-e^{-2\gamma \tau k_B T/E} \right),
\end{equation}
which de-saturates excitations at sufficiently small $\gamma \tau $. Indeed, we will discuss the sensitivity range of the thermometer more thoroughly in Sec.~\ref{sec:sensitivity range} and demonstrate the resulting increase in high-temperature accuracy for small $\gamma\tau$ in Sec.~\ref{sec:noneqResults} below.

\subsection{Likelihood and Jeffreys' prior}

The excitation probability of each qubit probe represents an independent coin toss with ``winning'' probability $q_\tau (T)$, and the likelihood for $n$ excitations in $N$ trials follows a Bernoulli chain,
\begin{equation}
    P_\tau(n|T) = \binom{N}{n} q_\tau^n (T) \left[ 1 - q_\tau (T) \right]^{N-n}, \label{eq:Bernoulli}
\end{equation}
which converges to a fair coin toss in the asymptotic limit $T\to \infty$.
The corresponding Jeffreys prior does not depend on the number of trials,
\begin{equation} \label{eq:JeffreyQubit}
    P^{(0)}_\tau (T) = \frac{2 \partial_T q_\tau (T)}{\pi \sqrt{q_\tau (T) [1-q_\tau(T)]}}.
\end{equation}
We omit the lengthy expression of the $T$-derivative here. The prior is correctly normalized, but already its first moment in $T$ diverges, because of the asymptotic behaviour $P_\tau^{(0)} (T) \sim T^{-2}$ for $T \to \infty$.
Crucially, this implies that the first posterior moments $\langle T\rangle$ as well as the higher ones diverge for any outcome $n$, such that the mean estimator $\boldsymbol{\vartheta}^{(2)}$ and the error measures $\boldsymbol{\varepsilon}^{(1)}$ and $\boldsymbol{\varepsilon}^{(2)}$ are no longer well-defined. If one tries to remove the divergence by restricting the allowed temperature range to $T \in [0,T_{\max}]$, then the respective estimates and errors will become sensitive to the chosen upper bound $T_{\max}$. Hence the mean estimator $\boldsymbol{\vartheta}^{(2)}$ and the two moment-based error measures $\boldsymbol{\varepsilon}^{(1)}$ and $\boldsymbol{\varepsilon}^{(2)}$ are not suitable for noninformative qubit thermometry.
The median estimator $\boldsymbol{\vartheta}^{(1)}$ in combination with the credibility region $\boldsymbol{\varepsilon}^{90\%}$, on the other hand, would not diverge, and neither do the relative estimators and associated errors in Sec.~\ref{sub:estimators}.

Picking a simpler prior with worse asymptotic behaviour should be avoided as it adds to the complications. For example, the $1/T$-prior cannot be normalized, and it invalidates not only the mean estimator $\boldsymbol{\vartheta}^{(2)}$, but also the median estimate $\vartheta^{(1)}_0$ when zero excitations are measured, due to divergence at $T\to 0$. Alternatively, if a finite $T$-range is imposed a priori, the estimates will be sensitive to the chosen temperature bounds.
Estimation of bounded parameters, as studied in quantum phase metrology, is typically not plagued by such problems.

\subsection{Sensitivity range}
\label{sec:sensitivity range}

While the uninformed approach does not exclude temperatures prior to measurement, any thermometer will only be accurate within a certain temperature range determined by the likelihood function. In the qubit case, temperatures much smaller (greater) than $E/k_B$ are no longer distinguishable from $T=0$ ($T=\infty$).

We propose to estimate the sensible range of temperatures by comparing the likelihoods in the low- and high-temperature limit. Given the $N$ bits of information the measurement provides in our scenario, we deem two temperatures $T_1, T_2$ barely distinguishable if the relative base-2 entropy between their likelihoods,
\begin{equation}
    D(T_1 \| T_2) = \sum_{n=0}^N P(n|T_1) \log_2 \frac{P(n|T_1)}{P(n|T_2)},
\end{equation}
measures no less than 1\,bit. It quantifies the amount of discriminating information for outcomes sampled from either likelihood. For the Bernoulli chain \eqref{eq:Bernoulli}, we get
\begin{eqnarray}
    D(T_1 \| T_2) &=& N q_\tau (T_1) \log_2 \frac{q_\tau (T_1)}{q_\tau (T_2)}  \\
    &&+ N [1-q_\tau (T_1)] \log_2 \frac{1-q_\tau (T_1)}{1-q_\tau (T_2)}.\nonumber
\end{eqnarray}
It is zero only if $T_1=T_2$, and otherwise a positive number of at most $N$ bits. We can thus restrict our view on temperatures between the boundaries $T_0$ and $T_{\infty}$ at which $D(0 \| T_0) = 1$ and $D (\infty \| T_{\infty}) = 1$, that is,
\begin{equation}\label{eq:Tbounds}
    q_\tau (T_0) = 1-2^{-1/N}, \quad q_\tau (T_{\infty}) = \frac{1-\sqrt{1-4^{-1/N}}}{2} .
\end{equation}
Temperatures outside this range will not be discernible by the $N$-probe measurement.

\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{T_range.pdf}
\caption{\label{fig:Trange} (a) Lowest and highest detectable temperature, $T_0$ (blue, lower) and $T_{\infty}$ (red,upper) in units $E/k_B$, as well as $T_{\infty}/T_0$ (dashed) as a function of thermalization time $\gamma\tau$ for 200 qubits. (b) Highest attainable temperature resolution relative to equilibrium, $\max_T I_\tau (T)/\max_T I_\infty (T)$  versus $\gamma \tau$.}
\end{figure}

Figure \ref{fig:Trange}(a) depicts the boundary values $T_{\infty}$ and $T_0$ as well as the ratio between them as a function of $\gamma\tau$ for $N=200$ qubits. The right end of the diagram corresponds to the equilibrium thermometry limit, $\gamma \tau \gg 1$. We observe that non-equilibrium thermometry at $\tau < \infty$ shifts and extends the \emph{range} of detectable temperatures towards higher values with decreasing thermalization time $\tau$---a useful feature when nothing is known about the expected temperature scale a priori.

However, the increased range comes at the price of reduced local temperature \emph{accuracy}, i.e.~lower sensitivity of the likelihood function to changes in the true $T$-value. This amounts to an overall lower Fisher information $I_\tau (T)$ of the likelihood with respect to $T$ for a given $\tau$. As a figure of merit, we plot the highest value relative to the equilibrium case, $\max_T I_\tau (T)/\max_T I_\infty (T)$ in Fig.~\ref{fig:Trange}(b). It decreases monotonically with decreasing $\gamma \tau$, which implies that the highest local temperature sensitivity is always achieved at equilibrium.
Exemplary detailed comparisons between equilibrium and non-equilibrium estimation are provided in Sec.~\ref{sec:noneqResults}.

\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{eqm_est_comp_corr.pdf}
\caption{\label{fig:comparison equlibrium} (Color online) Comparison of the Bayesian estimators from Sec.~\ref{sec:bayes} and their associated error measures \eqref{eq:errors} as a function of the true temperature $T$ for 200 fully thermalized qubits.
(a) Outcome-averaged estimated temperatures $\bar\vartheta (T)$ from \eqref{eq:avgEstErr} relative to $T$; (b) average RMS deviations \eqref{eq:RMSerr} relative to $T$; (c) average error measures $\bar\varepsilon (T)$ from \eqref{eq:avgEstErr} relative to the RMS deviations, excluding the mode estimator. The grey regions indicate temperatures outside the detectable range given by \eqref{eq:Tbounds}. }
\end{figure*}

Assuming tunable probe parameters, the trade-off between temperature range and local accuracy can be exploited in a two-stage measurement protocol: The first measurement stage would be carried out with a rapid sequence of qubits at small $\gamma \tau$ in order to efficiently narrow down the reservoir's temperature scale and choose an appropriate energy scale $E$ for optimal sensitivity in the second stage of equilibrium thermometry.

\section{Numerical assessment}\label{sec:results}

We assess the performance of different temperature estimators in qubit thermometry. For the most part, we consider thermal equilibrium, achieved in the limit of long qubit-reservoir coupling times, $\gamma\tau \gg 1$. Nonequilibrium results at shorter times are presented in Sec.~\ref{sec:noneqResults}.

Local and global figures of merit for the average performance over a broad range of temperatures are discussed in Secs.~\ref{sec:equilResults} and \ref{sec:globalScaling}, respectively.
Additionally, we assess the influence of different chosen priors in Sec.~\ref{sec:priorCompare}, showing that Jeffreys' prior yields more accurate estimates than biased priors.

In our computations, we discretized the temperature support with a step size of $1 \cdot 10^{-3}$ in the range $k_B T/E \in [0.01,200]$, which was chosen so that the cutoffs were well outside of the sensitive range dictated by \eqref{eq:Tbounds}.

\subsection{Comparison of estimators}\label{sec:equilResults}

We compare the various estimators introduced in Sec.~\ref{sec:bayes} in terms of the bias and the associated error measure. To this end, we shall employ the weighted averages over all outcomes at a given true temperature $T$,
\begin{equation}\label{eq:avgEstErr}
    \bar\vartheta (T) = \sum_n P(n|T) \vartheta_n, \quad \bar\varepsilon (T) = \sum_n P(n|T) \varepsilon_n.
\end{equation}
In an actual  experiment, only one out of $N+1$ random outcomes would be observed, leading to a random estimate $\vartheta_n$. It is therefore crucial that the error measures $\varepsilon_n$ accurately capture the spread of random outcomes, as quantified by the 90\% confidence interval, and the deviation from the true $T$-value. The latter can be measured in terms of the root-mean-square (RMS) deviation,
\begin{equation}\label{eq:RMSerr}
    \bar\varepsilon_{\text{RMS}} (T) = 3.29\sqrt{\sum_n P(n|T) (\vartheta_n-T)^2},
\end{equation}
once again scaled to match 90\% credibility.

Figure \ref{fig:comparison equlibrium} plots (a) the outcome-averaged biases $\bar\vartheta (T)$, (b) the RMS deviations $ \bar\varepsilon_{\text{RMS}} (T)$, and (c) the measured errors $\bar\varepsilon (T)$ of the various estimators against $T$. The results were evaluated for a rather coarse measurement with $N=200$ qubits at equilibrium, and using Jeffreys' prior. The grey shaded regions mark temperatures outside the detectable range $[T_0,T_\infty]$ defined in Sec.~\ref{sec:sensitivity range}.

Considering Fig.~\ref{fig:comparison equlibrium}(a) alone, one might rule out the mean estimator $\boldsymbol{\vartheta}^{(2)}$ based purely on the large bias at high temperatures. However, it is unclear whether the median $\boldsymbol{\vartheta}^{(1)}$ and the logarithmic $\boldsymbol{\vartheta}^{\rm{(2l)}}$ are better than the relative estimators $\boldsymbol{\vartheta}^{\rm{(2r)}}$ and $\boldsymbol{\vartheta}^{\rm{(1r)}}$ or the mode $\boldsymbol{\vartheta}^{\rm{(md)}}$.

Bias is not the most important figure of merit when assessing the estimators since it only provides information about how good the estimator is on average and not how much the estimator will vary from experiment to experiment. The latter is better represented by the average RMS deviation of the estimates relative to the true temperature, as shown in Fig.~\ref{fig:comparison equlibrium}(b). A similar performance is seen for all estimators in the intermediate temperature range $k_B T/E \lesssim 1$ at which the thermometer is most sensitive. At higher temperatures, on the other hand, the mean, the median, and the logarithmic mean perform significantly worse than the mode and the two relative estimators. The latter are slightly worse close to the low end of the detectable $T$-spectrum, but generally perform best overall.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{gallery_corr.pdf}
\caption{\label{fig:gallery} Temperature estimates and errors for $N=200$ thermal qubits, complementing Fig.~\ref{fig:comparison equlibrium}. Each two-panel column shows the results for the respective Bayesian estimator referenced by the label on top: the mode in (a,g), the median in (b,h), the relative median in (c,i), the mean in (d,j), the relative mean in (e,k), and the logarithmic mean in (f,l). In the top row (a-f), the solid curves show the average biases $\bar \vartheta (T)$, the dashed lines delimit the 90\% credible regions, and the green shaded regions from light to dark mark the 100\%, 90\%, and 50\% confidence intervals, all relative to the true temperature $T$.
The bottom row (g-l) shows the RMS deviation from $T$ (solid lines), the credible region widths $\bar\varepsilon^{90 \%}$ (dashed), the average measured errors $\bar\varepsilon(T)$ (dotted), and the 90\% confidence intervals (green shades), all given in \% with respect to $T$. The vertical grey bars delimit the detectable temperature range.}
\end{figure*}

A good estimator is of limited use if it lacks an associated error measure that the experimenter can evaluate without knowing the true $T$. The error measures discussed in \eqref{eq:errors} depend only on the posterior distribution and thus require no knowledge of the true temperature. For the estimator to be useful, the outcome-averaged associated error measure $\bar\varepsilon (T)$ should match the actual deviation $\bar\varepsilon_{\text{RMS}} (T)$ at a given true $T$. We plot the ratio of both quantities in Fig.~\ref{fig:comparison equlibrium}(c).
Once again, the relative estimators stand out over the whole detectable $T$-range. The mode estimator, which performs equally well in terms of bias and RMS deviation, lacks a meaningful error measure and is therefore absent. We will omit it for the rest of our analysis.

Further detail is seen in Fig.~\ref{fig:gallery}. Here, the top row depicts the estimators' biases (solid curves), their 90\%-credible regions (between the dashed curves), as well as their associated 100\%, 90\%, and 50\% confidence intervals (light to dark green shades). In the bottom row, we compare the relative RMS deviations $\bar\varepsilon_{\text{RMS}}(T)/T$ (solid) to the 90\% credibility ranges $\bar \varepsilon^{90\%}$ (dashed), the measured errors $\bar\varepsilon/T$ (dotted), and the 90\% confidence interval (green shade).

Both the RMS deviations and the measured errors capture the actual spread of outcomes, i.e.~confidence intervals, quite well within the sensitive region of temperatures. The confidence intervals depend on the estimator and vary in size quite notably.
Unfortunately, the 90\%-credibility range $\bar \varepsilon^{90\%}$ from \eqref{eq:error90}---a universal and robust error measure that does not depend on the chosen estimator---consistently overestimates the actual uncertainties at higher temperatures and is thus not a viable measure.
The error measures associated with the median and the mean estimator also rise quickly at high $T$, and they would diverge if we included arbitrarily high temperatures in our assessment. Hence, their high-temperature values are unreliable as they vary with increasing numerical temperature cutoff. Considering all these limitations, we conclude that the relative mean and median are the best temperature estimators.

\subsection{Global error scaling}\label{sec:globalScaling}

Next, we discuss how the estimators' underlying temperature-averaged cost functions \eqref{eq:costFunc} and averaged errors scale with an increasing number of qubits. This has emerged as a figure of merit to quantify the ``global'' performance of temperature estimators in the Bayesian framework \cite{rubio2020global,alves2020bayesian}. Given that the $T$-moments diverge when integrating over all true $T\geq 0$, we shall restrict the integration to a finite range of relevant temperatures, $0< T_1 \leq T \leq T_2 < \infty$, for a fair comparison between all estimators. That is, we restrict \eqref{eq:costFunc} to
\begin{equation}\label{eq:finite_c}
    \mathcal{C}_{\text{fin}} (\boldsymbol{\theta}) =  \frac{1}{\mathcal{M}} \int_{T_1}^{T_2} \!\! \mathrm{d} T\, P^{(0)}(T) \sum_{n=0}^N P(n|T) c(\theta_n, T),
\end{equation}
where we renormalize the prior \eqref{eq:JeffreyQubit} accordingly by
\begin{equation}
    \mathcal{M}
    = \frac{4}{\pi} \left[ \arcsin \sqrt{q_\tau (T_2)} - \arcsin \sqrt{q_\tau (T_1)} \right].
\end{equation}

The scaling of our considered average cost functions with $N$ is shown in Fig.~\ref{fig:scaling_cost} for a temperature range covering two orders of magnitude around the reference scale, $k_BT/E\in [0.1,10]$, as depicted in the previous figures. Naively, one may think that, because the cost functions of the logarithmic and relative mean estimators scale best, their associated estimators perform better overall. However, this claim is simply invalid since the plotted cost functions are all based on different parameterizations in $T$ and thus differ in dimension.

\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{scaling_cost.pdf}
\caption{\label{fig:scaling_cost} (Color online) Average costs \eqref{eq:finite_c} over the temperature range $k_B T/E\in [0.1,10]$ as a function of probe number $N$, evaluated for the listed estimators at thermal equilibrium. As the cost functions vary in dimension of $T$, we set $E/k_B \equiv 1$. }
\end{figure}

For a proper global comparison of the overall estimated uncertainty from low to high temperatures at given $N$, it is more reasonable to take the measured \emph{relative} temperature errors $\varepsilon_n/\vartheta_n$ and average them over all outcomes,
\begin{equation} \label{eq:globalE}
    \mathcal{E}_{\text{fin}}(\mathbf{\boldsymbol{\vartheta}}) =\sum_{n=0}^N P(n) \frac{\varepsilon_n}{\vartheta_n}.
\end{equation}
Here, the sub-index in $\mathcal{E}_{\text{fin}}$ denotes that the individual outcomes $(\vartheta_n,\varepsilon_n)$ taken from \eqref{eq:errors} be evaluated consistently, assuming the same restricted temperature range $[T_1,T_2]$ when computing the posterior \eqref{eq:posterior}. The resulting quantities $\mathcal{E}_{\text{fin}}$ are dimensionless figures of merit for the characteristic relative temperature uncertainties obtained from the estimators within that range.

\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{scaling_err_corr.pdf}
\caption{\label{fig:scaling_error} (Color online) Outcome-averaged relative errors \eqref{eq:globalE} as a function of probe number $N$, assuming the finite temperature range $k_BT/E\in [0.1,10]$. The listed estimators are compared to the $T$-averaged Cram\'er-Rao bound \eqref{eq:globalCRB} (light shade) and the global error benchmark \eqref{eq:vanTreesBoundEQ} (dark) at thermal equilibrium.}
\end{figure}

In Fig.~\ref{fig:scaling_error}, we plot the $N$-scaling of the overall relative errors \eqref{eq:globalE} averaged over the same range as before, $k_BT/E\in [0.1,10]$. At small $N$, the various estimators show different scaling behaviours, but from about $10^3$ probes onwards, all coincide and scale asymptotically like $1/\sqrt{N}$. This is expected from the Bernstein-von Mises theorem, and it matches the local Cram\'er-Rao bound for the (90\%-equivalent) RMS deviation of unbiased estimators, $\bar\varepsilon_{\text{CRB}}(T) = 3.29/\sqrt{I(T)}$. For explicit comparison, we shall convert it to the relative deviation $\bar\varepsilon_{\text{CRB}} (T)/T$ and take the prior-weighted integral over $[T_1,T_2]$,
\begin{equation}\label{eq:globalCRB}
    \mathcal{E}_{\text{CRB}} = \int_{T_1}^{T_2} \!\! \mathrm{d} T \frac{P^{(0)}(T) \bar\varepsilon_{\text{CRB}}(T)}{\mathcal{M} T} =  \frac{ 6.58 \, \ln \left( T_2/T_1 \right)}{\pi \sqrt{N} \mathcal{M}}.
\end{equation}
This asymptotic figure of merit is marked by the light shade in Fig.~\ref{fig:scaling_error}. Indeed, our simulations indicate that the various estimators exhibit decreasing average bias across the chosen $T$-range for sufficiently large $N$, and so the relative errors match \eqref{eq:globalCRB} asymptotically.

While the Cram\'er-Rao deviation is a good figure of merit in the asymptotic limit, it does not capture the error scaling at finite $N$ where the estimators' biases are significant. Our detailed error analysis as a function of $T$ in Fig.~\ref{fig:comparison equlibrium} has shown that the error measures typically overestimate the actual RMS deviation \eqref{eq:RMSerr} from the true $T$-value. Let us therefore consider the square-root of the prior-weighted integral over the \emph{relative} square deviation $\bar\varepsilon^2_{\text{RMS}}/T^2$ as a global error benchmark,
\begin{equation}\label{eq:globalErRMS}
    \mathcal{E}_{\text{rRMS}} (\boldsymbol{\vartheta}) = 3.29 \sqrt{ \mathcal{C}^{\rm{(2r)}} (\boldsymbol{\vartheta}) } = \sqrt{ \int_0^{\infty} \!\! \mathrm{d} T\, P^{(0)}(T) \frac{\bar\varepsilon^2_{\text{RMS}}(T)}{T^2} }.
\end{equation}
Its dimensionless value depends on the chosen estimator through \eqref{eq:RMSerr}, but we can arrive at an estimator-independent lower bound with help of a modified van Trees inequality \cite{Gill1995},
\begin{eqnarray} \label{eq:vanTreesBound}
    \mathcal{E}_{\text{rRMS}} &\geq& \frac{3.29}{\sqrt{ \int_0^{\infty}\!\!\mathrm{d} T\, P^{(0)}(T) T^2 I(T) + I_0 }}, \\
    I_0 &=& \int_0^{\infty}\!\!\mathrm{d} T\, P^{(0)}(T) T^2 \left[ \partial_T \ln P^{(0)}(T) \right]^2, \nonumber
\end{eqnarray}
see App.~\ref{app:RMSbound}. Here we integrate over the full temperature range for a truly global reference bound, which does not depend on the chosen energy scale $E$ either. The square root in the denominator comprises the Fisher information of the likelihood, integrated over a modified prior distribution, and the offset contribution $I_0$ measuring the $T$-sensitivity of the prior. The latter is rendered negligible by sufficient data, as the former term grows with $I(T) \propto N$. Hence we obtain a temperature-averaged global version of the Cram\'er-Rao bound in the asymptotic limit $N\to\infty$.

However, the global RMS error \eqref{eq:vanTreesBound} is no longer a tight bound, as exemplified by the dark-shaded region in Fig.~\ref{fig:scaling_error}. The errors associated to the relative estimators are well approximated for small $N$, but then consistently underestimated as $N \to \infty$. For this equilibrium case, the integrals in \eqref{eq:vanTreesBound} have explicit analytic solutions, and the global benchmark simplifies to
\begin{equation} \label{eq:vanTreesBoundEQ}
    \mathcal{E}_{\text{rRMS}} \geq \frac{3.29}{\sqrt{ (N+1)\pi^2/8 - N + 1 }} \approx \frac{6.81}{\sqrt{N+9.56}},
\end{equation}
regardless of the qubit energy $E$.

\subsection{Influence of the prior}\label{sec:priorCompare}

\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{rms_prior_comp.pdf}
\caption{\label{fig:comparison prior} (Color online) Influence of the prior distribution on the estimators' accuracies for 200 fully thermalized qubits. We compare the RMS deviations from the true temperature $T$ for (a) the prior $P^{(0)}(T) \propto 1/T$ and (b) the constant prior over the range $k_B T/E \in [0.01,200]$. See Fig.~\ref{fig:comparison equlibrium}(b) for Jeffreys' prior. The grey bars delimit the detectable temperatures.}
\end{figure}

Up to this point in our assessment, we have based our analysis on Jeffreys' prior. However, simpler priors over a finite temperature range are often used in the literature. Here we consider the scale-invariant $1/T$-prior \cite{rubio2020global} and the constant prior \cite{alves2020bayesian} for comparison. The resulting RMS deviations of our estimators are shown in Fig~\ref{fig:comparison prior}(a) and (b), respectively, and should be compared to Fig.~\ref{fig:comparison equlibrium}(b) for Jeffreys' prior.

The results for the different priors agree only within a small range of temperatures $k_BT/E \lesssim 1$ at which the thermometer scheme is most sensitive. There the posterior distributions after 200 qubits have properly converged to a sharp quasi-Gaussian distribution and ``forgotten'' about the prior, which also explains why estimators all agree. However, the situation changes drastically outside the small optimal $T$-regime: both the $1/T$-prior in Fig~\ref{fig:comparison prior}(a) and the constant prior in (b) amplify the errors. For this reason, Jeffreys' prior is clearly a better choice leading to more accurate results in uninformed qubit thermometry.

\subsection{Nonequilibrium results} \label{sec:noneqResults}

So far, we have studied qubit thermometry at thermal equilibrium and found that the relative mean and median estimators consistently achieve the most accurate estimates and the most reliable errors. This remains the case when considering nonequilibrium probes due to short coupling times $\tau$, which mainly affects the high-temperature behavior of the likelihood function via \eqref{eq:qtau}.

\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{steady_rms_comp_1r.pdf}
\caption{\label{fig:comparison non-eqm}RMS deviation relative to temperature for the relative median estimator with 200 qubits. We compare the case of fully thermalized probes (dark blue line) to finite thermalization times, $\gamma \tau = 0.1$ (medium blue) and $\gamma \tau = 0.01$ (light blue). The solid stretches extend over the respective detectable temperature ranges given by \eqref{eq:Tbounds}. }
\end{figure}

Figure \ref{fig:comparison non-eqm} compares our exemplary 200-qubit equilibrium measurement ($\gamma\tau \to \infty$) with short-time non-equilibrium measurements ($\gamma\tau = 0.1$ and $0.01$) in terms of the average RMS deviation for the relative median estimator $\boldsymbol{\vartheta}^{\rm{(1r)}}$. The solid stretch of each curve extends over the respective range of detectable temperatures from \eqref{eq:Tbounds}. As predicted in Sec.~\ref{sec:sensitivity range}, we find that with shorter $\tau$, the detectable range widens and shifts towards higher temperatures, while the lowest achievable error grows. This trade-off could be controlled and exploited in adaptive estimation protocols.

\section{Conclusion} \label{sec:conclusion}

We have presented an uninformed Bayesian approach to quantum thermometry when a single $N$-probe measurement is performed without prior information about the expected temperature range. Focusing on the case of fully or partially thermalized qubit probes of fixed energy gap, we have compared known and novel temperature estimators and their associated error measures.

Within the range of temperatures accessible to the measurement, which can be gauged by evaluating the relative entropies of the likelihood function, we have observed striking differences in the attainable accuracy: the widely used mean estimator with its variance-based error measure gives profusely inaccurate predictions and errors at high temperatures. While the median and the recently proposed logarithmic mean estimator improve the accuracy, they do not rectify the exaggerated error.

Our study shows one can achieve a consistently better accuracy and matching error measure by means of either the relative mean or the relative median estimator. The former takes the posterior mean of $1/T$ divided by that of $1/T^2$ as the temperature estimate for a given outcome, while the latter uses the median of the posterior distribution divided by $T$ (and normalized accordingly). Our Bayesian procedure is based on the uninformed Jeffreys prior, which exhibits diverging moments in $T$. Hence the conventional mean and median estimators depend on a high-temperature cutoff, whereas the logarithmic and our relative estimates are always finite. Commonly used alternative priors such as the flat or the scale-invariant one aggravate the divergence problem and generally worsen the estimates.

The temperature-averaged relative errors for most estimators adhere to the likewise averaged Cram\'{e}r-Rao bound in the asymptotic limit $N\to\infty$. At small probe numbers, we find that their scaling is better described by a modified van Trees bound for the relative root-mean-square deviation between estimated and true temperature. This bound is not tight, but universal in that it holds for any temperature range and depends only on $N$.

Finally, we have demonstrated the trade-off between local accuracy and temperature coverage in nonequilibrium thermometry of bosonic reservoirs. By shortening the coupling time of each probe and thereby preventing full thermalization, one can vastly broaden the range of detectable temperatures to higher values at the cost of greater overall errors. This could be a useful control feature for adaptive temperature estimation protocols.

\emph{Acknowledgments.---}
S.S. acknowledges funding from the Swiss National Science Foundation (NCCR SwissMAP). J.B. acknowledges support from the House of Young Talents (HYT) of Siegen University.

redpajama_set_name: RedPajamaArXiv


Drivers are software modules, which can be loaded onto the integrated framework (such as TridiumAX® or North ObSys) to provide intelligent interfacing to proprietary 3rd party systems. They provide the ability to control, monitor and log elements within a sub-system as well as creating and receiving alarms. Drivers can be downloaded or emailed and installed onto the different platforms and are licensed according to your requirements. We have a range of over 250 drivers available for different applications and for different platforms.
The NiagaraAX Framework® allows NiagaraAX® certified development partners such as Tyrrell Products to build bespoke products from the native framework, targeted at specific applications. Typically using the JACE® hardware platform, appliances can be built such as chiller controls or sequencing, lighting controls, or automated interfaces and solutions. Appliances focus NiagaraAX® towards making engineering and interaction more friendly, whilst the business of setting up drivers and framework specifics is handled automatically by the appliance software. We have a range of appliances available as key elements of our major products such as the Advanced System Interface for Branch Monitoring, eBMS/HVAC™ or the Apollo Open Connect gateway product and the configuration tool for eBMS/Door Entry™.
We offer a full service for Original Equipment Manufacturers to design, develop and deliver drivers and appliances. If you have an application to consider, get in touch with our team today.


text:
Africa trip, 1957 14
Birmingham, Ala
''Religion and Peace of Soul''
Schweitzer, Albert
1 John 14:27
Foreword, ''The Negro Leadership Class''
Englewood Cliffs, N.J.
Englewood Cliffs, N.J.: Prentice-Hall, 1963.
Thompson, Daniel C.
Coordinating Council of Greater New Orleans
Coppin State Teachers College (Baltimore, Md.)
Jones, Alvin [Xavier University (New Orleans, La.)]
Times-Picayune (New Orleans, La.)
Congress of Industrial Organizations (U.S.)
Xavier University (New Orleans, La.)
Immaculate Conception Church (New Orleans, La.)
United States. House of Representatives. Committee on Un-American Activities
Bronx Citizens to Elect Kennedy
Opelousas, La.
Draft, ''The Good Samaritan''
On Being a Good Neighbor
Form letter to Martin Luther King, Jr.
Harriman, N.Y.
Martin Luther King, Jr.--membership in organizations
UIAD-MiDW-AL
UAW International Affairs Department, Victor Reuther Files, 1962-1968, Wayne State University, Walter P. Reuther Library of Labor and Urban Affairs, Detroit, Mich.
Letter to Ralph J. (Ralph Johnson) Bunche
Martin Luther King, Jr.- Membership in organizations
RJBPC-CLU-AR
Ralph J. Bunche Collection (2051), University of California, Los Angeles, Charles E. Young Research Library, Special Collections, Los Angeles, Calif.
''The Parable of Dives and Lazarus,'' Sermon Delivered at Ebenezer Baptist Church
Putnam, Carleton
Race and Reason
''The Parable of Dives and Lazarus,'' Sermon at Ebenezer Baptist Church
Philharmonic Orchestra
Carnegie Hall (New York, N.Y.)
University of California (Los Angeles, Calif.)
5/9/2012 kpp
Draft, ''The Splitting South and the Kennedy Administration''
Guihard, Paul
Gunter, Ray
Russell, Donald S. [South Carolina. Office of the Governor]
Sanford, Terry [North Carolina. Office of the Governor]
Sanders, Carl Edward [Georgia. Office of the Governor]
Allen, Ivan [Atlanta (Ga.). Office of the Mayor]
Weltner, Charles L. [United States. Congress. House of Representatives]
Johnson, Leroy R. [Georgia. State Senate]
Lincoln, Abraham [United States. White House]
United States. National Labor Relations Board
Atlanta Constitution
North Carolina. Office of the Governor
South Carolina. Office of the Governor
Georgia. Office of the Governor
Georgia. State Senate

redpajama_set_name: RedPajamaCommonCrawl


text:
Wednesday, 23 February 2022 / Published in Immigration Law_news, News
The National Immigration Authority established a new procedure for domestic worker visa applications.
The Resolution No. 2579 of February 10, 2022, published in the Official Gazette on February 21, 2022 (hereinafter the "Resolution") established a new procedure for Domestic Worker Visa applications, as follow:
The registration and filed application of Domestic Worker Visa was authorized, prior to the applicant entry to Panama, which must be done through an attorney. The power of attorney must be authenticated by apostille or by the Panamanian Consulate in the country in which is granted.
The National Immigration Authority will proceed with the registration and opening of the process, prior to the payment of the cost of the immigration process and immigration Id card.
Once the application is filed, the applicant can enter to Panama, and he must come to the National Immigration Office to continue with the process.
Foreigners who are citizens from a country with a waiver of visa agreement or that do not require visa, may apply for Domestic Worker Visa as established in this Resolution.
Foreigners who are citizen from a country that require visa to entry to Panama, must obtain first the approval of their tourist visa in order to apply with the procedure established in this Resolution.
This Resolution will take effect from its promulgation.
With this Resolution the National Immigration Authority authorized to file the Domestic Worker Visa application, prior the entry to Panama. Before, the foreigner had to be in Panama in order to file the application to the Nation Immigration Authority.
Once the applicant entre to Panama he must come to the National Immigration Authority to continue with the process and to obtain his immigration Id card.
Through this Resolution, it is established that foreigners, who are citizen from a country with a waiver of visa agreement or that do not require visa to enter to Panama, can directly file the Domestic Work Visa application, however, foreigners, that require visa to entry to Panama, must request their tourist visa and obtain the approval of it before they file the Domestic Worker Visa application.
This Resolution will take effect from February 21, 2022.


SYB type filter feed pump is dedicated by the company in connection with the introduction of foreign technology and manufacturer of the filter and the actual site conditions, well design and development of a new generation of leak-free filter pumps series.
Hot water boilers. Grundfos' single-stage and multistage pumps for hot water boilers are available for temperatures up to 140°C. All pumps can be supplied with motors with a wall-mounted or integrated frequency converter (up to 22 kW), ensuring that the pressure and temperature in the system are constant.
This Asia-Pacific Boiler Feed Water Pump Market report studies the global Asia-Pacific Boiler Feed Water Pump market status and forecast, categorizes the global Asia-Pacific Boiler Feed Water Pump market size (value & volume) by key players, type, application, and region.
A full range of tanks, pumps, motors, electrical parts, valves, fittings, control equipment, steam traps, PRV's, safety valves, boiler tubes, gaskets, packing, refractory material, laggings and many other items is maintained to ensure availability for our customers.
China product high pressure electic boiler feed water pump clean water GC series high pressure electic boiler feed water pump is horizontal single-suction multi-stage segmental vertical centrifugal pump, mainly used for boiler feed water, so also known as boiler feed pump.
PRA Peripheral Pumps. The Ebara PRA range of peripheral water pumps are available from 0.37kW to 1.5kW and are useful when water flows required are small but with high pressure. For example a Boiler feed application.
The aim of Southern Pumps is to develop reliable pumps geared to market and application requirements, by making use of advanced technology and a modern flexible production organisation. Throughout the production process, until the final delivery, the pumps are subject to stringent quality control.
The feed pump pumps this water back to the boiler and the cycle repeats itself. Apart from these four main parts, the system also includes a drain tank that collect the condensate from the condenser and provides a suction head to the feed pump for transferring the water back to the boiler.
Boiler. Production lines depend heavily on efficient and reliable boiler operation. This is why Grundfos' robust feeding pumps for steam boilers, hot water boilers and thermal oil boilers have been specially designed to offer optimum performance in even the most extreme conditions.
A steam turbine driven feedwater pump is most often applied because of the abundance of available steam. Boiler Optimization. With a boiler's firing rate held constant, feedwater pressure and flow affect boiler pressure. As electricity demands change, so do associated boiler loads.
Download the Boilers and Systems white paper Choosing the right method of level control in boiler feed systems is a major step towards operational efficiency. Learn about the advantages and disadvantages of level control options, and how Grundfos iSOLUTIONS can increase system performance.
Boiler Feed Pumps Page 1 Industrial boiler feed water pumps for clean water & steam Industrial boilers, for example on board a marine vessel, in a large industrial factory or power plant, are fed either fresh water or steam by a high pressure boiler feed pump.
This USA Boiler Feed Water Pump Market report studies the global USA Boiler Feed Water Pump market status and forecast, categorizes the global USA Boiler Feed Water Pump market size (value & volume) by key players, type, application, and region.


text:
Hanger – facebook
Hanger – twitter
Hanger – instagram
Hublot Big Bang Unico Sapphire
By Celeste Goh • Mar 2nd, 2016 in Tick Talk
When it comes to Hublot's Big Bang Unico series, it has always been about adding innovation and expertise with materials to the history of watchmaking firsts. This year, it's the Big Bang Unico Sapphire shining in the spotlight.
Sapphire, a material almost as hard and scratch-resistant as diamond, has until now been an extremely difficult material to machine, and is reserved for one-of-a-kind pieces or very private collections. But after the ambitious MP05 'LaFerrari' Sapphire project, there is nothing Hublot can't achieve when it comes to creating a collection of absolute transparency that consecrates the acclaimed architecture of the Big Bang model, and the performances of the Unico movement.
The case middle, bezel, and back of the Big Bang Unico Sapphire are cut from blocks of sapphire, and only a few pieces forming the spine of the timepiece are made from titanium; the signature H-shaped screws, the crown over-moulded with silicon, and the deployable buckle. The skeleton dial is made from transparent resin, revealing all the fears of the proprietary Unico HUB1242 movement and its finishes.
www.hublot.com


My client, a boutique independent investor specialising in Acquisitions and Disposals, are seeking an experienced Investment and development Consultant to join their solid team.
As part of organic and continued growth, they would like to welcome a further member to work closely with the Director, on Commercial and Residential deals.
My client are particularly active in large scale residential transactions (with and without planning permission) and extensive residential investments,on behalf of private and institutional clients, from both the UK and overseas.
The suitable consultant will have industry relevant experience, market knowledge, established contacts and a proven track record in brokering property investment and development deals in many sectors.
You will be confident to challenge decisions, bring on new ideas and add new life to the company.
If you are well presented, with excellent communication and negotiation skills, ambitious, self-motivated and extremely driven this could be the right opportunity for you.
If you see yourself as a real "deal maker" and feel it's time to be properly rewarded for your successes, whilst keen to embark on a career without limitations, get in touch.
The role will be based from Marylebone, with unrestricted sourcing territory.
This role is highly confidential.


text:
[AutoCorres][] is a tool which can automatically produce a monadic abstraction
of a C program, together with an [Isabelle/HOL][Isabelle] proof of
correspondence between the abstraction and a [low-level embedding][c-parser] of
the C program. AutoCorres was developed primarily by [David Greenaway][] during
his [PhD][] at [NICTA][] and [UNSW][].

[AutoCorres]: http://ssrg.nicta.com.au/projects/TS/autocorres/
[Isabelle]: http://isabelle.in.tum.de/
[c-parser]: http://ssrg.nicta.com.au/software/TS/c-parser/
[David Greenaway]: http://www.davidgreenaway.com/
[PhD]: http://ssrg.nicta.com.au/publications/nictaabstracts/8758.pdf
[NICTA]: https://www.nicta.com.au/
[UNSW]: https://www.cse.unsw.edu.au/

AutoCorres is currently distributed in the [tools][] directory of the [l4v][]
verifcation of the [seL4][] microkernel.

[tools]: https://github.com/seL4/l4v/tree/master/tools
[l4v]: https://github.com/seL4/l4v
[seL4]: http://sel4.systems/

This repository contains a couple of example proofs using AutoCorres, which I
reworked for my own educational purposes.

redpajama_set_name: RedPajamaGithub


Filipa works for Transition Network, the small charity which aims to inspire, encourage, connect, support and train communities world-wide as they self-organise around the Transition model.
She is a Portuguese forest scientist by training. After 10 years working in EU Institutions and organisations related to sustainable development, she quit her job to dedicate herself fully to the Transition Movement.
Filipa is International Coordinator of the Transition National Hubs, exploring with people all over the world how to develop the social infrastructure, collaborative culture and new forms of leadership capable of supporting a distributed movement for social change. She also acts as a bridge between Transition Network and the EU (and other Brussels-based organisations).
She is co-founder of a Transition Initiative in Portalegre, Portugal and initiator of AJUDADA, an experimental community empowerment international event, under the theme of gift economy.
I feel excited and interested to join the Lush Spring Prize judging panel, seeing this as an opportunity to contribute in a different way to the systemic change many of us are dreaming of.


text:
The Best Mutual Fund Sectors
By: Patrick Gleeson, Ph. D.,
Some market sectors do comparatively better in bull markets, others in bear markets.
George Doyle/Stockbyte/Getty Images
1. The Difference Between Consumer Staples and Consumer Discretionary ETFs
2. Influences on Cyclical Stocks
3. How Do I Find Bullish Sector Stocks in a Bear Market?
When purchasing mutual funds, you can buy broad-based funds that invest across market sectors -- such as a fund that holds companies listed in the S&P 500 -- or you can buy sector funds that invest in a specific economic area, including consumer discretionary, consumer staples, energy, financial, health care, industrials, information technology, telecommunication services and utilities. There is no one sector that will always have the best returns. At different times in the market, some sectors will tend to do better than others.
The Market Cycle
The stock market moves forward in a never-ending cycle of economic expansion and investor confidence (bull market) followed by economic contraction and investor pessimism (bear market) Early in a bull market, growth stocks tend to outperform value stocks; later in a bull market and through a bear market, value stocks tend to do better than growth. Growth stocks are companies with recent better-than-average price, earnings and sales momentum. Value stocks are companies that analysts may under-rate and investors undervalue; in other words, they trade at a lower value than they're actually worth. Characteristics of value stocks are low price-to-earnings and price-to-book ratios and high dividend-to-share-price ratio. Book value is the value of the company's physical assets.
Relation of Sector to Cycle
In several instances, a sector will tend to do better in one part of the cycle than in another. Early in a bull market, consumer discretionaries often enjoy an advantage. Later in the cycle, consumer staples may do better. Consumer discretionary sector funds invest in companies that manufacture and sell the things we want to have, but do not absolutely need: for example, automobiles, furniture, recreational equipment and services, and designer clothing and goods. Consumer staples manufacture and market the things we need, including food, clothing and household items.
Necessity vs. Desire
The same distinction between consumer discretionaries and consumer staples -- things we'd like to have versus things we must have -- can be made for other sectors as well. Health care, for example, is clearly not optional and enjoys an advantage later in the cycle. Industrials and materials -- depending upon the companies chosen by the fund -- may do better during the expansion part of the cycle than later, when consumers tend to delay optional purchases. The energy and utility sectors manufacture and deliver goods and services we always need, and also may outperform later in the market cycle.
Other Trends
Other, non-cyclical trends may suggest fund sectors (and within the sector, funds devoted to a specialization) that will outperform others. Renewable energy funds, for example, will likely do comparatively better with a Democratic president and congressional majority; funds specializing in oil-, gas- and coal-related activities may do comparatively better with a Republican president and congressional majority. Health care is a complex issue, but also maybe analyzed in terms of political outcomes to determine, for example, which health care services will outperform with an expansion of medicare, and which may be relatively unaffected.
NYU Stern: Investor Sentiment and the Cross-Section of Stock Returns
Knowledge Wharton: Value Stocks vs. Growth Stocks: Timing Counts
Fidelity: How to Invest in Sectors Using the Business Cycle
Cornerstone: U.S. Energy Subsidies in Perspective
CBS News: The Fight Over Healthcare: Still Political?
I am a retired Registered Investment Advisor with 12 years experience as head of an investment management firm. I also have a Ph.D. in English and have written more than 4,000 articles for regional and national publications.
Do Rising Interest Rates Negatively Affect Stocks?
How to Invest in the Baltic Dry Index
Correlation of Stocks & Bonds
Cyclical Trends in the Stock Market
How do I Find Dirt Cheap Value Stocks?
How to Invest in Crude Oil Stock
Simple Explanation of Stock Market
What Is the Relationship Between the Value of a Company's Stock & Its Stock Price?
Why Do Stock Prices Drop?


text:
No money for seed funding in Southern Norway
Minister of Education and Research, Torbjørn Røe Isaksen (right) presented the national budget in Kristiansand Wednesday. Left: Mayor of Kristiansand elect Harald Furre.
Private investors are willing to put MNOK 150 into seed funding for the Agder region, but the government will not match the number.
"We are disappointed that seed funding is not part of the national budget for 2016. This would be an important restructuring measure for our region, which is the only one in Norway without public seed funding", says CEO Anne-Grete Ellingsen at GCE NODE.
Minister of Education and Research, Torbjørn Røe Isaksen, presented the national budget on behalf of the government in Kristiansand Wednesday morning.
"The Minister understands the issue and presented a strong argument for seed funding. Strangely, he forgot to bring the money for it", says Ellingsen.
She has noted that the Christian Democrats and the Liberals have expressed support for seed funding in southern Norway, as an important counter-cyclical measure to a situation where many lose their jobs.
"I am still hopeful we can find a good solution", says Ellingsen.


Umziki Chalets - Mkuze Accommodation.
Set in the picturesque northern KwaZulu-Natal;Pongolapoort Dam. Umziki Chalets are found on the Jozini Game Ranch which along with Hlambanyathi Game Reserve belonging to the Senekal family covering a total area of 15 000 ha resulting in a variety of habitat types, animal species and record bird species.
There are seven thatched chalets with two, three and four bed options, accommodating up to 23 guests. All the chalets are air-conditioned and have en-suite bathrooms. Linen, cutlery, crockery and domestic services are all provided. A central kitchen, lapa and swimming pool provides for recreational purposes.
There are self-drive 4x4 drives through the game farm or alternatively bring your mountain bike or quad bike and see what you can see.
Boat-hire can be arranged with our sister camp Nkonkoni Fishing Camp if tiger-fishing is your game or bring your own boat.
Game drives in Hlambanyathi Game Reserve can be arranged for our guests.
There is an option of catering if you don't feel like doing your own cooking.
All you have to do when you get to Umziki is unpack put your feet up and enjoy the experience.
Umziki Chalets is centrally positioned to many sort after destinations. One can take a day trip into Swaziland or go salt-water fly fishing at Kosi Bay..... The options are endless, with several KZN game parks and Maputaland jewels waiting to be explored, all within reasonable driving distance.
25378 page views since 02 Nov, 2007.
Last updated on 26 Nov, 2018.


The new Siemens Gas Cooktops with stepFlame Technology.
Now with nine different levels of hot. With stepFlame Technology.
No more leaning over to check the flame size, no more hoping that it is at the right heat level. All this guesswork has been removed from cooking with gas. From now on, whatever level you choose – you'll get exactly the heat and the flame size you need. Just as easy to operate as it sounds. That is stepFlame Technology: A groundbreaking innovation to precisely control the flame. Your meals will be delicious, and your guests will be delighted.
The best is always a good choice.
During the development of each new technology, engineers are faced with the challenge of combining usefulness with pleasure. The Gas Cooktops' new design satisfies both: In such a glamorous setting, you will want to amaze guests every day.
Flawless down to the smallest detail.
Watch our clip to discover more about the design.
Expect one thing from a cooktop more than anything: uncompromising performance. So that you do not need to wait longer to enjoy your perfectly prepared meals, the new Gas Cooktops offer an enhanced performance range for impressive dishes – regardless of how many portions you cook.
More possibilities for more joy.
Another treat for more flexibility in the kitchen: The dualWok Burner offers many possibilities – from keeping meals warm to preparing a wide variety of food in any quantity. And with its enlarged perfomance spectrum, you always get the desired results.
Watch our clip to discover more about dualWok Burner.
Cooking at the highest level. The new Gas Cooktops.
From groundbreaking technology to flawless design: The Siemens Gas Cooktops set new standards from all angles and provide more space for creative cooking.
Full flexibility and intuitive operation to quickly prepare various meals.
Automatic ventilation and easy operation for a breath of fresh air in the kitchen.
Fast and precise baking thanks to innovative features.


There's a photo of my dad as a young boy dressed in a cowboy hat with gun holster, trying to look like an outlaw. :) I think that would be perfect!
Congrats Deb! I'm emailing you now!
You have 48 hours to respond or another winner will be drawn!
Remember, you can get your own Poster Printing and Canvas Printing done for a great price!
Don't forget about my Summer Soiree' coming up beginning Monday, June 15th!


The Speedy Stitcher Sewing Awl is a handy sewing tool that creates the perfect lock-stitch, every time. It can be used for assembly and repair projects such as making leather crafts, repairing ripped canvas tarps, climbing and camping gear, bookbinding repair, saddlery and tack items, athletic equipment, and even shoes and belts. Including with the basic sewing awl are two different needles and a bobbin of thread. Below we demonstrate the proper techniques for threading a Speedy Stitcher, prepping for stitching and when to use the different kinds of thread.
1. Pull out the end cap (with bobbin inside). Push the end of bobbin thread out of the hole in the handle. Replace the end cap, securing the bobbin. Wrap thread end around tension post. Pass thread end "under" thread. This ensures that it will feed properly.
2. Pass thread along the groove in the handle and through the hole in the ferrule.
3. Select one of the needles. Insert the needle in the threaded post. Align the groove on the needle with the groove on the post.
4. Run thread along the groove on the post and into the eye of the needle.
5. Screw chuck lock back into place and pull the thread out to the length needed to complete the project.
On leather: To help keep stitches straight, first use a Stitching Groover to mark the path for stitching or a stylus to lightly mark a guideline.
On Canvas & Upholstery: Draw guidelines with chalk or a pencil.
Overstitch Wheel: Mark stitching hole placement with an Overstitch Wheel or ruler and stylus.
Stitching Pony: To help hold your project while stitching, use a Realeather Stitching Pony.
***Important: The length of the thread needed to stitch the project is equal to the length of the stitch line to be done plus 3".
Learn how to use a Speedy Stitcher by reading through our step-by-step instructions here!


text:
Home Teasers Naagin 4 Teasers – July 2022
Naagin 4 Teasers – July 2022
To acquire the highly coveted Coral pearl, Vishaka maliciously manipulates Dev into believing that Brinda has feelings for Rajat. Brinda rescues Dev before it's too late and also announces that she is pregnant.
Brinda refuses to keep Dev's family in the dark and gathers everyone to reveal the truth about the child in her womb. Vishaka transforms into Rajat and claims that he is the father of the child.
Friday 3 June 2022
Dev gets kidnapped by Vishaka's goons, with Brinda in hot pursuit. She encounters the kidnappers and prepares to take them on single-handedly.
Brinda assumes that Dev is behind the disappearance of the serpent pearl and rushes to save Manyata from him. Brinda vows to kill Dev for murdering Manyata.
Tuesday 7 June 2022
Vishaka uses magic to give a new face to Nayantara and joins hands with her to destroy Brinda. After a year passes, Dev marries Shalakha due to the pressure by his family.
As the Parekh family celebrates the festival of Holi, they are shocked when Brinda make a surprise entrance in the party and announces her return.
Brinda is surprised to see Nayantara enter her bedroom and help Dev, thereby completely ignoring her. Nayantara throws a card against Brinda by revealing the child in front of whole family.
Vishaka tells Shalakha that she has to get Dev away from Brinda if she wants to make a place in his life and replace Brinda. Shalakha behaves very romantically with Dev and convinces him to go on a honeymoon with her.
After Shalaka lays out a well-knitted plan to prove Brinda a killer in Dev's eyes, she goes on to brag about her conspiracy in front of Mundika.
After learning the vicious truth about Vishaka, Brinda exposes her in front of the family and Shalakha traps Vishaka in such a way that she is framed to be the killer. Hence, she gets arrested for murder.
Fed up with all the drama in the house, Dev plans to leave along with Brinda. Although she tries to convince him to stay as his family needs him, Dev refuses to pay heed and decides to go ahead with his plan.
Shalakha plans to get close to Dev at any cost and sees Brinda as an obstacle in her way. She uses the fragment of Dev's soul to manipulate his thoughts and actions so that he breaks up with Brinda.
Brinda is puzzled when she gets a call from her mother in a frantic state. Unaware that Dev is not himself, Brinda takes him with her to meet her mother.
On witnessing Dev shoot and murder her mother Swarand about his plan to abandon her so that he can marry Shalaka, Brinda swears to take revenge on him and the entire family.
Season 4 finale!
As the joyous occasion of Hardik's wedding takes place, Brinda makes a sudden appearance and reveals that she plans to kill everyone responsible for the death of her family members.
Naagin 5 follows immediately after Season 4 ends – we'll add the teasers as soon as they're available.
Please note: Naagin moves to 17h40 for its new season.
Premiere episodes of Naagin air on eExtra from Mondays to Fridays at 16h45.


Culture matters here: from Demo Days and Current Events Talks to our mentorship program and community involvement, we pride ourselves on the workplace environment we've created. Our products AND our culture get attention from the media—we've been featured in outlets like the Wall Street Journal, New York Times, TechCrunch, Forbes, and Crain's.
"Before I joined Avant, I worried that my non-traditional tech background could keep me from building the career I wanted. But here, my diverse skills and experience are actually considered an asset - I've felt like a real contributor from day one!"
"It's great to work in an environment so full of talented, driven and fun people. Avant will help you become the best version of yourself."
"At Avant, I work alongside some of the smartest, most innovative people I know. I feel empowered to make decisions that have a real impact on our customers and our business."


text:
César Marie, marquis de Talaru et de Chalmazel, comte de Chamarande (-1794) appartient à une très vieille famille noble du comté de Forez. En , il est fait brigadier des armées du roi, puis devient gouverneur de Strasbourg et Phalsbourg en , avant de participer à l'expédition de Minorque, la première bataille navale de la guerre de Sept Ans. Après la victoire française, il est nommé commandement en second de l'île. Minorque ne sera rendu à la Grande-Bretagne que lors du traité de Paris, en échange de la restitution des Antilles françaises et de Belle-Île.

Après avoir été fait Maréchal de camp en 1761, il devient Premier maître d'hôtel de la reine Marie-Antoinette, dont il administre à ce titre les domaines. Il possède deux hôtels particuliers rue de Richelieu, ce qui, par un étrange enchaînement de faits, conduira à son arrestation, puis à son exécution sous la Révolution française.

Vie avant la Révolution française

Les débuts
César-Marie de Talaru appartient à une très vieille famille noble du comté de Forez, dont le nom de Talaru provient du château du même nom, qui appartenait à la famille depuis l'an 1084.

Fils de Louis de Talaru et de sa seconde épouse, Marie-Marthe de Bonneval, il avait un frère cadet, Louis-François (1729-1782), vicomte de Talaru, dont le fils, Louis Justin Marie, dernier marquis de Talaru, fut membre de la Chambre des pairs sous la Restauration et la Monarchie de Juillet. César-Marie épouse Marie-Justine de Sassanage en 1750, dont il a deux enfants.

Le marquis devient brigadier des armées du roi le . Puis, le , il est fait gouverneur de Strasbourg et Phalsbourg.

Guerre de Sept Ans

Il participe ensuite à l'expédition de Minorque, en 1756.
Lors de cette bataille navale, la toute première de la guerre de Sept Ans, la flotte française remporte une courte victoire, et contraint la flotte anglaise commandée par l'amiral Byng à se replier sur Gibraltar, permettant ainsi la prise de l'île de Minorque. César Marie de Talaru est alors nommé commandant en second, et « chargé du détail de la garde de l'ile », sous les ordres de Hyacinthe Gaëtan, comte de Lannion. Il est inspecteur des troupes stationnées sur l'ile jusqu'en , date à laquelle il rentre en France et prend le commandement du régiment de Talaru. Les Français parviendront à tenir l'île jusqu'à la fin de la guerre de Sept Ans, et elle ne sera finalement rendue à l'Angleterre que lors du traité de Paris, en échange de la restitution des Antilles françaises et de Belle-Île.

Envoyé en Allemagne il devient inspecteur des troupes en Hesse. Il est fait inspecteur général de l'infanterie le . C'est par brevêt du  qu'il est nommé Maréchal de camp ; il conservera son nouveau régiment jusqu'au .

La cour de France, après la guerre
Enfin, en  il obtient la charge de Premier maitre d'hôtel de la Reine, dont il gère les domaines à ce titre. Il conservera cette charge jusqu'en 1770. Il est fait lieutenant général en 1780.

Révolution française

Réquisition sous la Terreur
Au moment de la Terreur, il possède deux hôtels particuliers situés rue de Richelieu, aux numéros 60 et 62. Il en donne un en location à un homme qui souhaite ouvrir un hôtel. L'immeuble est réquisitionné par les autorités de la section le  pour en faire une maison d'arrêt, et Talaru en devient le premier pensionnaire. Il se trouve payer, pour une seule pièce de sa propre maison, plus cher que le tout ne lui rapportait en loyer. On y met jusqu'à deux cents prisonniers.

Arrestation par méprise
À la suite d'une méprise, il est guillotiné au printemps suivant. L'ordre du tribunal révolutionnaire est « d'amener La Borde, logé dans la chambre de Boutin, chez Talaru ». Mais on amène au tribunal tous les noms marqués, c'est-à-dire La Borde, Boutin et Talaru, et le tribunal trouve plus simple de modifier l'acte d'accusation que de les renvoyer.

On peut aussi y rencontrer le député Jean-Marie Boscary de Romaine ; Jeanne de Lavaulx, veuve du maréchal duc de Richelieu, petit neveu du cardinal dont les mémoires nous sont restés ; François-Nicolas-Henri Racine de Monville, le créateur du célèbre Désert de Retz à Chambourcy, en forêt de Marly, dont il dessine lui-même les plans ; Paul-Henri Marron, pasteur protestant, membre du club des Feuillants ; Antoine d'Andlau, gendre du philosophe Helvétius ; Louis Gérard, abbé et écrivain, ancien chanoine de l'église Saint-Louis-du-Louvre ; Rousselin, qui était allé terroriser Troyes comme commissaire de la Convention, et que Robespierre fait jeter en prison.

Chute de Robespierre et exécution

Les détenus, au lendemain du 9 thermidor (date de la chute de Robespierre), ont la surprise de voir arriver une bonne partie de la famille Duplay, chez qui Robespierre séjournait depuis 1791. De même Marie Brunet, veuve du conventionnel Georges Couthon, guillotiné avec Robespierre. De même Paul Auguste Taschereau de Fargues, opulent armateur qui avait été membre des Jacobins et ami de Robespierre. De même Dossonville, policier qui avait particulièrement tourmenté les détenus, et Rousseville, l'informateur "personnel" de Robespierre. On y met ensuite des réfugiés de Saint-Domingue qui plaident pour que la République écrase la révolte des Noirs et leur rende leurs plantations.

Au reste, hormis MM. de la Borde, Talaru et Boutin, qui seront donc décapités en 1794, ils survivront tous à cette épreuve.

Sort de l'hôtel
La maison devient l'hôtel meublé des Colonnes en 1816, puis abrite l'agence Havas à partir de 1825. De 1831 à 1907, la joaillerie « Nitot Étienne et fils » le loue pour y placer ses magasins. Elle se transporte place Vendôme, où elle existe encore sous le nom de « Bijouterie Chaumet ».
L'hôtel de Talaru est démoli en 1914 pour permettre l'édification de locaux plus modernes destinés à l'agence Havas.

Notes et références
Notes

Références

Articles connexes
 Expédition de Minorque
 Révolution française
 Robespierre

Naissance en juin 1725
Décès en juillet 1794
Personnalité guillotinée durant la Révolution française
Général français du XVIIIe siècle
Décès à 69 ans

redpajama_set_name: RedPajamaWikipedia