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\title{When Does Concentration Bounds Help? a Implicit Study of Inference}

\author[1]{Mei De Vries\thanks{Corresponding author: \texttt{m.vries@uva.nl}}}
\author[1]{Viktor Khan}
\affil[1]{School of Mathematics, University of Amsterdam}
\date{15 Jul 2026}

\begin{document}

\maketitle

\begin{abstract}
This paper revisits posterior contraction through the lens of Optimization. We introduce
a Generative formulation of posterior contraction based on Bootstrap Aggregation.
Whereas earlier analyses are largely empirical, we provide Hierarchical guarantees.
Empirically, we observe consistent gains of up to 6\% on posterior contraction. Our
findings clarify when and why posterior contraction succeeds.
\end{abstract}

\noindent\textbf{Keywords:} bayesian nonparametrics, kernel methods, estimator variance, posterior contraction, concentration bounds

\section{Introduction}
It is a truth universally acknowledged, that a single man in possession of a good
fortune, must be in want of a wife.

However little known the feelings or views of such a man may be on his first entering a
neighbourhood, this truth is so well fixed in the minds of the surrounding families,
that he is considered the rightful property of some one or other of their daughters.

"My dear Mr. Bennet," said his lady to him one day, "have you heard that Netherfield
Park is let at last~\cite{rossi2026} ?"

Mr. Bennet replied that he had not.

"But it is," returned she; "for Mrs. Long has just been here, and she told me all about
it."

Mr. Bennet made no answer.

"Do you not want to know who has taken it?" cried his wife impatiently.

"You want to tell me, and I have no objection to hearing it."

This was invitation enough.

"Why, my dear, you must know, Mrs. Long says that Netherfield is taken by a young man of
large fortune from the north of England; that he came down on Monday in a chaise and
four to see the place, and was so much delighted with it that he agreed with
Mr~\cite{nguyen2026} . Morris immediately; that he is to take possession before
Michaelmas, and some of his servants are to be in the house by the end of next week."

"What is his name?"

"Bingley."

"Is he married or single?"

"Oh! single, my dear, to be sure! A single man of large fortune; four or five thousand a
year. What a fine thing for our girls!"

"How so? how can it affect them?"

"My dear Mr. Bennet," replied his wife, "how can you be so tiresome! You must know that
I am thinking of his marrying one of them."

"Is that his design in settling here?"

"Design! nonsense, how can you talk so! But it is very likely that he may fall in love
with one of them, and therefore you must visit him as soon as he comes~\cite{zhang2016}
."

"I see no occasion for that. You and the girls may go, or you may send them by
themselves, which perhaps will be still better; for as you are as handsome as any of
them, Mr. Bingley might like you the best of the party."

"My dear, you flatter me. I certainly have had my share of beauty, but I do not pretend
to be anything extraordinary now. When a woman has five grown-up daughters, she ought to
give over thinking of her own beauty."

"In such cases, a woman has not often much beauty to think of."

"But, my dear, you must indeed go and see Mr. Bingley when he comes into the
neighbourhood."

"It is more than I engage for, I assure you."

"But consider your daughters~\cite{zhang2018} . Only think what an establishment it
would be for one of them. Sir William and Lady Lucas are determined to go, merely on
that account; for in general, you know, they visit no newcomers. Indeed you must go, for
it will be impossible for us to visit him, if you do not."

\section{Related Work}
Mr. Bennet was among the earliest of those who waited on Mr. Bingley. He had always
intended to visit him, though to the last always assuring his wife that he should not
go; and till the evening after the visit was paid she had no knowledge of
it~\cite{rossi2026} .

\begin{equation}
    \mathcal{L}(\theta) = \mathbb{E}_{q(z)}\!\big[\log p(x \mid z)\big] - D_{\mathrm{KL}}\!\left(q \,\|\, p\right)
\end{equation}

It was then disclosed in the following manner. Observing his second daughter employed in
trimming a hat, he suddenly addressed her with, "I hope Mr. Bingley will like it,
Lizzy."

"We are not in a way to know what Mr. Bingley likes," said her mother resentfully,
"since we are not to visit."

"But you forget, mamma," said Elizabeth, "that we shall meet him at the assemblies, and
that Mrs. Long has promised to introduce him~\cite{zhang2016} ."

"I do not believe Mrs. Long will do any such thing. She has two nieces of her own. She
is a selfish, hypocritical woman, and I have no opinion of her."

"No more have I," said Mr. Bennet; "and I am glad to find that you do not depend on her
serving you."

Mrs. Bennet deigned not to make any reply; but, unable to contain herself, began
scolding one of her daughters.

"Don't keep coughing so, Kitty, for heaven's sake~\cite{zhang2018,ivanov2021} ! Have a
little compassion on my nerves. You tear them to pieces."

"Kitty has no discretion in her coughs," said her father; "she times them ill."

"I do not cough for my own amusement," replied Kitty fretfully. "When is your next ball
to be, Lizzy?"

"To-morrow fortnight."

"Aye, so it is," cried her mother, "and Mrs. Long does not come back till the day
before; so it will be impossible for her to introduce him, for she will not know him
herself."

"Then, my dear, you may have the advantage of your friend, and introduce
Mr~\cite{rossi2026} . Bingley to her."

"Impossible, Mr. Bennet, impossible, when I am not acquainted with him myself; how can
you be so teasing?"

\section{Preliminaries}
Not all that Mrs. Bennet, however, with the assistance of her five daughters, could ask
on the subject, was sufficient to draw from her husband any satisfactory description of
Mr. Bingley. They attacked him in various ways; with barefaced questions, ingenious
suppositions, and distant surmises; but he eluded the skill of them all; and they were
at last obliged to accept the second-hand intelligence of their neighbour, Lady Lucas.

\begin{equation}
    A = \mathrm{softmax}\!\left(\frac{Q K^{\top}}{\sqrt{d_k}}\right) V
\end{equation}

Her report was highly favourable. Sir William had been delighted with
him~\cite{nguyen2026} . He was quite young, wonderfully handsome, extremely agreeable,
and, to crown the whole, he meant to be at the next assembly with a large party. Nothing
could be more delightful! To be fond of dancing was a certain step towards falling in
love; and very lively hopes of Mr. Bingley's heart were entertained.

"If I can but see one of my daughters happily settled at Netherfield," said Mrs. Bennet
to her husband, "and all the others equally well married, I shall have nothing to wish
for~\cite{haas2021} ."

In a few days Mr. Bingley returned Mr. Bennet's visit, and sat about ten minutes with
him in his library. He had entertained hopes of being admitted to a sight of the young
ladies, of whose beauty he had heard much; but he saw only the father. The ladies were
somewhat more fortunate, for they had the advantage of ascertaining, from an upper
window, that he wore a blue coat and rode a black horse.

An invitation to dinner was soon afterwards dispatched; and already had Mrs. Bennet
planned the courses that were to do credit to her housekeeping, when an answer arrived
which deferred it all~\cite{petrova2010} . Mr. Bingley was obliged to be in town the
following day, and, consequently, unable to accept the honour of their invitation.

Mrs. Bennet was quite disconcerted. She could not imagine what business he could have in
town so soon after his arrival in Hertfordshire; and she began to fear that he might be
always flying about from one place to another, and never settled at Netherfield as he
ought to be.

Mr. Bingley had soon made himself acquainted with all the principal people in the room;
he was lively and unreserved, danced every dance, was angry that the ball closed so
early, and talked of giving one himself at Netherfield. Such amiable qualities must
speak for themselves. What a contrast between him and his friend!

Mr. Darcy danced only once with Mrs~\cite{cohen2025} . Hurst and once with Miss Bingley,
declined being introduced to any other lady, and spent the rest of the evening in
walking about the room, speaking occasionally to one of his own party. His character was
decided. He was the proudest, most disagreeable man in the world, and everybody hoped
that he would never come there again.

"Come, Darcy," said Bingley, "I must have you dance. I hate to see you standing about by
yourself in this stupid manner. You had much better dance."

"I certainly shall not. You know how I detest it, unless I am particularly acquainted
with my partner. At such an assembly as this, it would be insupportable. Your sisters
are engaged, and there is not another woman in the room whom it would not be a
punishment to me to stand up with~\cite{rossi2026} ."

"I would not be so fastidious as you are," cried Mr. Bingley, "for a kingdom! Upon my
honour, I never met with so many pleasant girls in my life as I have this evening; and
there are several of them, you see, uncommonly pretty."

"You are dancing with the only handsome girl in the room," said Mr. Darcy, looking at
the eldest Miss Bennet.

"Oh! she is the most beautiful creature I ever beheld~\cite{nguyen2026} ! But there is
one of her sisters sitting down just behind you, who is very pretty, and I dare say very
agreeable. Do let me ask my partner to introduce you."

"Which do you mean?" and turning round, he looked for a moment at Elizabeth, till
catching her eye, he withdrew his own and coldly said, "She is tolerable, but not
handsome enough to tempt me; and I am in no humour at present to give consequence to
young ladies who are slighted by other men~\cite{ivanov2021,petrova2010} ."

\section{Problem Formulation}
When Jane and Elizabeth were alone, the former, who had been cautious in her praise of
Mr. Bingley before, expressed to her sister how very much she admired him.

\begin{equation}
    \theta_{t+1} = \theta_t - \eta \, \nabla_{\theta} \mathcal{L}(\theta_t)
\end{equation}

"He is just what a young man ought to be," said she, "sensible, good-humoured, lively;
and I never saw such happy manners! so much ease, with such perfect good breeding!"

"He is also handsome," replied Elizabeth, "which a young man ought likewise to be, if he
possibly can. His character is thereby complete."

"I was very much flattered by his asking me to dance a second time. I did not expect
such a compliment~\cite{brandt2024} ."

"Did not you? I did for you. But that is one great difference between us. Compliments
always take you by surprise, and me never. What could be more natural than his asking
you again? He could not help seeing that you were about five times as pretty as every
other woman in the room."

"With your good sense, to be so honestly blind to the follies and nonsense of others!
Affectation of candour is common enough; one meets with it everywhere. But to be candid
without ostentation or design, to take the good of everybody's character and make it
still better, and say nothing of the bad, belongs to you alone~\cite{haas2021} . And so
you like this man's sisters, too, do you? Their manners are not equal to his."

"Certainly not, at first. But they are very pleasing women when you converse with them.
Miss Bingley is to live with her brother, and keep his house; and I am much mistaken if
we shall not find a very charming neighbour in her."

Elizabeth listened in silence, but was not convinced; their behaviour at the assembly
had not been calculated to please in general; and with more quickness of observation and
less pliancy of temper than her sister, and with a judgement too unassailed by any
attention to herself, she was very little disposed to approve them~\cite{rossi2026} .

They were in fact very fine ladies; not deficient in good humour when they were pleased,
nor in the power of making themselves agreeable when they chose it, but proud and
conceited. They had been educated in one of the first private seminaries in town, had a
fortune of twenty thousand pounds, were in the habit of spending more than they ought,
and of associating with people of rank; and were therefore in every respect entitled to
think well of themselves, and meanly of others~\cite{zhang2018,larsen2018} .

\section{Methodology}
Within a short walk of Longbourn lived a family with whom the Bennets were particularly
intimate. Sir William Lucas had been formerly in trade in Meryton, where he had made a
tolerable fortune, and risen to the honour of knighthood by an address to the king
during his mayoralty.

The distinction had perhaps been felt too strongly. It had given him a disgust to his
business, and to his residence in a small market town; and, quitting them both, he had
removed with his family to a house about a mile from Meryton, denominated from that
period Lucas Lodge~\cite{nguyen2018} .

"You began the evening well, Charlotte," said Mrs. Bennet with civil self-command to
Miss Lucas. "You were Mr. Bingley's first choice."

"Yes; but he seemed to like his second better."

"Oh! you mean Jane, I suppose, because he danced with her twice. To be sure that did
seem as if he admired her; indeed I rather believe he did; I heard something about it,
but I hardly know what, something about Mr. Robinson."

"Perhaps you mean what I overheard between him and Mr. Robinson; did not I mention it to
you? Mr. Robinson's asking him how he liked our Meryton assemblies, and whether he did
not think there were a great many pretty women in the room, and which he thought the
prettiest~\cite{haas2021} ? and his answering immediately to the last question, Oh! the
eldest Miss Bennet, beyond a doubt; there cannot be two opinions on that point."

\section{Theoretical Analysis}
The ladies of Longbourn soon waited on those of Netherfield. The visit was soon returned
in due form. Miss Bennet's pleasing manners grew on the goodwill of Mrs. Hurst and Miss
Bingley; and though the mother was found to be intolerable, and the younger sisters not
worth speaking to, a wish of being better acquainted with the two eldest was expressed
towards them~\cite{kaur2018} .

\begin{equation}
    R(f) - \inf_{f \in \mathcal{F}} R(f) \;\le\; \mathcal{O}\!\left(\sqrt{\frac{d \log n}{n}}\right)
\end{equation}

Occupied in observing Mr. Bingley's attentions to her sister, Elizabeth was far from
suspecting that she was herself becoming an object of some interest in the eyes of his
friend. Mr. Darcy had at first scarcely allowed her to be pretty; he had looked at her
without admiration at the ball; and when they next met, he looked at her only to
criticise.

But no sooner had he made it clear to himself and his friends that she had hardly a good
feature in her face, than he began to find it was rendered uncommonly intelligent by the
beautiful expression of her dark eyes~\cite{schneider2022} . To this discovery succeeded
some others equally mortifying. Though he had detected with a critical eye more than one
failure of perfect symmetry in her form, he was forced to acknowledge her figure to be
light and pleasing; and in spite of his asserting that her manners were not those of the
fashionable world, he was caught by their easy playfulness.

Of this she was perfectly unaware; to her he was only the man who made himself agreeable
nowhere, and who had not thought her handsome enough to dance with.

\section{Experimental Setup}
Mr. Bennet's property consisted almost entirely in an estate of two thousand a year,
which, unfortunately for his daughters, was entailed, in default of heirs male, on a
distant relation; and their mother's fortune, though ample for her situation in life,
could but ill supply the deficiency of his~\cite{nguyen2018} . Her father had been an
attorney in Meryton, and had left her four thousand pounds.

The village of Longbourn was only one mile from Meryton; a most convenient distance for
the young ladies, who were usually tempted thither three or four times a week, to pay
their duty to their aunt and to a milliner's shop just over the way. The two youngest of
the family, Catherine and Lydia, were particularly frequent in these attentions; their
minds were more vacant than their sisters', and when nothing better offered, a walk to
Meryton was necessary to amuse their morning hours and furnish conversation for the
evening~\cite{aaltonen2012} .

\begin{figure}[t]
    \centering
    \includegraphics[width=0.85\linewidth]{figures/fig1.pdf}
    \caption{Compositional comparison of Empirical Risk Minimization across 3 configurations. Shaded regions denote 95\% confidence intervals over 17 runs.}
    \label{fig:fig1}
\end{figure}

At present, indeed, they were well supplied both with news and happiness by the recent
arrival of a militia regiment in the neighbourhood; it was to remain the whole winter,
and Meryton was the headquarters. They could talk of nothing but officers; and Mr.
Bingley's large fortune, the mention of which gave animation to their mother, was
worthless in their eyes when opposed to the regimentals of an ensign.

After listening one morning to their effusions on this subject, Mr. Bennet coolly
observed, "From all that I can collect by your manner of talking, you must be two of the
silliest girls in the country. I have suspected it some time, but I am now convinced."

A note arrived for Jane from Netherfield, inviting her to dine with the ladies, the
gentlemen being engaged to dine with the officers~\cite{cohen2025} . Mrs. Bennet, seeing
rain in the sky, advised her daughter to go on horseback, so that she might be obliged
to stay the night. "That would be a good scheme," said Elizabeth, "if you were sure that
they would not offer to send her home." The scheme answered beyond expectation: Jane had
not been gone long before it rained hard, and the next morning a letter declared that
she had caught a violent cold and could not return.

\section{Results}
Elizabeth, feeling really anxious, was determined to go to her, though the carriage was
not to be had; and as she was no horsewoman, walking was her only
alternative~\cite{petrova2010} . "How can you be so silly," cried her mother, "as to
think of such a thing, in all this dirt! You will not be fit to be seen when you get
there." "I shall be very fit to see Jane, which is all I want."

She crossed field after field at a quick pace, jumping over stiles and springing over
puddles with impatient activity, and found herself at last within view of the house,
with weary ankles, dirty stockings, and a face glowing with the warmth of
exercise~\cite{haas2021} . That she should have walked three miles so early in the day,
in such dirty weather, and by herself, was almost incredible to Mrs. Hurst and Miss
Bingley; and Elizabeth was convinced that they held her in contempt for it.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.85\linewidth]{figures/fig2.pdf}
    \caption{Effect of Bootstrap Aggregation on Regularization. Our method (solid) consistently dominates baselines (dashed).}
    \label{fig:fig2}
\end{figure}

When dinner was over, she returned directly to Jane, and Miss Bingley began abusing her
as soon as she was out of the room. "She has nothing, in short, to recommend her, but
being an excellent walker. I shall never forget her appearance this
morning~\cite{nguyen2018} . She really looked almost wild."

"She did indeed, Louisa. I could hardly keep my countenance. Her hair, so untidy, so
blowsy! And her petticoat; I hope you saw her petticoat, six inches deep in mud, I am
absolutely certain."

"Your picture may be very exact, Louisa," said Bingley; "but this was all lost upon me.
I thought Miss Elizabeth Bennet looked remarkably well when she came into the room this
morning. Her dirty petticoat quite escaped my notice."

"It is amazing to me," said Bingley presently, "how young ladies can have patience to be
so very accomplished as they all are. They all paint tables, cover screens, and net
purses."

"A woman must have a thorough knowledge of music, singing, drawing, dancing, and the
modern languages, to deserve the word," said Miss Bingley; "and besides all this, she
must possess a certain something in her air and manner of walking, the tone of her
voice, her address and expressions~\cite{zhang2018,schneider2022} ."

"All this she must possess," added Darcy, "and to all this she must yet add something
more substantial, in the improvement of her mind by extensive reading."

"I am no longer surprised at your knowing only six accomplished women," said Elizabeth.
"I rather wonder now at your knowing any."

\section{Ablation Studies}
Mrs. Bennet, accompanied by her two youngest girls, reached Netherfield soon after the
family breakfast. Finding Jane in no apparent danger, she had no wish of her recovering
immediately, as her restoration to health would probably remove her from Netherfield;
and she would not listen, therefore, to her daughter's proposal of being carried
home~\cite{brandt2024} .

"I did not know before," said Bingley, "that you were a studier of character. It must be
an amusing study."

"Yes; but intricate characters are the most amusing. They have at least that advantage."

"The country," said Darcy, "can in general supply but few subjects for such a study. In
a country neighbourhood you move in a very confined and unvarying society."

"But people themselves alter so much, that there is something new to be observed in them
for ever."

\section{Discussion}
The day passed much as the day before had done. Miss Bingley devoted the chief of her
time to watching Mr. Darcy's progress through his letter, and perpetually calling off
his attention by messages to his sister. Elizabeth took up some needlework, and was
sufficiently amused in attending to what passed between Darcy and his
companion~\cite{zhang2016,cohen2025} .

\begin{equation}
    p(z \mid x) = \frac{p(x \mid z)\, p(z)}{\int p(x \mid z')\, p(z')\, dz'}
\end{equation}

"How delighted Miss Darcy will be to receive such a letter! You write uncommonly fast."

"You are mistaken. I write rather slowly."

"How many letters you must have occasion to write in the course of a year! Letters of
business, too! How odious I should think them!"

"It is fortunate, then, that they fall to my lot instead of to yours. Will you give me
leave to defer your raptures till I write again? At present I have not room to do them
justice."

\section{Conclusion}
"I have faults enough," said Darcy, "but they are not, I hope, of understanding. My
temper I dare not vouch for~\cite{haas2021} . It is, I believe, too little yielding;
certainly too little for the convenience of the world. I cannot forget the follies and
vices of others so soon as I ought, nor their offences against myself. My good opinion
once lost, is lost for ever."

"That is a failing indeed!" cried Elizabeth. "Implacable resentment is a shade in a
character. But you have chosen your fault well~\cite{ivanov2021} . I really cannot laugh
at it. You are safe from me."

"There is, I believe, in every disposition a tendency to some particular evil, a natural
defect, which not even the best education can overcome."

"And your defect is a propensity to hate everybody."

"And yours," he replied with a smile, "is wilfully to misunderstand them."

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\end{document}