Laplace Distribution Puns

In Differential Privacy, the required noise addition is often achieved by sampling values from the Laplace distribution (ie, the 'Laplace mechanism').

This means we usually think about the average relative error of a counting query as: [var(Lap(scale))] / [number of records]. One thing to note is that if you divide both the scale and the number of records by the same amount (ie, applying a record partitioning or subsampling trick to reduce sensitivity), you shouldn't see any improvement in that error metric: [var(Lap(scale/k))] / ([number of records]/k) = [var(Lap(scale))] / [number of records].

However, I've been wondering whether variance (or even mean absolute difference) is actually the correct way to think about the impact of noise addition. The Laplace distribution is a long tail distribution and, for small scale parameters, it's sharply skewed towards the origin. Ie, lots of small noise values, a few very large noise values. This is interesting because these noisy query results feed into other algorithms that build models, post-process, etc, to produce a final privatized analytic or data product... and this post processing may be more or less tolerant of different distributions of added noise. For example, if most noise values are very small, and only a few randomly sampled values are very large, it can be possible to use publicly known properties of the data space to do smoothing and reduce the impact of the large noise values. While that same trick might not be successful with a less sharply skewed distribution of noise values, even if the average noise value stayed the same.

So hopefully that's enough interesting motivation to justify me posting a fairly mundane question to r/math: Does anyone know the equation for the median absolute difference of the Laplace distribution?

I was just looking at a machine learning package for Python/R called 'Prophet' by Facebook which is making a bit of noise in the machine learning/data science world due to its simplicity, especially in Python. Here's a summary:

• Time series algorithm, automatically standardises time and predicted variable

• Automatically de-trending on three components -- long term (over the entire data?), monthly/weekly/days of the week, and holidays (comes with holidays data for different regions)

• By default, it fits 25 linear or logistic models over the first 80% time of the data

• By default, assumes Laplace distribution for the covariates

Prior to reading this, it always bothered me when papers don't mention what type of distribution each of the covariates are, unless of course, graphically shown individually (so stuff like Poisson distribution is easier to tell). Now that I'm reading above linked article, it took me by surprise that the package assumes Laplace distribution for all the covariates by default.

So then my question is two-fold. Is it ok to assume normal distribution if the paper doesn't mention anything for each of the covariates (some papers mention them for few to many covariates, and I think some of them may be obvious so no need to mention, though maybe not obvious to all)? On top of this, why would one choose to assume Laplace instead of normal distribution? What would be the advantages/disadvantages of such decision, and what would be the result on the estimates/errors/bias?

Economical models seem dominated with Gaussian distribution, which has very thin exp(-x^2) tails. I have recently worked on ROI of Dow Jones daily averages sequence, and its empirical distribution fits well CDF of Laplace distribution instead - with heavier exp(-|x|) tails, what means larger risks. Plot and sources:

https://economics.stackexchange.com/questions/23967/is-gaussian-distribution-the-proper-choice-for-step-in-financial-time-series-e

As our financial institutions operate on these models, for stability of economy it is crucial that they don't underestimate the risks - is this disagreement discussed in literature, handled in real models?

I'm dealing with a set of data that is Laplace distributed. The trouble is that my current algorithm with this problem can only work well with gaussian-like distributed data. I know there are some transformations like box-cox or yeo-johnson that work for data exponentially distributed but can find any for Laplace. Do we have any such transformation function since exponential and Laplace distribution is quite similar in the way that Laplace is in fact just like a double exponential?

I tried finding text/research paper/books that shows the KLD of laplace distribution between two pdf(probability density function) let say P(λ1,μ1) and q(λ2,μ2). If you happen to know then please share it with me.

Thank you.

Because I can't type in LateX here I decided to write the question through TeXmaker, and post the PDF here.

Thanks in advance!

For the control theory class I'm teaching to my graduate students in mathematics, I've been tracking down literature that will give the appropriate mathematical depth for some of the concepts we are going to be exploring. This is starting with the Laplace transform.

The Laplace transform is so easy to define for functions of exponential type, with the caveat that the integral might not converge for s with small real part, but I honestly, hadn't seen just how many layers there were between there and giving a proper mathematical definition for distributions.

There are the obvious steps that agree with the Fourier transform, like defining a Schwartz like space, then the dual space on that space using a collection of semi-norms. What surprised me was that in the definition of the Laplace transform on distributions, you have to use an approximate method, where you find a sequence of functions in your space that converge to e^{-st} in some fashion.

After flipping back and forth between two textbooks, I put this video together, if anyone is curious. I left some simple exercises and extra reading in the description that I'm giving to my students to help them get the hang of the space.

https://youtu.be/F-xS27UfF30

Cheers! I'm happy for any feedback!

Hello everyone! And happy new year!

This spring I am teaching a graduate course on Control Theory in the mathematics department at my university, and this is going to be a great learning experience for me. I got my PhD almost a decade ago studying functional analysis and operator theory, but then I went on to do two postdocs focusing on nonlinear controls.

I've always felt that I missed out on learning linear controls, and so I'm using this class to really dive into the subject. I have about 10 books I'm pulling from, where I am trying to strike the balance between advanced mathematical material and some more boots on the ground (for a mathematician) control theory.

It's surprisingly difficult to find a textbook that covers both the mathematics and the control theory well. Sontag's text does a decent job, but some of the topics I want to cover (like H infinity control) aren't in there. However, Doyle, Francis, and Tannenbaum's textbook covers H infinity controls, but only mention the essential mathematics in passing.

And none seem to really go deep enough to give a rigorous definition of the Laplace transform on Distributions (like the delta function)! Yamamoto's textbook From Vector Space to Function Spaces does a half way decent job, but then pushes the important proofs off into references. So I have a whole library I'm using to teach a single class.

H infinity control theory is a great little space to explore the interconnection between some operator theory and controls. It rests on the mathematical framework of Nevanlinna Pick interpolation, which concerns operators over reproducing Kernel Hilbert Spaces (specifically the Hardy space of the half plane). But I'll also go into PID controllers, cover Nyquist's stability theorem, and other fundamental concepts from controls.

This video here is my introduction to the course, and I'm currently editing the lecture on the definition of the Laplace transform for distributions. It should be a lot of fun!

Let me know if you have any pointers, references, or advice. I'm happy to learn as much as I can :)

For the latest version with inline equations and images, read this article on my wiki.

In this article, we compare several ways of creating a ternary outcome system. We'll use the Ironsworn terminology for the three outcomes: strong hit, weak hit, and miss.

A typical description of the three outcomes is:

• Strong hit: you succeed at your task and get away without consequences.
• Weak hit: you succeed at your task but suffer a consequence.
• Miss: you fail at your task and suffer a consequence.

Though this will vary from game to game and possibly even between different situations within the same game.

2dN, count successes versus target number (Modiphius 2d20 without Focus)

Roll two dice, counting them individually against a target number (usually roll-under, but mathematically you could make a roll-over system with the same probabilities).

• You score a strong hit if both dice succeed.
• You score a weak hit if one die succeeds.
• You miss if neither of the dice succeeds.

An example is the simplest case of Modiphius 2d20 (without Focus) where you roll the eponymous 2d20 against a single target number.

Image.

The curves are beta distributions.

The tails of this system are relatively short---once the target number reaches the end of a single die, the outcome is guaranteed.

You can scale the curves horizontally (or equivalently, change the granularity) by changing the die size.

Further reading: roll-and-keep dice pool

2dN + modifier versus two thresholds (Powered by the Apocalypse)

Roll 2dN and add a modifier.

• You score a strong hit if the total reaches an upper threshold.
• You score a weak hit if the total reaches a lower threshold.
• You miss if the total reaches neither threshold.

An example is Powered by the Apocalypse, where you roll 2d6 + modifier against an upper threshold of 10 and a lower threshold of 7.

Image.

The curves are triangular distributions.

The tails of this type of system are longer than for the Modiphius 2d20-style system above, though they still reach a guaranteed

I came across this quote from Karl Pearson " Many years ago [in 1893] I called the Laplace-Gaussian curve the normal curve, which name, while it avoids the international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another abnormal "

And recently I also came across this tweet "Originally, Gauss (in 1823) used the term "normal" (in the sense of "orthogonal") referring to the geometric interpretation of a system of linear equations from which the distribution bearing his name is derived "

So my question is two fold :

1. Did Gauss originally use the term 'Normal' referring to the geometric interpretation of a system of linear equations ?

2. Was Pearson influenced by Gauss' s original usage of the word 'Normal' and therefore used the word 'Normal curve' or 'Normal Distribution'.

For the latest version with inline equations and images, read this article on my wiki.

"Swinginess" is a term often thrown around when talking about dice, and in particular, it is commonly asserted that the d20 is particularly "swingy". What could this mean, and to what extent is this actually true?

In this article, I'll focus on fixed-die + modifier systems with binary outcomes. This is not to say that this is the only or best type of system for a RPG, nor the only type worth analyzing; however, it is frequently encountered, it is the easiest to analyze, and it can be used as a building block for more complex systems in both design and analysis.

"Binary outcomes can't be swingy"

Another reason for focusing on binary-outcome systems is that it's not as clear that they can be "swingy" in the first place, thus making for a more interesting question. Contrast systems that are not binary-outcome: for example D&D-style damage rolls, where 1d12 damage is obviously "swingier" than 2d6 but damage rolls; or the (in)famous concept of critical hit/fumble tables.

The argument against binary-outcome-swinginess goes something like this: the function of a dice roll in a binary-outcome system is to determine a chance of success. Once that chance of success is determined, the procedure used to determine it does not matter; if you replaced the die roll with any other die roll with the same chance, nobody would be the wiser in a blind test. Therefore, the shape of the probability distribution does not matter at all for binary outcomes.

This is true---but only in the very narrow sense of a single contest in isolation. Consider this question:

• A beats B 25% of the time.
• B beats C 25% of the time.
• What is the chance of A beating C?

Having fixed the probabilities of A beating B and B beating C, the chance of A beating C is completely determined by the shape of the probability distribution, and it is not the same for different shapes:

• The uniform distribution^1 says: 0.00%
• The normal distribution says: 8.87%
• The logistic distribution says: 10.00%
• The Laplace distribution says: 12.50%

Thus, having fixed the chances for two contests in a chain, the shape of the distribution can make the difference between something being literally impossible for the lowest underdog, and that lowest underdog having a 1-in-8 chance of winning.^2

You may or may not regard this differ

Looks like corpse Lyndon got N wrong, just like the rest of Darwinian Nietzscheans. Have only skimmed his 2 "books", but he seems to carry on a lot about Alpha/ beta males, weak vs strong, on a purely physical le vel. Here are some quotes from his two scribblings.

“This whole society is sick, and it needs healed or euthanized. And you are the man to do it. But, you are alone my friend and we need a few dozen version of you to make this work. Selfishness

in the Nietzschean sense: great men being great so that weaker men will rise to their highest possible heights merely so they may get a look at him over the berm; over the wall. No longer will

weak men admire weaker men,” Isaiah said.

earth- can be radically altered by one man with the capacity to reproduce himself one million times, a man with a singular purpose and mindset, with no desire for money or fortune or

fame, a man with cathexis for only one thing: the re-instantiation of his race of men. Science and technology has reached a point where it can undo the mindset that brought it about, like

Nietzsche’s Christianity undid itself with Truth. Science itself will destroy science. Science itself will return us to the time of magic and spells.

See, as you know, art and poetry and literature and myth and ritual are all innate needs of complete men, as Nietzsche put it, of complete beasts . And the reason I never liked these white

nationalist types was they were illiterate hillbillie fucks who had never read Shakespeare or Milton or the Sagas . And these men would call men like us who like such things, well, they’d call us

names, let’s say.

“But the Spartans knew -as you know, as I know- that a true warrior is a poet too; he knows the reason he unsheathes his claymore or tanto blade, the reason he hardens his fists and focuses his

eye in battle. He knows the reason is to protect his people’s language, and culture and germline; that all three are one: like the guts, the heart and the lungs.

“Why language? Well, language is how we cut swaths in the inner-forest, the internal landscape of a man is made capacious, made grand by his use, his command of language.

Nietzsche saw men as he saw species of animals, some men are lambs and some eagles; the lambs are weak by nature, their

only values are the negation of values, they spend their moral energy lamenting the eagle’s predatory ways; they grumble about how if they were eagles themselves well, they wouldn’t be

so mean and blah b

Hi everyone,

I'm conducting a post hoc power analysis in G*Power for a Wilcoxon Signed Ranks test. However, this requires a parent distribution input (normal, min ARE, laplace or logistic). I haven't come across this term before, and can't find a clear answer as to what a parent distribution actually is, or what these different inputs mean.

I am guessing that a parent distribution refers to the distribution of the data I used for the test - is this right? I used a Wilcoxon test because the score differences between conditions were not normally distributed, so if my guess is correct, I can't input 'normal'. I have read a lot of pages where people have suggested using min ARE as default but I can't find an explanation as to why.

Please can anyone help me understand this?

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Laplace with Neumann Boundaries, dirac-distribution, Einstein notation,... I have so hard troubles understanding all these topics. Im in my third semester of my bachelor in Europe and it feels like this is the hardest semester by now. Complex Analysis seems to cover so much more topics than the real one and four-momentum and co fuck my brain with notations.
This post is mainly a rant, but any tips on how to pass this? I would need an ELI5 of everything in complex analysis or a complex analysis for dummies. Same of everything Maxwell.

There are multiple sources that release data pertaining to the allocation of therapeutics across the United States. I'm posting this so people are aware that this is publicly available information, and to get an idea of what may be in the state. These numbers are not real-time and often are the amount that has been allocated and not the amount that was delivered or used. So please keep that in mind. Finally, look at the case rates, look at the allocated therapeutics, they are not even comparable. If you know anyone who may be on the fence about vaccination with the belief that therapeutics will be available to them, please let them know how severely limited supplies are.

• Louisiana, Total Allocation: 2,582
  • Regeneron Co-form: 360
  • Regeneron 11.1 mL Roche Packs: 348
  • Regeneron 8 Vial 2 Dose Packs: 0
  • bamlanivimab/etesevimab: 590
  • Sotrovimab: 612
  • Evusheld: 672

Date: January 3-9, 2022 Source

By HHS Data:

#Evuhshield

• 1,488 total courses (~672 reported available)
  • 76% of course are in New Orleans (1,128 courses)
  • Link of table for mobile

#Paxlovid

• 860 total courses (~356 reported available)
  • 23% of courses are in Monroe, (200 courses)
  • Link of table for mobile

Hi all! I've been stumped on this problem for about a week now, so I thought I would turn to reddit.

I am supposed to find a numerical solution to the integral of the Poisson pmf times the Gamma pdf (from 0 to +infinity). This supposedly results in the pmf for the Negative Binomial distribution. The

I'm supposed to use at least two numerical integration methods such as quadrature (gauss-hermite, newton-cotes, etc), the method of Laplace, etc. Whichever ones are most appropriate.

My problem is that the integral I'm supposed to solve has many parameters such as lambda, alpha, beta, and since this is a numerical solution I need to fill those parameters in with.. numbers. I don't know what to do about this. Do I choose random numbers for them? Do I create the Lambda using something like dgamma() in R?

I just cannot find anything online about numerical methods for integrals like this, so I feel super super lost. Any guidance at all would help. Thank you!

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

Played around with the lore book and the memory section and wound up creating this really cool, rather well fleshed out universe to play in. I am stunned how awesome this AI is and how descriptive and detailed it can be. I've done some minor editing to make the lore a bit more coherent, but other than that the AI did most of the detailing in this. Enjoy! I'll post the scenario file in a minute in case anyone wants to play with this scenario.

Tell me about Brevasha...

The city of Brevasha was founded by humans from Earth over 500 years ago. In those days, Brevasha was little more than a small village, and had no name whatsoever. The natives of Brevasha were simple peasants and farmers. But then came the great fire of 1782. A terrible conflagration destroyed most of Brevasha, killing thousands of people and burning the town to ashes. When the smoke cleared, Brevasha stood before them like a shining beacon of hope. And so Brevasha was reborn. The new Brevasha became a center for trade and commerce, and quickly grew into one of the greatest cities in the world. Over the centuries since then, the city has grown immensely in size and population. Today, Brevasha is a thriving metropolis filled with rich merchants and nobles, as well as impoverished beggars and homeless orphans.

Tell me about the districts of Brevasha...

Districts are areas within Brevasha that are governed by their own governor or council. Each district has its own unique characteristics and personalities. While each district does share certain traits, there are differences between districts as well. Zlotopol - The central district of Brevasha, Zlotopol is where all the major landmarks are located. Zlotopol is the heart of Brevasha, and is home to many of the noble families. Zlotopol is also the place where the majority of the wealthiest residents live. Zlotopol is ruled by the Archduke Ferdinand, and he governs with an iron fist. Zlotopol is the location of the Archduke's palace, as well as the Archduke's private gardens. The main road through Zlotopol is called the Noble's Walk, and is lined with grand mansions belonging to the archdukes and their relatives. The Nobles' Walk is considered the very height of luxury and opulence, and many wealthy aristocrats travel from far away to see what lies beyond the walls of the Palais de Justice. Rikoviz - Rikoviz is the district directly west of Zlotopol. It is also known as "Little Russia", because of the large number of Russian immigrants that ha

It really does, I swear!

They’re on standbi

Pilot on me!!

Because she wanted to see the task manager.

Nothing, he was gladiator.

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

I won't be doing that today!

[Removed]

Hi. I need to generate some random numbers from different distributions (normal, Laplace, ...). What would be a good (simple, reliable) library to use?

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "