#Bayesian probability

[ontologicalevil]

Bayesians

I can't wait to watch you happen to yourself.

[byexbyez]

SJSJJSJEJEH not for long i’m afraid

[omegaphilosophia]

The Philosophy of Statistics

The philosophy of statistics explores the foundational, conceptual, and epistemological questions surrounding the practice of statistical reasoning, inference, and data interpretation. It deals with how we gather, analyze, and draw conclusions from data, and it addresses the assumptions and methods that underlie statistical procedures. Philosophers of statistics examine issues related to probability, uncertainty, and how statistical findings relate to knowledge and reality.

Key Concepts:

Probability and Statistics:

Frequentist Approach: In frequentist statistics, probability is interpreted as the long-run frequency of events. It is concerned with making predictions based on repeated trials and often uses hypothesis testing (e.g., p-values) to make inferences about populations from samples.

Bayesian Approach: Bayesian statistics, on the other hand, interprets probability as a measure of belief or degree of certainty in an event, which can be updated as new evidence is obtained. Bayesian inference incorporates prior knowledge or assumptions into the analysis and updates it with data.

Objectivity vs. Subjectivity:

Objective Statistics: Objectivity in statistics is the idea that statistical methods should produce results that are independent of the individual researcher’s beliefs or biases. Frequentist methods are often considered more objective because they rely on observed data without incorporating subjective priors.

Subjective Probability: In contrast, Bayesian statistics incorporates subjective elements through prior probabilities, meaning that different researchers can arrive at different conclusions depending on their prior beliefs. This raises questions about the role of subjectivity in science and how it affects the interpretation of statistical results.

Inference and Induction:

Statistical Inference: Philosophers of statistics examine how statistical methods allow us to draw inferences from data about broader populations or phenomena. The problem of induction, famously posed by David Hume, applies here: How can we justify making generalizations about the future or the unknown based on limited observations?

Hypothesis Testing: Frequentist methods of hypothesis testing (e.g., null hypothesis significance testing) raise philosophical questions about what it means to "reject" or "fail to reject" a hypothesis. Critics argue that p-values are often misunderstood and can lead to flawed inferences about the truth of scientific claims.

Uncertainty and Risk:

Epistemic vs. Aleatory Uncertainty: Epistemic uncertainty refers to uncertainty due to lack of knowledge, while aleatory uncertainty refers to inherent randomness in the system. Philosophers of statistics explore how these different types of uncertainty influence decision-making and inference.

Risk and Decision Theory: Statistical analysis often informs decision-making under uncertainty, particularly in fields like economics, medicine, and public policy. Philosophical questions arise about how to weigh evidence, manage risk, and make decisions when outcomes are uncertain.

Causality vs. Correlation:

Causal Inference: One of the most important issues in the philosophy of statistics is the relationship between correlation and causality. While statistics can show correlations between variables, establishing a causal relationship often requires additional assumptions and methods, such as randomized controlled trials or causal models.

Causal Models and Counterfactuals: Philosophers like Judea Pearl have developed causal inference frameworks that use counterfactual reasoning to better understand causation in statistical data. These methods help to clarify when and how statistical models can imply causal relationships, moving beyond mere correlations.

The Role of Models:

Modeling Assumptions: Statistical models, such as regression models or probability distributions, are based on assumptions about the data-generating process. Philosophers of statistics question the validity and reliability of these assumptions, particularly when they are idealized or simplified versions of real-world processes.

Overfitting and Generalization: Statistical models can sometimes "overfit" data, meaning they capture noise or random fluctuations rather than the underlying trend. Philosophical discussions around overfitting examine the balance between model complexity and generalizability, as well as the limits of statistical models in capturing reality.

Data and Representation:

Data Interpretation: Data is often considered the cornerstone of statistical analysis, but philosophers of statistics explore the nature of data itself. How is data selected, processed, and represented? How do choices about measurement, sampling, and categorization affect the conclusions drawn from data?

Big Data and Ethics: The rise of big data has led to new ethical and philosophical challenges in statistics. Issues such as privacy, consent, bias in algorithms, and the use of data in decision-making are central to contemporary discussions about the limits and responsibilities of statistical analysis.

Statistical Significance:

p-Values and Significance: The interpretation of p-values and statistical significance has long been debated. Many argue that the overreliance on p-values can lead to misunderstandings about the strength of evidence, and the replication crisis in science has highlighted the limitations of using p-values as the sole measure of statistical validity.

Replication Crisis: The replication crisis in psychology and other sciences has raised concerns about the reliability of statistical methods. Philosophers of statistics are interested in how statistical significance and reproducibility relate to the notion of scientific truth and the accumulation of knowledge.

Philosophical Debates:

Frequentism vs. Bayesianism:

Frequentist and Bayesian approaches to statistics represent two fundamentally different views on the nature of probability. Philosophers debate which approach provides a better framework for understanding and interpreting statistical evidence. Frequentists argue for the objectivity of long-run frequencies, while Bayesians emphasize the flexibility and adaptability of probabilistic reasoning based on prior knowledge.

Realism and Anti-Realism in Statistics:

Is there a "true" probability or statistical model underlying real-world phenomena, or are statistical models simply useful tools for organizing our observations? Philosophers debate whether statistical models correspond to objective features of reality (realism) or are constructs that depend on human interpretation and conventions (anti-realism).

Probability and Rationality:

The relationship between probability and rational decision-making is a key issue in both statistics and philosophy. Bayesian decision theory, for instance, uses probabilities to model rational belief updating and decision-making under uncertainty. Philosophers explore how these formal models relate to human reasoning, especially when dealing with complex or ambiguous situations.

Philosophy of Machine Learning:

Machine learning and AI have introduced new statistical methods for pattern recognition and prediction. Philosophers of statistics are increasingly focused on the interpretability, reliability, and fairness of machine learning algorithms, as well as the role of statistical inference in automated decision-making systems.

The philosophy of statistics addresses fundamental questions about probability, uncertainty, inference, and the nature of data. It explores how statistical methods relate to broader epistemological issues, such as the nature of scientific knowledge, objectivity, and causality. Frequentist and Bayesian approaches offer contrasting perspectives on probability and inference, while debates about the role of models, data representation, and statistical significance continue to shape the field. The rise of big data and machine learning has introduced new challenges, prompting philosophical inquiry into the ethical and practical limits of statistical reasoning.

[lindahall]

John Canton – Scientist of the Day

John Canton, a British mathematician and schoolmaster, was born July 31, 1718, in Stroud, Gloucestershire. 

read more...

[gynoidgearhead]

[Image caption for above addition: two comics, each headlined "Mental Gymnastics". The first is a woman or girl stepping a few steps on a gymnastic mat and then happily extending her arms, with a caption reading "I guess fairies are real."

The second has a man energetically swinging from a bar, first from his hands and then his feet; then swinging from rings; then using a pommel horse; then jumping over a flaming car while in a superhero outfit. The various steps are captioned: "A walrus has escaped a zoo or decided to leave its territory for some reason then traveled hundreds of miles of waterways and rivers without being noticed." / It then decided to leave the water and start traveling on land specifically in my city, traveling across miles of streets without being run into or stopped." / "Then it came to my house in particular, made its way to the door, then knocked on it and politely waited." End caption.]

Or someone decided to pull a very illegal prank on you and deliver a walrus to your doorstep, in the manner of xkcd's "instead of office chair, package contained bobcat. would not buy again".

I've asked this question before and been surprised by the results, now I have access to more weirdos it's your problem:

It is the middle of a Sunday afternoon. You have nothing on, and aren't expecting visitors, deliveries or post.

Unexpectedly, there is a knock at the door.

[raffaellopalandri]

Advanced Methodologies for Algorithmic Bias Detection and Correction

I continue today the description of Algorithmic Bias detection. Photo by Google DeepMind on Pexels.com The pursuit of fairness in algorithmic systems necessitates a deep dive into the mathematical and statistical intricacies of bias. This post will provide just a small glimpse of some of the techniques everyone can use, drawing on concepts from statistical inference, optimization theory, and…

[ukdailymail]

'Britain's Bill Gates' - Mike Lynch's tips for building a business

[psycheapuleius]

“Even if the mind is a probabilistic or logical calculating engine, it may not be possible to engage that engine with verbally, symbolically, or numerically stated probabilistic or logical puzzles, which it is presumably not adapted to handle. This point is no deeper than the observation that, although the early visual processes in the retina may compute elaborate convolutions and decorrelations of the image, this does not mean that people can thereby readily apply this machinery to solve mathematics problems concerning convolution or decorrelation.”

— Oaksford & Chater: Précis of Bayesian Rationality, BEHAVIORAL AND BRAIN SCIENCES (2009) 32, 69–120.

[usunezukoinezu]

''For Bayesianists, the world is in shades of gray. The reasons are as follows:

No one can predict the future in this complex world.

Everything is changing with the passage of time.

Even with the aforementioned uncertainties, the world is difficult to predict precisely, but we can still use probabilities to describe it.

From the perspective of worldly success or failure, winners only need to gain a relative advantage locally to surpass their competitors. Therefore, many winners can succeed as long as the probability of winning is a few percentage points higher.

Cognitive and judgment processes based on probability are a continuous approximation and evolving process.

The acceptance and understanding of uncertainty are at the core of Bayesian thinking. We need to embrace the uncertainty of things and use probability to describe and comprehend it.

Faced with uncertainty, Bayesian thinking encourages us not to fear making mistakes, to try new things, learn from failures, and adjust strategies. This aligns well with the process of personal growth.''

-crypto_chanshi

[otaku553]

Flames (Part 1)

(NEXT) ->

(Spade Pirate Sabo AU Masterpost)

This one took AGES i hope you all enjoy!! It's probably a 2-parter but pending how my finals week is looking I might take a hiatus for next week. As I'm writing this post I've just finished my stat mech final which is great! (I wish unspeakable terrors on the person who decided that stat mech was required for the neurobiology-focused major. i get that it's useful for some bayesian statistics but like,,,,, come on i dont need to know thermodynamics)

I did originally have 2 more pages to post with this but scrapped them unfortunately, but I do hope the revised version, which will come out with the next part, will be enjoyable. I was also waffling briefly on whether to color this comic since it turned out so long but there were some panels that I envisioned so clearly in color that I just decided I might as well color everything. I'm especially proud of how the fruit turned out, I want to eat it

Feel free to send in asks if you want annotations of different things I decided on and details I added in each page :) I'm quite proud of this one hehe

[sleepynoons]

nagumo yoichi x gn!reader, sfw, not beta read

cw: slight suggestive content, explicit language

notes: wait if you catch several typos/grammar mistakes, that's not on me, that's on ellipsus for constantly glitching out today and preventing me from making edits. this is a drabble, too, and i don't proofread those oops. anyway, i thought it'd be funny if nagumo also got into a relationship the same way sakamoto and aoi got together. i also think it's hilarious that the npcs in this series don't give a fuck lol. nagumo also comes off as weird af at first LMAO wait this was kinda meant to satisfy my belief that nagumo has a sleeper build iykwim - wait i'm realizing there are several references + tidbits in this piece so it'd be funny if y'all catch anything hehe

"WELCOME!"

greeting customers is arguably the least rewarding thing about your job. most people who walk in ignore you, some even look annoyed, and you hate public speaking in the first place. you think you lose five minutes of your life every single time you raise your voice, and those five minutes have probably accumulated to years by now.

you sigh. it can't be helped. another part-timer recently quit, and you can't possibly let the manager of this convenience store, an elderly man in his 60s, take on additional night shifts when he's already handling the early mornings.

besides, there are some pros. since the store is located near a university and a residential area, there are familiar faces. there's a group of computer science students that often drop by, and they play the occasional harmless prank on you. there's also that mother-daughter pair that buys frozen taiyakis every saturday as a reward for the daughter for finishing her weekly violin lesson. and perhaps the most intriguing of them all is a man that pops by every three days around midnight.

he wears the same tan trench coat, along with a loose patterned button-up and black pants. before winter set in, he always went straight to the freezer to fish out a popsicle, bar already in his mouth as he walked over to pay, but in the past two weeks, he's been opting for a cup of hot coffee and small packets of candy instead.

it seems he's craving sour gummies today. with a swift swipe of your arm, you grab and scan the barcode on the back of the plastic bag, and type in the amount for his drink.

"your total's ¥600."

"no discounts for your most loyal customer?"

startled, you freeze, determined to avoid eye contact. you've had conversations with other customers before, but never with him. he's always left as quickly as he came, so you're caught off-guard by this unexpected interaction.

"u-uh, not this time, sorry. i can ask the manager if we have a loyalty program, if you want."

the man hums as he nods happily and hands you two ¥500 coins. his unbothered smile unnerves you a bit, so you count the difference and return the loose change in personal record time.

but he doesn't leave, and instead, asks, "any thoughts on getting hitched?"

your spit-take's almost comical, but the absurdity of the situation takes precedent. "w-what now?"

"one of my co-workers recently got married to a convenience store worker, so i'd thought i'd give it a try, too!"

you're practically shaking from how anxious and overwhelmed this person's making you feel. it doesn't help that he's clearly not disturbed at all, which almost makes you doubt your own ethics and gut instincts. but, the more you think about it, the more you're sure there's something wrong with this man and not you.

"i-i, uh, well, i'm not interested in-in getting married right now."

"oh, that's a shame! guess i'll try again tomorrow!”

–

you wake up with a jolt, almost knocking the crown of your head into nagumo's chin. though, of course, there's no actual need to worry about that.

"hm, what's wrong?"

with a workbook on bayesian statistics in one hand, a pen resting on his ear, and his other arm folded behind his head, he looks down at you curiously. despite having just woken up, your head's never been clearer, and you sit up between his legs before looking behind your shoulder and shooting a glare at him.

you ask, "can i punch your face?"

nagumo laughs, probably already imagining your futile attempts. "sure! but can i ask why?"

"i dreamt about our first conversation, and it reminded me that you're kinda fucked up."

your boyfriend chuckles more, amused by your moral qualms. "you could say that."

the thought that your relationship is weird has never left you. you're (still) a simple convenience store cashier, and nagumo gets filthy rich by murdering people. you were never that interested in the world around you, having been too busy paying back student loans and applying to other jobs throughout your early adolescent years to care about other things, so when he told you about the JAA and the establishment of the assassin industry as a whole, you were shocked. but that's always as far down into the rabbit hole as you let yourself go.

from this view, with nagumo spread out before you, he doesn't look dangerous at all. if anything, he resembles a nerdy graduate student, thanks to his obvious passions for mathematics and reading. moreover, his short-sleeved t-shirt exposes his tattoo-riddled arms, and the bottom of it has ridden up, giving you a pleasurable view of his hip bones and happy trail. in fact, when the two of you got into bed together for the first time (don't ask how he succeeded in seducing you), you were surprised by his physique. his outside clothes certainly don't do his abs or biceps justice.

anyway, the point is, he looks like your fantasy of a dreamy, hot, geeky boyfriend, not your local professional hitman-for-hire.

you sigh. you're not going to punch his stupidly attractive face. you lie back down onto his chest, burrowing your nose into the crook of his neck. you do let yourself get away with a pinch to his cheek.

then, you mutter, "don't hurt me."

"i won't," he chirps.

nagumo presses the knuckles of his free hand into the knots around your shoulder blades and flips his book back open.

he knows you mean more than in the literal sense.

[king-of-men]

For each point, the new probability is the integral of the Bayes formula from 0 to 1.

oh an interesting scenario might be if you have $100 and can place repeated bets on the flip of a biased coin, but you don't know how it's biased; what strategy would you adopt to maximise your winnings?

[riseofthecommonwoodpile]

funniest thing that the rationalist community does is misuse Bayesian probability to make arbitrary statistical claims that look objective to people who don’t know what they’re doing. some AI researcher will be like “there’s a 64.7% likelihood of a Chinese-developed AGI causing an anti-human technological singularity versus only 38% for US-developed AGI” and you try to figure out what their methods for developing priors are and it’s just “trust me bro”

[max1461]

Bayesianism vs. frequentism... let's put these guys through a quick and dirty test on the basis of the physicalist nominalist structuralism thing that I think.

I sincerely doubt something like a measurable, quantitative "degree of belief" is cognitively real. Can you point me to the "degree of belief" in your brain? No. Bayesianism is out. On the other hand, you can't do infinitely many trials of things, at least as we currently understand the laws of physics. So the limit of a bunch of trials? Not real. Point at it. Not real.

But frequentism is closer. If I took every event in past and future history, and made a big list which ones count as "a coin flip" and which ones don't, and then looked at just the coin flips and, well, I bet about 50% would be heads and about 50% tails. Not exactly but about. That's what probability really is, I claim confidently in this post, it's literally just a proportion of two finite numbers. All the infinity shit is just an approximation. It's evident that the "true" probability of a coin coming up heads is not 50%, it's slightly different. Because if you count up all coin flips in past and future history it's not exactly 50/50. Saying it's a 50% probability is just an approximation, you don't have to do philosophical contortions to make it literally true.

Infinities just approximate large numbers well. Maybe some infinities are actually real, that's a question for physicists. Well for those I don't know.

[ceescedasticity]

There is the opportunity for a trick question along the lines of: You have tossed a coin 99 times and it has come up heads every time. What do you predict for the next toss? Heads; tails; equally likely.

With the correct answer being 'heads'. Because with a fair coin toss they're equally likely, but! Do you know what the global rate of unfair coins is? I don't, but it isn't zero. Trick coins exist. Even ruling that out, the primary purpose of coins is not equal-probability randomization and I'm pretty sure no one is validating their aerodynamics and/or density to make sure they're fair.

So you modify your previous idea of the probability distribution of the coin toss outcomes, and figure the next toss is more likely to be heads.

Actually that's a fun idea for a Bayesian statistics exploration — what does your posterior distribution do as you accumulate more and more results from what is in fact a two-head coin?

[canmom]

i probably would call myself a consequentialist, but not a utilitarian. my objection to utilitarianism is similar to my objection to the absolutist Bayesianism practiced in That Subculture: it's a philosophy that claims to be based around a certain computation, but actually performing that computation is completely intractable. there's no way to actually update your probability assignments of all possible statements in response to new information, any more than it's possible to aggregate the total happiness/suffering/whatever across the entire future for each imaginable course of action.

so this calculation is entirely notional. what you're actually doing is coming up with verbal arguments and vague heuristics for how you think this notional calculation would work. perhaps it's as good an entry point as any. but the supposed mathematical rigour is just rhetoric! you can talk about utilons this and QALYs that, but there is no way to calculate this shit, it's just a mathematical coat of paint.

the second objection is the 'seeing like a state' objection (or seeing like a company/NGO): the 'utility function' is a construct used to make economic models. it doesn't model humans particularly well, who have a variety of competing impulses that don't lend themselves to nice formalisms. and to demand that you should live according to a utility function is accordingly to strip the world of its complexity to make it more tractable. instead of specific people with specific desires and needs and relationships into which you fit, which aren't necessarily commensurable, you have abstract fungible units of pleasure or suffering or whatever else you're trying to optimise.

this worldview appealed to me as a teenager. I imagined that you could model an agent as a some kind of surface between it and the world - a sphere, perhaps, inside your head; the course of your life would be the movement of particles in and out of this sphere, and theoretically there would be a pattern for every instant of time that would lead to the best possible impact on the world, solving 'life' much like a tool assisted speedrun solves a game. the goal would be then to approximate this optimal run as much as possible. then I'd think of problems with this model: couldn't you just spawn high energy photons on the sphere to melt shit like a laser? we'd have to put some restrictions on it, obviously. what if the optimal run was really close to a harmful run, so a small mistake would lead to disaster? perhaps you'd be better to find a stable local maximum instead. and so on.

I'm not sure what good it did me to imagine this funny (or if you prefer, terminally STEM-brained) thought experiment, but it was very nice and mathematical-looking, and back then I really wanted my philosophy to be impossibly demanding for some reason. some weird combo of depression and autism and a self image very much dependent on being told i was good?

these days my feeling is that the pretense of mathematical rigour where it doesn't exist is untrustworthy, and particularly where people are concerned, abstracting too much loses important information. I'm not a court of law where strict consistency matters for the sake of stability or whatever, nor a government trying to figure out which levers to pull to create the ideal society - I'm an organism embedded in a bewilderingly complex system, and I can take each situation as it comes. treating the people I interact with well is important to me. I still sometimes think along utilitarianish lines sometimes - particularly 'this person could use this money more than me' - but I make no pretense to rigour or optimisation with it.