multiplicative   50

Complexity no Bar to AI -
Critics of AI risk suggest diminishing returns to computing (formalized asymptotically) means AI will be weak; this argument relies on a large number of questionable premises and ignoring additional resources, constant factors, and nonlinear returns to small intelligence advantages, and is highly unlikely. (computer science, transhumanism, AI, R)
created: 1 June 2014; modified: 01 Feb 2018; status: finished; confidence: likely; importance: 10
ratty  gwern  analysis  faq  ai  risk  speedometer  intelligence  futurism  cs  computation  complexity  tcs  linear-algebra  nonlinearity  convexity-curvature  average-case  adversarial  article  time-complexity  singularity  iteration-recursion  magnitude  multiplicative  lower-bounds  no-go  performance  hardware  humanity  psychology  cog-psych  psychometrics  iq  distribution  moments  complement-substitute  hanson  ems  enhancement  parable  detail-architecture  universalism-particularism  neuro  ai-control  environment  climate-change  threat-modeling  security  theory-practice  hacker  academia  realness  crypto  rigorous-crypto  usa  government 
april 2018 by nhaliday
Altruism in a volatile world | Nature
The evolution of altruism—costly self-sacrifice in the service of others—has puzzled biologists1 since The Origin of Species. For half a century, attempts to understand altruism have developed around the concept that altruists may help relatives to have extra offspring in order to spread shared genes2. This theory—known as inclusive fitness—is founded on a simple inequality termed Hamilton’s rule2. However, explanations of altruism have typically not considered the stochasticity of natural environments, which will not necessarily favour genotypes that produce the greatest average reproductive success3,4. Moreover, empirical data across many taxa reveal associations between altruism and environmental stochasticity5,6,7,8, a pattern not predicted by standard interpretations of Hamilton’s rule. Here we derive Hamilton’s rule with explicit stochasticity, leading to new predictions about the evolution of altruism. We show that altruists can increase the long-term success of their genotype by reducing the temporal variability in the number of offspring produced by their relatives. Consequently, costly altruism can evolve even if it has a net negative effect on the average reproductive success of related recipients. The selective pressure on volatility-suppressing altruism is proportional to the coefficient of variation in population fitness, and is therefore diminished by its own success. Our results formalize the hitherto elusive link between bet-hedging and altruism4,9,10,11, and reveal missing fitness effects in the evolution of animal societies.
study  bio  evolution  altruism  kinship  stylized-facts  models  intricacy  random  signal-noise  time  order-disorder  org:nat  EGT  cooperate-defect  population-genetics  moments  expectancy  multiplicative  additive 
march 2018 by nhaliday
Variance of product of multiple random variables - Cross Validated
prod_i (var[X_i] + (E[X_i])^2) - prod_i (E[X_i])^2

two variable case: var[X] var[Y] + var[X] (E[Y])^2 + (E[X])^2 var[Y]
nibble  q-n-a  overflow  stats  probability  math  identity  moments  arrows  multiplicative  iidness  dependence-independence 
october 2017 by nhaliday
Lecture 14: When's that meteor arriving
- Meteors as a random process
- Limiting approximations
- Derivation of the Exponential distribution
- Derivation of the Poisson distribution
- A "Poisson process"
nibble  org:junk  org:edu  exposition  lecture-notes  physics  mechanics  space  earth  probability  stats  distribution  stochastic-processes  closure  additive  limits  approximation  tidbits  acm  binomial  multiplicative 
september 2017 by nhaliday
[1502.05274] How predictable is technological progress?
Recently it has become clear that many technologies follow a generalized version of Moore's law, i.e. costs tend to drop exponentially, at different rates that depend on the technology. Here we formulate Moore's law as a correlated geometric random walk with drift, and apply it to historical data on 53 technologies. We derive a closed form expression approximating the distribution of forecast errors as a function of time. Based on hind-casting experiments we show that this works well, making it possible to collapse the forecast errors for many different technologies at different time horizons onto the same universal distribution. This is valuable because it allows us to make forecasts for any given technology with a clear understanding of the quality of the forecasts. As a practical demonstration we make distributional forecasts at different time horizons for solar photovoltaic modules, and show how our method can be used to estimate the probability that a given technology will outperform another technology at a given point in the future.

- p_t = unit price of tech
- log(p_t) = y_0 - μt + ∑_{i <= t} n_i
- n_t iid noise process
preprint  study  economics  growth-econ  innovation  discovery  technology  frontier  tetlock  meta:prediction  models  time  definite-planning  stylized-facts  regression  econometrics  magnitude  energy-resources  phys-energy  money  cost-benefit  stats  data-science  🔬  ideas  speedometer  multiplicative  methodology  stochastic-processes  time-series  stock-flow  iteration-recursion  org:mat 
april 2017 by nhaliday
Beta function - Wikipedia
B(x, y) = int_0^1 t^{x-1}(1-t)^{y-1} dt = Γ(x)Γ(y)/Γ(x+y)
one misc. application: calculating pdf of Erlang distribution (sum of iid exponential r.v.s)
concept  atoms  acm  math  calculation  integral  wiki  reference  identity  AMT  distribution  multiplicative 
march 2017 by nhaliday
254A, Supplement 4: Probabilistic models and heuristics for the primes (optional) | What's new
among others, the Cramér model for the primes (basically kinda looks like primality is independently distributed w/ Pr[n is prime] = 1/log n)
gowers  mathtariat  lecture-notes  exposition  math  math.NT  probability  heuristic  models  cartoons  nibble  org:bleg  pseudorandomness  borel-cantelli  concentration-of-measure  multiplicative 
february 2017 by nhaliday

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