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107-517: The concept originates in Robin Hanson 's argument that the failure to find any extraterrestrial civilizations in the observable universe implies that something is wrong with one or more of the arguments (from various scientific disciplines) that the appearance of advanced intelligent life is probable; this observation is conceptualized in terms of a "Great Filter" which acts to reduce the great number of sites where intelligent life might arise to

214-425: A Bayesian would not be concerned with such issues, but it can be important in this situation. For example, one would want any decision rule based on the posterior distribution to be admissible under the adopted loss function. Unfortunately, admissibility is often difficult to check, although some results are known (e.g., Berger and Strawderman 1996). The issue is particularly acute with hierarchical Bayes models ;

321-406: A beta distribution to model the distribution of the parameter p of a Bernoulli distribution , then: In principle, priors can be decomposed into many conditional levels of distributions, so-called hierarchical priors . An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach

428-558: A DARPA project to implement a market for betting on future developments in the Middle East. Hanson has expressed great disappointment in DARPA's cancellation of its related FutureMAP project, and he attributes this to the controversy surrounding the related Total Information Awareness program. He also created and supports a proposed system of government called futarchy , in which policies would be determined by prediction markets. In

535-501: A basis for induction in very general settings. Practical problems associated with uninformative priors include the requirement that the posterior distribution be proper. The usual uninformative priors on continuous, unbounded variables are improper. This need not be a problem if the posterior distribution is proper. Another issue of importance is that if an uninformative prior is to be used routinely , i.e., with many different data sets, it should have good frequentist properties. Normally

642-441: A case, the scale group is the natural group structure, and the corresponding prior on X is proportional to 1/ x . It sometimes matters whether we use the left-invariant or right-invariant Haar measure. For example, the left and right invariant Haar measures on the affine group are not equal. Berger (1985, p. 413) argues that the right-invariant Haar measure is the correct choice. Another idea, championed by Edwin T. Jaynes ,

749-473: A controversial 2018 blog post on the incel movement, Hanson appeared to agree with the incel movement's likening of the distribution of job opportunities to "access to sex". He wrote that he found it puzzling that similar concern had not been shown for incels as for low-income individuals. Some journalists, such as Alexandra Scaggs in the Financial Times , criticized Hanson for discussing sex as if it

856-444: A current assumption, theory, concept or idea is founded. A strong prior is a type of informative prior in which the information contained in the prior distribution dominates the information contained in the data being analyzed. The Bayesian analysis combines the information contained in the prior with that extracted from the data to produce the posterior distribution which, in the case of a "strong prior", would be little changed from

963-414: A die is thrown) to the total number of events—and these considered purely deductively, i.e. without any experimenting. In the case of the die if we look at it on the table without throwing it, each elementary event is reasoned deductively to have the same probability—thus the probability of each outcome of an imaginary throwing of the (perfect) die or simply by counting the number of faces is 1/6. Each face of

1070-491: A fairly detailed discussion of Hanson's views: Robin has strange ideas ... My other friend and colleague Bryan Caplan put it best: "When the typical economist tells me about his latest research, my standard reaction is 'Eh, maybe.' Then I forget about it. When Robin Hanson tells me about his latest research, my standard reaction is 'No way! Impossible!' Then I think about it for years." Nate Silver , in his book The Signal and

1177-544: A galaxy filled with intelligent extraterrestrial civilizations that have failed to colonize Earth . Perhaps the aliens lacked the intent and purpose to colonize or depleted their resources, or maybe the galaxy is colonized but in a heterogeneous manner, or the Earth could be located in a "galactic backwater". Although absence of evidence generally is only weak evidence of absence , the absence of extraterrestrial megascale engineering projects, for example, might point to

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1284-417: A huge number of replicas of this system, one obtains what is called a microcanonical ensemble . It is for this system that one postulates in quantum statistics the "fundamental postulate of equal a priori probabilities of an isolated system." This says that the isolated system in equilibrium occupies each of its accessible states with the same probability. This fundamental postulate therefore allows us to equate

1391-578: A matter wave is explicitly ψ ∝ sin ⁡ ( l π x / L ) sin ⁡ ( m π y / L ) sin ⁡ ( n π z / L ) , {\displaystyle \psi \propto \sin(l\pi x/L)\sin(m\pi y/L)\sin(n\pi z/L),} where l , m , n {\displaystyle l,m,n} are integers. The number of different ( l , m , n ) {\displaystyle (l,m,n)} values and hence states in

1498-472: A number of states in quantum (i.e. wave) mechanics, recall that in quantum mechanics every particle is associated with a matter wave which is the solution of a Schrödinger equation. In the case of free particles (of energy ϵ = p 2 / 2 m {\displaystyle \epsilon ={\bf {p}}^{2}/2m} ) like those of a gas in a box of volume V = L 3 {\displaystyle V=L^{3}} such

1605-495: A particular economic or policy outcome, like whether Israel will go to war with Iran, or how much global temperatures will rise because of climate change. His argument for these is pretty simple: They ensure that we have a financial stake in being accurate when we make forecasts, rather than just trying to look good to our peers. Hanson is credited with originating the concept of the Policy Analysis Market (PAM),

1712-405: A particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable . In Bayesian statistics , Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution , which is the conditional distribution of the uncertain quantity given new data. Historically,

1819-464: A prior by the likelihood, and an empty product is just the constant likelihood 1. However, without starting with a prior probability distribution, one does not end up getting a posterior probability distribution, and thus cannot integrate or compute expected values or loss. See Likelihood function § Non-integrability for details. Examples of improper priors include: These functions, interpreted as uniform distributions, can also be interpreted as

1926-595: A prior for the running speed of a runner who is unknown to us. We could specify, say, a normal distribution as the prior for his speed, but alternatively we could specify a normal prior for the time he takes to complete 100 metres, which is proportional to the reciprocal of the first prior. These are very different priors, but it is not clear which is to be preferred. Jaynes' method of transformation groups can answer this question in some situations. Similarly, if asked to estimate an unknown proportion between 0 and 1, we might say that all proportions are equally likely, and use

2033-461: A prior might also be called a not very informative prior , or an objective prior , i.e. one that is not subjectively elicited. Uninformative priors can express "objective" information such as "the variable is positive" or "the variable is less than some limit". The simplest and oldest rule for determining a non-informative prior is the principle of indifference , which assigns equal probabilities to all possibilities. In parameter estimation problems,

2140-441: A priori weighting is L Δ p / h {\displaystyle L\Delta p/h} . In customary 3 dimensions (volume V {\displaystyle V} ) the corresponding number can be calculated to be V 4 π p 2 Δ p / h 3 {\displaystyle V4\pi p^{2}\Delta p/h^{3}} . In order to understand this quantity as giving

2247-451: A subject of philosophical controversy, with Bayesians being roughly divided into two schools: "objective Bayesians", who believe such priors exist in many useful situations, and "subjective Bayesians" who believe that in practice priors usually represent subjective judgements of opinion that cannot be rigorously justified (Williamson 2010). Perhaps the strongest arguments for objective Bayesianism were given by Edwin T. Jaynes , based mainly on

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2354-541: A system in dynamic equilibrium (i.e. under steady, uniform conditions) with (2) total (and huge) number of particles N = Σ i n i {\displaystyle N=\Sigma _{i}n_{i}} (this condition determines the constant ϵ 0 {\displaystyle \epsilon _{0}} ), and (3) total energy E = Σ i n i ϵ i {\displaystyle E=\Sigma _{i}n_{i}\epsilon _{i}} , i.e. with each of

2461-556: A uniform prior. Alternatively, we might say that all orders of magnitude for the proportion are equally likely, the logarithmic prior , which is the uniform prior on the logarithm of proportion. The Jeffreys prior attempts to solve this problem by computing a prior which expresses the same belief no matter which metric is used. The Jeffreys prior for an unknown proportion p is p (1 −  p ) , which differs from Jaynes' recommendation. Priors based on notions of algorithmic probability are used in inductive inference as

2568-882: Is Chief Scientist ), and has conducted research on signalling . He also proposed the Great Filter hypothesis. Hanson received a BS in physics from the University of California, Irvine in 1981, an MS in physics and an MA in Conceptual Foundations of Science from the University of Chicago in 1984, and a PhD in social science from Caltech in 1997 for his thesis titled Four puzzles in information and politics: Product bans, informed voters, social insurance, and persistent disagreement . Before getting his PhD he researched artificial intelligence , Bayesian statistics and hypertext publishing at Lockheed , NASA , and elsewhere. In addition, he started

2675-579: Is a measure of the fraction of states actually occupied by electrons at energy ϵ i {\displaystyle \epsilon _{i}} and temperature T {\displaystyle T} . On the other hand, the a priori probability g i {\displaystyle g_{i}} is a measure of the number of wave mechanical states available. Hence n i = f i g i . {\displaystyle n_{i}=f_{i}g_{i}.} Since n i {\displaystyle n_{i}}

2782-514: Is a quasi-KL divergence ("quasi" in the sense that the square root of the Fisher information may be the kernel of an improper distribution). Due to the minus sign, we need to minimise this in order to maximise the KL divergence with which we started. The minimum value of the last equation occurs where the two distributions in the logarithm argument, improper or not, do not diverge. This in turn occurs when

2889-611: Is an associate professor of economics at George Mason University and a former research associate at the Future of Humanity Institute of Oxford University . He is known for his work on idea futures and markets, and he was involved in the creation of the Foresight Institute's Foresight Exchange and DARPA 's FutureMAP project. He invented market scoring rules like LMSR ( Logarithmic Market Scoring Rule ) used by prediction markets such as Consensus Point (where Hanson

2996-530: Is an ellipse of area ∮ d p θ d p ϕ = π 2 I E 2 I E sin ⁡ θ = 2 π I E sin ⁡ θ . {\displaystyle \oint dp_{\theta }dp_{\phi }=\pi {\sqrt {2IE}}{\sqrt {2IE}}\sin \theta =2\pi IE\sin \theta .} By integrating over θ {\displaystyle \theta } and ϕ {\displaystyle \phi }

3103-547: Is an observer with limited knowledge about the system. As a more contentious example, Jaynes published an argument based on the invariance of the prior under a change of parameters that suggests that the prior representing complete uncertainty about a probability should be the Haldane prior p (1 −  p ) . The example Jaynes gives is of finding a chemical in a lab and asking whether it will dissolve in water in repeated experiments. The Haldane prior gives by far

3210-405: Is clear that the same result would be obtained if all the prior probabilities P ( A i ) and P ( A j ) were multiplied by a given constant; the same would be true for a continuous random variable . If the summation in the denominator converges, the posterior probabilities will still sum (or integrate) to 1 even if the prior values do not, and so the priors may only need to be specified in

3317-779: Is constant under uniform conditions (as many particles as flow out of a volume element also flow in steadily, so that the situation in the element appears static), i.e. independent of time t {\displaystyle t} , and g i {\displaystyle g_{i}} is also independent of time t {\displaystyle t} as shown earlier, we obtain d f i d t = 0 , f i = f i ( t , v i , r i ) . {\displaystyle {\frac {df_{i}}{dt}}=0,\quad f_{i}=f_{i}(t,{\bf {v}}_{i},{\bf {r}}_{i}).} Expressing this equation in terms of its partial derivatives, one obtains

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3424-499: Is given by K L = ∫ p ( t ) ∫ p ( x ∣ t ) log ⁡ p ( x ∣ t ) p ( x ) d x d t {\displaystyle KL=\int p(t)\int p(x\mid t)\log {\frac {p(x\mid t)}{p(x)}}\,dx\,dt} Here, t {\displaystyle t} is a sufficient statistic for some parameter x {\displaystyle x} . The inner integral

3531-629: Is here θ , ϕ {\displaystyle \theta ,\phi } ), i.e. E = 1 2 I ( p θ 2 + p ϕ 2 sin 2 ⁡ θ ) . {\displaystyle E={\frac {1}{2I}}\left(p_{\theta }^{2}+{\frac {p_{\phi }^{2}}{\sin ^{2}\theta }}\right).} The ( p θ , p ϕ ) {\displaystyle (p_{\theta },p_{\phi })} -curve for constant E and θ {\displaystyle \theta }

3638-404: Is independent of time—you can look at the die on the table as long as you like without touching it and you deduce the probability for the number 6 to appear on the upper face is 1/6. In statistical mechanics, e.g. that of a gas contained in a finite volume V {\displaystyle V} , both the spatial coordinates q i {\displaystyle q_{i}} and

3745-403: Is limited and a vast amount of prior knowledge is available. In these methods, either an information theory based criterion, such as KL divergence or log-likelihood function for binary supervised learning problems and mixture model problems. Philosophical problems associated with uninformative priors are associated with the choice of an appropriate metric, or measurement scale. Suppose we want

3852-459: Is part of the prior and, as more evidence accumulates, the posterior is determined largely by the evidence rather than any original assumption, provided that the original assumption admitted the possibility of what the evidence is suggesting. The terms "prior" and "posterior" are generally relative to a specific datum or observation. A strong prior is a preceding assumption, theory, concept or idea upon which, after taking account of new information,

3959-461: Is proportional to the (asymptotically large) sample size. We do not know the value of x ∗ {\displaystyle x*} . Indeed, the very idea goes against the philosophy of Bayesian inference in which 'true' values of parameters are replaced by prior and posterior distributions. So we remove x ∗ {\displaystyle x*} by replacing it with x {\displaystyle x} and taking

4066-633: Is the 'true' value. Since this does not depend on t {\displaystyle t} it can be taken out of the integral, and as this integral is over a probability space it equals one. Hence we can write the asymptotic form of KL as K L = − log ⁡ ( 1 k I ( x ∗ ) ) − ∫ p ( x ) log ⁡ [ p ( x ) ] d x {\displaystyle KL=-\log \left(1{\sqrt {kI(x^{*})}}\right)-\,\int p(x)\log[p(x)]\,dx} where k {\displaystyle k}

4173-1125: Is the KL divergence between the posterior p ( x ∣ t ) {\displaystyle p(x\mid t)} and prior p ( x ) {\displaystyle p(x)} distributions and the result is the weighted mean over all values of t {\displaystyle t} . Splitting the logarithm into two parts, reversing the order of integrals in the second part and noting that log [ p ( x ) ] {\displaystyle \log \,[p(x)]} does not depend on t {\displaystyle t} yields K L = ∫ p ( t ) ∫ p ( x ∣ t ) log ⁡ [ p ( x ∣ t ) ] d x d t − ∫ log ⁡ [ p ( x ) ] ∫ p ( t ) p ( x ∣ t ) d t d x {\displaystyle KL=\int p(t)\int p(x\mid t)\log[p(x\mid t)]\,dx\,dt\,-\,\int \log[p(x)]\,\int p(t)p(x\mid t)\,dt\,dx} The inner integral in

4280-482: Is the negative expected value over t {\displaystyle t} of the entropy of x {\displaystyle x} conditional on t {\displaystyle t} plus the marginal (i.e. unconditional) entropy of x {\displaystyle x} . In the limiting case where the sample size tends to infinity, the Bernstein-von Mises theorem states that

4387-505: Is the number of standing waves (i.e. states) therein, where Δ q {\displaystyle \Delta q} is the range of the variable q {\displaystyle q} and Δ p {\displaystyle \Delta p} is the range of the variable p {\displaystyle p} (here for simplicity considered in one dimension). In 1 dimension (length L {\displaystyle L} ) this number or statistical weight or

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4494-410: Is the standard normal distribution . The principle of minimum cross-entropy generalizes MAXENT to the case of "updating" an arbitrary prior distribution with suitable constraints in the maximum-entropy sense. A related idea, reference priors , was introduced by José-Miguel Bernardo . Here, the idea is to maximize the expected Kullback–Leibler divergence of the posterior distribution relative to

4601-452: Is the variance of the distribution. In this case therefore H = log ⁡ 2 π e N I ( x ∗ ) {\displaystyle H=\log {\sqrt {\frac {2\pi e}{NI(x^{*})}}}} where N {\displaystyle N} is the arbitrarily large sample size (to which Fisher information is proportional) and x ∗ {\displaystyle x*}

4708-497: Is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for that day of the year. This example has a property in common with many priors, namely, that the posterior from one problem (today's temperature) becomes the prior for another problem (tomorrow's temperature); pre-existing evidence which has already been taken into account

4815-472: Is to use the principle of maximum entropy (MAXENT). The motivation is that the Shannon entropy of a probability distribution measures the amount of information contained in the distribution. The larger the entropy, the less information is provided by the distribution. Thus, by maximizing the entropy over a suitable set of probability distributions on X , one finds the distribution that is least informative in

4922-415: The n i {\displaystyle n_{i}} particles having the energy ϵ i {\displaystyle \epsilon _{i}} . An important aspect in the derivation is the taking into account of the indistinguishability of particles and states in quantum statistics, i.e. there particles and states do not have labels. In the case of fermions, like electrons, obeying

5029-520: The A j . Statisticians sometimes use improper priors as uninformative priors . For example, if they need a prior distribution for the mean and variance of a random variable, they may assume p ( m ,  v ) ~ 1/ v (for v  > 0) which would suggest that any value for the mean is "equally likely" and that a value for the positive variance becomes "less likely" in inverse proportion to its value. Many authors (Lindley, 1973; De Groot, 1937; Kass and Wasserman, 1996) warn against

5136-505: The Fermi paradox : "They do exist, but we see no evidence". Other ideas include: it is too expensive to spread physically throughout the galaxy; Earth is purposely isolated; it is dangerous to communicate and hence civilizations actively hide, among others. Astrobiologists Dirk Schulze-Makuch and William Bains, reviewing the history of life on Earth, including convergent evolution , concluded that transitions such as oxygenic photosynthesis ,

5243-715: The Milky Way would be full of colonies. So perhaps step 9 is the unlikely one, and the only things that appear likely to keep us from step 9 are some sort of catastrophe , an underestimation of the impact of procrastination as technology increasingly unburdens existence, or resource exhaustion leading to the impossibility of making the step due to consumption of the available resources (for example highly constrained energy resources). So by this argument, finding multicellular life on Mars (provided it evolved independently) would be bad news, since it would imply steps 2–6 are easy, and hence only 1, 7, 8 or 9 (or some unknown step) could be

5350-508: The Pauli principle (only one particle per state or none allowed), one has therefore 0 ≤ f i F D ≤ 1 , whereas 0 ≤ f i B E ≤ ∞ . {\displaystyle 0\leq f_{i}^{FD}\leq 1,\quad {\text{whereas}}\quad 0\leq f_{i}^{BE}\leq \infty .} Thus f i F D {\displaystyle f_{i}^{FD}}

5457-542: The eukaryotic cell , multicellularity , and tool -using intelligence are likely to occur on any Earth-like planet given enough time. They argue that the Great Filter may be abiogenesis , the rise of technological human-level intelligence, or an inability to settle other worlds because of self-destruction or a lack of resources. Astronomer Seth Shostak of the SETI Institute argues that one can postulate

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5564-465: The likelihood function in the absence of data, but are not proper priors. While in Bayesian statistics the prior probability is used to represent initial beliefs about an uncertain parameter, in statistical mechanics the a priori probability is used to describe the initial state of a system. The classical version is defined as the ratio of the number of elementary events (e.g. the number of times

5671-399: The uncertainty relation , which in 1 spatial dimension is Δ q Δ p ≥ h , {\displaystyle \Delta q\Delta p\geq h,} these states are indistinguishable (i.e. these states do not carry labels). An important consequence is a result known as Liouville's theorem , i.e. the time independence of this phase space volume element and thus of

5778-541: The Bernoulli random variable. Priors can be constructed which are proportional to the Haar measure if the parameter space X carries a natural group structure which leaves invariant our Bayesian state of knowledge. This can be seen as a generalisation of the invariance principle used to justify the uniform prior over the three cups in the example above. For example, in physics we might expect that an experiment will give

5885-548: The Brain (2018), coauthored with Kevin Simler, looks at mental blind spots of society and individuals. Prior probability A prior probability distribution of an uncertain quantity, often simply called the prior , is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for

5992-401: The Earth, seems "dead"; Hanson states: Our planet and solar system, however, don't look substantially colonized by advanced competitive life from the stars, and neither does anything else we see. To the contrary, we have had great success at explaining the behavior of our planet and solar system, nearby stars, our galaxy, and even other galaxies, via simple "dead" physical processes, rather than

6099-507: The Great Filter at work. Does this mean that one of the steps leading to intelligent life is unlikely? According to Shostak: This is, of course, a variant on the Fermi paradox: We don't see clues to widespread, large-scale engineering, and consequently we must conclude that we're alone. But the possibly flawed assumption here is when we say that highly visible construction projects are an inevitable outcome of intelligence. It could be that it's

6206-454: The Noise (2012), writes: He is clearly not a man afraid to challenge the conventional wisdom. Instead, Hanson writes a blog called Overcoming Bias, in which he presses readers to consider which cultural taboos, ideological beliefs, or misaligned incentives might constrain them from making optimal decisions. Hanson ... is an advocate of prediction markets – systems where you can place bets on

6313-958: The a priori probability is effectively a measure of the degeneracy , i.e. the number of states having the same energy. In statistical mechanics (see any book) one derives the so-called distribution functions f {\displaystyle f} for various statistics. In the case of Fermi–Dirac statistics and Bose–Einstein statistics these functions are respectively f i F D = 1 e ( ϵ i − ϵ 0 ) / k T + 1 , f i B E = 1 e ( ϵ i − ϵ 0 ) / k T − 1 . {\displaystyle f_{i}^{FD}={\frac {1}{e^{(\epsilon _{i}-\epsilon _{0})/kT}+1}},\quad f_{i}^{BE}={\frac {1}{e^{(\epsilon _{i}-\epsilon _{0})/kT}-1}}.} These functions are derived for (1)

6420-507: The a priori probability to the degeneracy of a system, i.e. to the number of different states with the same energy. The following example illustrates the a priori probability (or a priori weighting) in (a) classical and (b) quantal contexts. Consider the rotational energy E of a diatomic molecule with moment of inertia I in spherical polar coordinates θ , ϕ {\displaystyle \theta ,\phi } (this means q {\displaystyle q} above

6527-662: The a priori probability. A time dependence of this quantity would imply known information about the dynamics of the system, and hence would not be an a priori probability. Thus the region Ω := Δ q Δ p ∫ Δ q Δ p , ∫ Δ q Δ p = c o n s t . , {\displaystyle \Omega :={\frac {\Delta q\Delta p}{\int \Delta q\Delta p}},\;\;\;\int \Delta q\Delta p=\mathrm {const.} ,} when differentiated with respect to time t {\displaystyle t} yields zero (with

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6634-779: The approximate number of states in the range dE is given by the degeneracy, i.e. Σ ∝ ( 2 n + 1 ) d n . {\displaystyle \Sigma \propto (2n+1)dn.} Thus the a priori weighting in the classical context (a) corresponds to the a priori weighting here in the quantal context (b). In the case of the one-dimensional simple harmonic oscillator of natural frequency ν {\displaystyle \nu } one finds correspondingly: (a) Ω ∝ d E / ν {\displaystyle \Omega \propto dE/\nu } , and (b) Σ ∝ d n {\displaystyle \Sigma \propto dn} (no degeneracy). Thus in quantum mechanics

6741-502: The big problem. Although steps 1–8 have occurred on Earth, any one of these may be unlikely. If the first seven steps are necessary preconditions to calculating the likelihood (using the local environment ) then an anthropically biased observer can infer nothing about the general probabilities from its ( pre-determined ) surroundings. In a 2020 paper, Jacob Haqq-Misra, Ravi Kumar Kopparapu, and Edward Schwieterman argued that current and future telescopes searching for biosignatures in

6848-682: The bleaker the future chances of humanity probably are. The idea was first proposed in an online essay titled "The Great Filter – Are We Almost Past It?". The first version was written in August 1996 and the article was last updated on September 15, 1998. Hanson's formulation has received recognition in several published sources discussing the Fermi paradox and its implications. There is no reliable evidence that aliens have visited Earth ; we have observed no intelligent extraterrestrial life with current technology, nor has SETI found any transmissions from other civilizations . The Universe , apart from

6955-526: The choice of priors was often constrained to a conjugate family of a given likelihood function , so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern. There are many ways to construct a prior distribution. In some cases, a prior may be determined from past information, such as previous experiments. A prior can also be elicited from

7062-428: The classical a priori weighting in the energy range d E {\displaystyle dE} is Assuming that the number of quantum states in a range Δ q Δ p {\displaystyle \Delta q\Delta p} for each direction of motion is given, per element, by a factor Δ q Δ p / h {\displaystyle \Delta q\Delta p/h} ,

7169-512: The complex purposeful processes of advanced life. Life is expected to expand to fill all available niches. With technology such as self-replicating spacecraft , these niches would include neighboring star systems and even, on longer time scales which are still small compared to the age of the universe, other galaxies. Hanson notes, "If such advanced life had substantially colonized our planet, we would know it by now." With no evidence of intelligent life in places other than Earth, it appears that

7276-637: The concept of entropy which, in the case of probability distributions, is the negative expected value of the logarithm of the probability mass or density function or H ( x ) = − ∫ p ( x ) log ⁡ [ p ( x ) ] d x . {\textstyle H(x)=-\int p(x)\log[p(x)]\,dx.} Using this in the last equation yields K L = − ∫ p ( t ) H ( x ∣ t ) d t + H ( x ) {\displaystyle KL=-\int p(t)H(x\mid t)\,dt+\,H(x)} In words, KL

7383-399: The consequences of symmetries and on the principle of maximum entropy. As an example of an a priori prior, due to Jaynes (2003), consider a situation in which one knows a ball has been hidden under one of three cups, A, B, or C, but no other information is available about its location. In this case a uniform prior of p ( A ) = p ( B ) = p ( C ) = 1/3 seems intuitively like

7490-423: The correct proportion. Taking this idea further, in many cases the sum or integral of the prior values may not even need to be finite to get sensible answers for the posterior probabilities. When this is the case, the prior is called an improper prior . However, the posterior distribution need not be a proper distribution if the prior is improper. This is clear from the case where event B is independent of all of

7597-481: The danger of over-interpreting those priors since they are not probability densities. The only relevance they have is found in the corresponding posterior, as long as it is well-defined for all observations. (The Haldane prior is a typical counterexample. ) By contrast, likelihood functions do not need to be integrated, and a likelihood function that is uniformly 1 corresponds to the absence of data (all models are equally likely, given no data): Bayes' rule multiplies

7704-415: The die appears with equal probability—probability being a measure defined for each elementary event. The result is different if we throw the die twenty times and ask how many times (out of 20) the number 6 appears on the upper face. In this case time comes into play and we have a different type of probability depending on time or the number of times the die is thrown. On the other hand, the a priori probability

7811-493: The distribution of x {\displaystyle x} conditional on a given observed value of t {\displaystyle t} is normal with a variance equal to the reciprocal of the Fisher information at the 'true' value of x {\displaystyle x} . The entropy of a normal density function is equal to half the logarithm of 2 π e v {\displaystyle 2\pi ev} where v {\displaystyle v}

7918-408: The engineering of the small, rather than the large, that is inevitable. This follows from the laws of inertia (smaller machines are faster, and require less energy to function) as well as the speed of light (small computers have faster internal communication). It may be—and this is, of course, speculation—that advanced societies are building small technology and have little incentive or need to rearrange

8025-536: The expected value of the normal entropy, which we obtain by multiplying by p ( x ) {\displaystyle p(x)} and integrating over x {\displaystyle x} . This allows us to combine the logarithms yielding K L = − ∫ p ( x ) log ⁡ [ p ( x ) k I ( x ) ] d x {\displaystyle KL=-\int p(x)\log \left[{\frac {p(x)}{\sqrt {kI(x)}}}\right]\,dx} This

8132-517: The first internal corporate prediction market at Xanadu in 1990. He is married to Peggy Jackson, a hospice social worker , and has two children. He is the son of a Southern Baptist preacher. Hanson has elected to have his brain cryonically preserved in the event of medical death. He was involved early on in the creation of the Rationalist community through online weblogs. Tyler Cowen 's book Discover Your Inner Economist includes

8239-445: The help of Hamilton's equations): The volume at time t {\displaystyle t} is the same as at time zero. One describes this also as conservation of information. In the full quantum theory one has an analogous conservation law. In this case, the phase space region is replaced by a subspace of the space of states expressed in terms of a projection operator P {\displaystyle P} , and instead of

8346-410: The interval [0, 1]. This is obtained by applying Bayes' theorem to the data set consisting of one observation of dissolving and one of not dissolving, using the above prior. The Haldane prior is an improper prior distribution (meaning that it has an infinite mass). Harold Jeffreys devised a systematic way for designing uninformative priors as e.g., Jeffreys prior p (1 −  p ) for

8453-419: The list were complete—must be improbable. If it is not an early step (i.e., in the past), then the implication is that the improbable step lies in the future and humanity's prospects of reaching step 9 (interstellar colonization) are still bleak. If the past steps are likely, then many civilizations would have developed to the current level of the human species. However, none appear to have made it to step 9, or

8560-497: The momentum coordinates p i {\displaystyle p_{i}} of the individual gas elements (atoms or molecules) are finite in the phase space spanned by these coordinates. In analogy to the case of the die, the a priori probability is here (in the case of a continuum) proportional to the phase space volume element Δ q Δ p {\displaystyle \Delta q\Delta p} divided by h {\displaystyle h} , and

8667-427: The most weight to p = 0 {\displaystyle p=0} and p = 1 {\displaystyle p=1} , indicating that the sample will either dissolve every time or never dissolve, with equal probability. However, if one has observed samples of the chemical to dissolve in one experiment and not to dissolve in another experiment then this prior is updated to the uniform distribution on

8774-1340: The number of states in the energy range dE is, as seen under (a) 8 π 2 I d E / h 2 {\displaystyle 8\pi ^{2}IdE/h^{2}} for the rotating diatomic molecule. From wave mechanics it is known that the energy levels of a rotating diatomic molecule are given by E n = n ( n + 1 ) h 2 8 π 2 I , {\displaystyle E_{n}={\frac {n(n+1)h^{2}}{8\pi ^{2}I}},} each such level being (2n+1)-fold degenerate. By evaluating d n / d E n = 1 / ( d E n / d n ) {\displaystyle dn/dE_{n}=1/(dE_{n}/dn)} one obtains d n d E n = 8 π 2 I ( 2 n + 1 ) h 2 , ( 2 n + 1 ) d n = 8 π 2 I h 2 d E n . {\displaystyle {\frac {dn}{dE_{n}}}={\frac {8\pi ^{2}I}{(2n+1)h^{2}}},\;\;\;(2n+1)dn={\frac {8\pi ^{2}I}{h^{2}}}dE_{n}.} Thus by comparison with Ω {\displaystyle \Omega } above, one finds that

8881-449: The only reasonable choice. More formally, we can see that the problem remains the same if we swap around the labels ("A", "B" and "C") of the cups. It would therefore be odd to choose a prior for which a permutation of the labels would cause a change in our predictions about which cup the ball will be found under; the uniform prior is the only one which preserves this invariance. If one accepts this invariance principle then one can see that

8988-493: The other hand, if finding that life is commonplace while technosignatures are absent, then this would increase the likelihood that the Great Filter lies in the future. Recently, paleobiologist Olev Vinn has suggested that the great filter may exist between steps 8 and 9 due to inherited behavior patterns (IBP) that initially occur in all intelligent biological organisms. These IBPs are incompatible with conditions prevailing in technological civilizations and could inevitably lead to

9095-935: The prior distribution is proportional to the square root of the Fisher information of the likelihood function. Hence in the single parameter case, reference priors and Jeffreys priors are identical, even though Jeffreys has a very different rationale. Reference priors are often the objective prior of choice in multivariate problems, since other rules (e.g., Jeffreys' rule ) may result in priors with problematic behavior. Objective prior distributions may also be derived from other principles, such as information or coding theory (see e.g. minimum description length ) or frequentist statistics (so-called probability matching priors ). Such methods are used in Solomonoff's theory of inductive inference . Constructing objective priors have been recently introduced in bioinformatics, and specially inference in cancer systems biology, where sample size

9202-526: The prior distribution. A weakly informative prior expresses partial information about a variable, steering the analysis toward solutions that align with existing knowledge without overly constraining the results and preventing extreme estimates. An example is, when setting the prior distribution for the temperature at noon tomorrow in St. Louis, to use a normal distribution with mean 50 degrees Fahrenheit and standard deviation 40 degrees, which very loosely constrains

9309-423: The prior. This maximizes the expected posterior information about X when the prior density is p ( x ); thus, in some sense, p ( x ) is the "least informative" prior about X. The reference prior is defined in the asymptotic limit, i.e., one considers the limit of the priors so obtained as the number of data points goes to infinity. In the present case, the KL divergence between the prior and posterior distributions

9416-408: The probability in phase space, one has the probability density Σ := P Tr ( P ) , N = Tr ( P ) = c o n s t . , {\displaystyle \Sigma :={\frac {P}{{\text{Tr}}(P)}},\;\;\;N={\text{Tr}}(P)=\mathrm {const.} ,} where N {\displaystyle N} is the dimensionality of

9523-406: The process of starting with a star and ending with "advanced explosive lasting life" must be unlikely. This implies that at least one step in this process must be improbable. Hanson's list, while incomplete, describes the following nine steps in an "evolutionary path" that results in the colonization of the observable universe: According to the Great Filter hypothesis, at least one of these steps—if

9630-554: The purely subjective assessment of an experienced expert. When no information is available, an uninformative prior may be adopted as justified by the principle of indifference . In modern applications, priors are also often chosen for their mechanical properties, such as regularization and feature selection . The prior distributions of model parameters will often depend on parameters of their own. Uncertainty about these hyperparameters can, in turn, be expressed as hyperprior probability distributions. For example, if one uses

9737-423: The region between p , p + d p , p 2 = p 2 , {\displaystyle p,p+dp,p^{2}={\bf {p}}^{2},} is then found to be the above expression V 4 π p 2 d p / h 3 {\displaystyle V4\pi p^{2}dp/h^{3}} by considering the area covered by these points. Moreover, in view of

9844-401: The same results regardless of our choice of the origin of a coordinate system. This induces the group structure of the translation group on X , which determines the prior probability as a constant improper prior . Similarly, some measurements are naturally invariant to the choice of an arbitrary scale (e.g., whether centimeters or inches are used, the physical results should be equal). In such

9951-725: The second part is the integral over t {\displaystyle t} of the joint density p ( x , t ) {\displaystyle p(x,t)} . This is the marginal distribution p ( x ) {\displaystyle p(x)} , so we have K L = ∫ p ( t ) ∫ p ( x ∣ t ) log ⁡ [ p ( x ∣ t ) ] d x d t − ∫ p ( x ) log ⁡ [ p ( x ) ] d x {\displaystyle KL=\int p(t)\int p(x\mid t)\log[p(x\mid t)]\,dx\,dt\,-\,\int p(x)\log[p(x)]\,dx} Now we use

10058-498: The self-destruction of civilization in multiple ways. In a specific formulation named the " Berserker hypothesis ", a filter exists between steps 8 and 9 in which each civilization is destroyed by a lethal Von Neumann probe created by a more advanced civilization. There are many alternative scenarios that might allow for the evolution of intelligent life to occur multiple times without either catastrophic self-destruction or glaringly visible evidence. These are possible resolutions to

10165-406: The sense that it contains the least amount of information consistent with the constraints that define the set. For example, the maximum entropy prior on a discrete space, given only that the probability is normalized to 1, is the prior that assigns equal probability to each state. And in the continuous case, the maximum entropy prior given that the density is normalized with mean zero and unit variance

10272-502: The stars in their neighborhoods, for instance. They may prefer to build nanobots instead. It should also be kept in mind that, as Arthur C. Clarke said, truly advanced engineering would look like magic to us—or be unrecognizable altogether. By the way, we've only just begun to search for things like Dyson spheres , so we can't really rule them out. Robin Hanson Robin Dale Hanson (born August 28, 1959 )

10379-459: The subspace. The conservation law in this case is expressed by the unitarity of the S-matrix . In either case, the considerations assume a closed isolated system. This closed isolated system is a system with (1) a fixed energy E {\displaystyle E} and (2) a fixed number of particles N {\displaystyle N} in (c) a state of equilibrium. If one considers

10486-411: The temperature to the range (10 degrees, 90 degrees) with a small chance of being below -30 degrees or above 130 degrees. The purpose of a weakly informative prior is for regularization , that is, to keep inferences in a reasonable range. An uninformative , flat , or diffuse prior expresses vague or general information about a variable. The term "uninformative prior" is somewhat of a misnomer. Such

10593-416: The tiny number of intelligent species with advanced civilizations actually observed (currently just one: human ). This probability threshold, which could lie in the past or following human extinction , might work as a barrier to the evolution of intelligent life, or as a high probability of self-destruction. The main conclusion of this argument is that the easier it was for life to evolve to the present stage,

10700-414: The topic. Hanson also coined the term Great Filter , referring to whatever prevents "dead matter" from becoming an expanding and observable intelligent civilization. He was motivated to seek his doctorate so that his theories would reach a wider audience. Hanson has written two books. The Age of Em (2016) concerns his views on brain emulation and its eventual impact on society. The Elephant in

10807-642: The total volume of phase space covered for constant energy E is ∫ 0 ϕ = 2 π ∫ 0 θ = π 2 I π E sin ⁡ θ d θ d ϕ = 8 π 2 I E = ∮ d p θ d p ϕ d θ d ϕ , {\displaystyle \int _{0}^{\phi =2\pi }\int _{0}^{\theta =\pi }2I\pi E\sin \theta d\theta d\phi =8\pi ^{2}IE=\oint dp_{\theta }dp_{\phi }d\theta d\phi ,} and hence

10914-410: The ultraviolet to near-infrared wavelengths could place upper bounds on the fraction of planets in the galaxy that host life. Meanwhile, the evolution of telescopes that can detect technosignatures at mid-infrared wavelengths could provide insights into the Great Filter. They say that if planets with technosignatures are abundant, then this can increase confidence that the Great Filter is in the past. On

11021-419: The uniform prior is the logically correct prior to represent this state of knowledge. This prior is "objective" in the sense of being the correct choice to represent a particular state of knowledge, but it is not objective in the sense of being an observer-independent feature of the world: in reality the ball exists under a particular cup, and it only makes sense to speak of probabilities in this situation if there

11128-400: The use of an uninformative prior typically yields results which are not too different from conventional statistical analysis, as the likelihood function often yields more information than the uninformative prior. Some attempts have been made at finding a priori probabilities , i.e. probability distributions in some sense logically required by the nature of one's state of uncertainty; these are

11235-767: The usual priors (e.g., Jeffreys' prior) may give badly inadmissible decision rules if employed at the higher levels of the hierarchy. Let events A 1 , A 2 , … , A n {\displaystyle A_{1},A_{2},\ldots ,A_{n}} be mutually exclusive and exhaustive. If Bayes' theorem is written as P ( A i ∣ B ) = P ( B ∣ A i ) P ( A i ) ∑ j P ( B ∣ A j ) P ( A j ) , {\displaystyle P(A_{i}\mid B)={\frac {P(B\mid A_{i})P(A_{i})}{\sum _{j}P(B\mid A_{j})P(A_{j})}}\,,} then it

11342-475: Was a commodity. Hanson has been criticized for his writings relating to sexual relationships and women. "If you’ve ever heard of George Mason University economist Robin Hanson, there’s a good chance it was because he wrote something creepy", Slate columnist Jordan Weissman wrote in 2018. In an article on bias against women in economics, Bloomberg columnist Noah Smith cited a blog post by Hanson comparing cuckoldry to "gentle silent rape", lamenting that there

11449-526: Was no retraction and no outcry from fellow economists. In The New Yorker , Jia Tolentino described Hanson's blog post as a "flippantly dehumanizing thought experiment". A 2003 article in Fortune examined Hanson's work, noting, among other things, that he is a proponent of cryonics and that his ideas have found some acceptance among extropians on the Internet . He has since written extensively on

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