AlphaGo Zero is a version of DeepMind's Go software AlphaGo. AlphaGo's team published an article in Nature in October 2017 introducing AlphaGo Zero, a version created without using data from human games, and stronger than any previous version. By playing games against itself, AlphaGo Zero: surpassed the strength of AlphaGo Lee in three days by winning 100 games to 0; reached the level of AlphaGo Master in 21 days; and exceeded all previous versions in 40 days. Training artificial intelligence (AI) without datasets derived from human experts has significant implications for the development of AI with superhuman skills, as expert data is "often expensive, unreliable, or simply unavailable." Demis Hassabis, the co-founder and CEO of DeepMind, said that AlphaGo Zero was so powerful because it was "no longer constrained by the limits of human knowledge". Furthermore, AlphaGo Zero performed better than standard deep reinforcement learning models (such as Deep Q-Network implementations) due to its integration of Monte Carlo tree search. David Silver, one of the first authors of DeepMind's papers published in Nature on AlphaGo, said that it is possible to have generalized AI algorithms by removing the need to learn from humans. Google later developed AlphaZero, a generalized version of AlphaGo Zero that could play chess and shōgi in addition to Go. In December 2017, AlphaZero beat the 3-day version of AlphaGo Zero by winning 60 games to 40, and with 8 hours of training it outperformed AlphaGo Lee on an Elo scale. AlphaZero also defeated a top chess program (Stockfish) and a top Shōgi program (Elmo). == Architecture == The network in AlphaGo Zero is a ResNet with two heads. The stem of the network takes as input a 17x19x19 tensor representation of the Go board. 8 channels are the positions of the current player's stones from the last eight time steps. (1 if there is a stone, 0 otherwise. If the time step go before the beginning of the game, then 0 in all positions.) 8 channels are the positions of the other player's stones from the last eight time steps. 1 channel is all 1 if black is to move, and 0 otherwise. The body is a ResNet with either 20 or 40 residual blocks and 256 channels. There are two heads, a policy head and a value head. Policy head outputs a logit array of size 19 × 19 + 1 {\displaystyle 19\times 19+1} , representing the logit of making a move in one of the points, plus the logit of passing. Value head outputs a number in the range ( − 1 , + 1 ) {\displaystyle (-1,+1)} , representing the expected score for the current player. -1 represents current player losing, and +1 winning. == Training == AlphaGo Zero's neural network was trained using TensorFlow, with 64 GPU workers and 19 CPU parameter servers. Only four TPUs were used for inference. The neural network initially knew nothing about Go beyond the rules. Unlike earlier versions of AlphaGo, Zero only perceived the board's stones, rather than having some rare human-programmed edge cases to help recognize unusual Go board positions. The AI engaged in reinforcement learning, playing against itself until it could anticipate its own moves and how those moves would affect the game's outcome. In the first three days AlphaGo Zero played 4.9 million games against itself in quick succession. It appeared to develop the skills required to beat top humans within just a few days, whereas the earlier AlphaGo took months of training to achieve the same level. According to Epoch.ai, training cost 3e23 FLOPs. For comparison, the researchers also trained a version of AlphaGo Zero using human games, AlphaGo Master, and found that it learned more quickly, but actually performed more poorly in the long run. DeepMind submitted its initial findings in a paper to Nature in April 2017, which was then published in October 2017. == Hardware cost == The hardware cost for a single AlphaGo Zero system in 2017, including the four TPUs, has been quoted as around $25 million. == Applications == According to Hassabis, AlphaGo's algorithms are likely to be of the most benefit to domains that require an intelligent search through an enormous space of possibilities, such as protein folding (see AlphaFold) or accurately simulating chemical reactions. AlphaGo's techniques are probably less useful in domains that are difficult to simulate, such as learning how to drive a car. DeepMind stated in October 2017 that it had already started active work on attempting to use AlphaGo Zero technology for protein folding, and stated it would soon publish new findings. == Reception == AlphaGo Zero was widely regarded as a significant advance, even when compared with its groundbreaking predecessor, AlphaGo. Oren Etzioni of the Allen Institute for Artificial Intelligence called AlphaGo Zero "a very impressive technical result" in "both their ability to do it—and their ability to train the system in 40 days, on four TPUs". The Guardian called it a "major breakthrough for artificial intelligence", citing Eleni Vasilaki of Sheffield University and Tom Mitchell of Carnegie Mellon University, who called it an impressive feat and an “outstanding engineering accomplishment" respectively. Mark Pesce of the University of Sydney called AlphaGo Zero "a big technological advance" taking us into "undiscovered territory". Gary Marcus, a psychologist at New York University, has cautioned that for all we know, AlphaGo may contain "implicit knowledge that the programmers have about how to construct machines to play problems like Go" and will need to be tested in other domains before being sure that its base architecture is effective at much more than playing Go. In contrast, DeepMind is "confident that this approach is generalisable to a large number of domains". In response to the reports, South Korean Go professional Lee Sedol said, "The previous version of AlphaGo wasn’t perfect, and I believe that’s why AlphaGo Zero was made." On the potential for AlphaGo's development, Lee said he will have to wait and see but also said it will affect young Go players. Mok Jin-seok, who directs the South Korean national Go team, said the Go world has already been imitating the playing styles of previous versions of AlphaGo and creating new ideas from them, and he is hopeful that new ideas will come out from AlphaGo Zero. Mok also added that general trends in the Go world are now being influenced by AlphaGo's playing style. "At first, it was hard to understand and I almost felt like I was playing against an alien. However, having had a great amount of experience, I’ve become used to it," Mok said. "We are now past the point where we debate the gap between the capability of AlphaGo and humans. It’s now between computers." Mok has reportedly already begun analyzing the playing style of AlphaGo Zero along with players from the national team. "Though having watched only a few matches, we received the impression that AlphaGo Zero plays more like a human than its predecessors," Mok said. Chinese Go professional Ke Jie commented on the remarkable accomplishments of the new program: "A pure self-learning AlphaGo is the strongest. Humans seem redundant in front of its self-improvement." == Comparison with predecessors == == AlphaZero == On 5 December 2017, DeepMind team released a preprint on arXiv, introducing AlphaZero, a program using generalized AlphaGo Zero's approach, which achieved within 24 hours a superhuman level of play in chess, shogi, and Go, defeating world-champion programs, Stockfish, Elmo, and 3-day version of AlphaGo Zero in each case. AlphaZero (AZ) is a more generalized variant of the AlphaGo Zero (AGZ) algorithm, and is able to play shogi and chess as well as Go. Differences between AZ and AGZ include: AZ has hard-coded rules for setting search hyperparameters. The neural network is now updated continually. Chess (unlike Go) can end in a tie; therefore AZ can take into account the possibility of a tie game. An open source program, Leela Zero, based on the ideas from the AlphaGo papers is available. It uses a GPU instead of the TPUs recent versions of AlphaGo rely on.
Order-independent transparency
Order-independent transparency (OIT) is a class of techniques in rasterisational computer graphics for rendering transparency in a 3D scene, which do not require rendering geometry in sorted order for alpha compositing. == Description == Commonly, 3D geometry with transparency is rendered by blending (using alpha compositing) all surfaces into a single buffer (think of this as a canvas). Each surface occludes existing color and adds some of its own color depending on its alpha value, a ratio of light transmittance. The order in which surfaces are blended affects the total occlusion or visibility of each surface. For a correct result, surfaces must be blended from farthest to nearest or nearest to farthest, depending on the alpha compositing operation, over or under. Ordering may be achieved by rendering the geometry in sorted order, for example sorting triangles by depth, but can take a significant amount of time, not always produce a solution (in the case of intersecting or circularly overlapping geometry) and the implementation is complex. Instead, order-independent transparency sorts geometry per-pixel, after rasterisation. For exact results this requires storing all fragments before sorting and compositing. == History == The A-buffer is a computer graphics technique introduced in 1984 which stores per-pixel lists of fragment data (including micro-polygon information) in a software rasteriser, REYES, originally designed for anti-aliasing but also supporting transparency. More recently, depth peeling in 2001 described a hardware accelerated OIT technique. With limitations in graphics hardware the scene's geometry had to be rendered many times. A number of techniques have followed, to improve on the performance of depth peeling, still with the many-pass rendering limitation. For example, Dual Depth Peeling (2008). In 2009, two significant features were introduced in GPU hardware/drivers/Graphics APIs that allowed capturing and storing fragment data in a single rendering pass of the scene, something not previously possible. These are, the ability to write to arbitrary GPU memory from shaders and atomic operations. With these features a new class of OIT techniques became possible that do not require many rendering passes of the scene's geometry. The first was storing the fragment data in a 3D array, where fragments are stored along the z dimension for each pixel x/y. In practice, most of the 3D array is unused or overflows, as a scene's depth complexity is typically uneven. To avoid overflow the 3D array requires large amounts of memory, which in many cases is impractical. Two approaches to reducing this memory overhead exist. Packing the 3D array with a prefix sum scan, or linearizing, removed the unused memory issue but requires an additional depth complexity computation rendering pass of the geometry. The "Sparsity-aware" S-Buffer, Dynamic Fragment Buffer, "deque" D-Buffer, Linearized Layered Fragment Buffer all pack fragment data with a prefix sum scan and are demonstrated with OIT. Storing fragments in per-pixel linked lists provides tight packing of this data and in late 2011, driver improvements reduced the atomic operation contention overhead making the technique very competitive. == Exact OIT == Exact, as opposed to approximate, OIT accurately computes the final color, for which all fragments must be sorted. For high depth complexity scenes, sorting becomes the bottleneck. One issue with the sorting stage is local memory limited occupancy, in this case a SIMT attribute relating to the throughput and operation latency hiding of GPUs. Backwards memory allocation (BMA) groups pixels by their depth complexity and sorts them in batches to improve the occupancy and hence performance of low depth complexity pixels in the context of a potentially high depth complexity scene. Up to a 3× overall OIT performance increase is reported. Sorting is typically performed in a local array, however performance can be improved further by making use of the GPU's memory hierarchy and sorting in registers, similarly to an external merge sort, especially in conjunction with BMA. == Approximate OIT == Approximate OIT techniques relax the constraint of exact rendering to provide faster results. Higher performance can be gained from not having to store all fragments or only partially sorting the geometry. A number of techniques also compress, or reduce, the fragment data. These include: Stochastic Transparency: draw in a higher resolution in full opacity but discard some fragments. Downsampling will then yield transparency. Adaptive Transparency, a two-pass technique where the first constructs a visibility function which compresses on the fly (this compression avoids having to fully sort the fragments) and the second uses this data to composite unordered fragments. Intel's pixel synchronization avoids the need to store all fragments, removing the unbounded memory requirement of many other OIT techniques. Weighted Blended Order-Independent Transparency replaced the over operator with a commutative approximation. Feeding depth information into the weight produces visually-acceptable occlusion. == OIT in Hardware == The Sega Dreamcast games console included hardware support for automatic OIT.
Markov chain central limit theorem
In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity. == Statement == Suppose that: the sequence X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } of random elements of some set is a Markov chain that has a stationary probability distribution; and the initial distribution of the process, i.e. the distribution of X 1 {\textstyle X_{1}} , is the stationary distribution, so that X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and g {\textstyle g} is some (measurable) real-valued function for which var ( g ( X 1 ) ) < + ∞ . {\textstyle \operatorname {var} (g(X_{1}))<+\infty .} Now let μ = E ( g ( X 1 ) ) , μ ^ n = 1 n ∑ k = 1 n g ( X k ) σ 2 := lim n → ∞ var ( n μ ^ n ) = lim n → ∞ n var ( μ ^ n ) = var ( g ( X 1 ) ) + 2 ∑ k = 1 ∞ cov ( g ( X 1 ) , g ( X 1 + k ) ) . {\displaystyle {\begin{aligned}\mu &=\operatorname {E} (g(X_{1})),\\{\widehat {\mu }}_{n}&={\frac {1}{n}}\sum _{k=1}^{n}g(X_{k})\\\sigma ^{2}&:=\lim _{n\to \infty }\operatorname {var} ({\sqrt {n}}{\widehat {\mu }}_{n})=\lim _{n\to \infty }n\operatorname {var} ({\widehat {\mu }}_{n})=\operatorname {var} (g(X_{1}))+2\sum _{k=1}^{\infty }\operatorname {cov} (g(X_{1}),g(X_{1+k})).\end{aligned}}} Then as n → ∞ , {\textstyle n\to \infty ,} we have n ( μ ^ n − μ ) → D Normal ( 0 , σ 2 ) , {\displaystyle {\sqrt {n}}({\hat {\mu }}_{n}-\mu )\ {\xrightarrow {\mathcal {D}}}\ {\text{Normal}}(0,\sigma ^{2}),} where the decorated arrow indicates convergence in distribution. == Monte Carlo Setting == The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following: Consider a simple hard spheres model on a grid. Suppose X = { 1 , … , n 1 } × { 1 , … , n 2 } ⊆ Z 2 {\displaystyle X=\{1,\ldots ,n_{1}\}\times \{1,\ldots ,n_{2}\}\subseteq Z^{2}} . A proper configuration on X {\displaystyle X} consists of coloring each point either black or white in such a way that no two adjacent points are white. Let χ {\displaystyle \chi } denote the set of all proper configurations on X {\displaystyle X} , N χ ( n 1 , n 2 ) {\displaystyle N_{\chi }(n_{1},n_{2})} be the total number of proper configurations and π be the uniform distribution on χ {\displaystyle \chi } so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if W ( x ) {\displaystyle W(x)} is the number of white points in x ∈ χ {\displaystyle x\in \chi } then we want the value of E π W = ∑ x ∈ χ W ( x ) N χ ( n 1 , n 2 ) {\displaystyle E_{\pi }W=\sum _{x\in \chi }{\frac {W(x)}{N_{\chi }{\bigl (}n_{1},n_{2}{\bigr )}}}} If n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are even moderately large then we will have to resort to an approximation to E π W {\displaystyle E_{\pi }W} . Consider the following Markov chain on χ {\displaystyle \chi } . Fix p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and set X 1 = x 1 {\displaystyle X_{1}=x_{1}} where x 1 ∈ χ {\displaystyle x_{1}\in \chi } is an arbitrary proper configuration. Randomly choose a point ( x , y ) ∈ X {\displaystyle (x,y)\in X} and independently draw U ∼ U n i f o r m ( 0 , 1 ) {\displaystyle U\sim \mathrm {Uniform} (0,1)} . If u ≤ p {\displaystyle u\leq p} and all of the adjacent points are black then color ( x , y ) {\displaystyle (x,y)} white leaving all other points alone. Otherwise, color ( x , y ) {\displaystyle (x,y)} black and leave all other points alone. Call the resulting configuration X 1 {\displaystyle X_{1}} . Continuing in this fashion yields a Harris ergodic Markov chain { X 1 , X 2 , X 3 , … } {\displaystyle \{X_{1},X_{2},X_{3},\ldots \}} having π {\displaystyle \pi } as its invariant distribution. It is now a simple matter to estimate E π W {\displaystyle E_{\pi }W} with w n ¯ = ∑ i = 1 n W ( X i ) / n {\displaystyle {\overline {w_{n}}}=\sum _{i=1}^{n}W(X_{i})/n} . Also, since χ {\displaystyle \chi } is finite (albeit potentially large) it is well known that X {\displaystyle X} will converge exponentially fast to π {\displaystyle \pi } which implies that a CLT holds for w n ¯ {\displaystyle {\overline {w_{n}}}} . == Implications == Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.
Adam Tauman Kalai
Adam Tauman Kalai is an American computer scientist who specializes in artificial intelligence and works at OpenAI. == Education and career == Kalai graduated from Harvard University in 1996 with a BA in computer science and received a MA and PhD, both in computer science, from Carnegie Mellon University in 1999 and 2001, respectively. His doctoral advisor was Avrim Blum. After graduation, Kalai did his postdoctoral research at Massachusetts Institute of Technology under Santosh Vempala until 2003. Kalai became a faculty member at the Toyota Technological Institute at Chicago from 2003 to 2006, followed by a stint as an assistant professor at Georgia Institute of Technology from 2007 to 2008. He joined Microsoft Research in 2008 and subsequently moved to OpenAI in 2023. == Contributions == Kalai is known for his algorithm for generating random factored numbers (see Bach's algorithm), for co-inventing the cooperative-competitive value (coco value), for efficiently learning learning mixtures of Gaussians, for the Blum-Kalai-Wasserman algorithm for learning parity with noise, and for the intractability of the folk theorem in game theory. More recently, Kalai is known for identifying and reducing gender bias in word embeddings, which are a representation of words commonly used in AI systems. In 2026, he coauthored a Nature paper on hallucinations in large language models. == Personal life == Kalai is the son of game theorist Ehud Kalai and is married to cryptographer Yael Tauman Kalai.
Baum–Welch algorithm
In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step. The Baum–Welch algorithm, the primary method for inference in hidden Markov models, is numerically unstable due to its recursive calculation of joint probabilities. As the number of variables grows, these joint probabilities become increasingly small, leading to the forward recursions rapidly approaching values below machine precision. == History == The Baum–Welch algorithm was named after its inventors Leonard E. Baum and Lloyd R. Welch. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the IDA Center for Communications Research, Princeton in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences. == Description == A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables. It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state. The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors. Let X t {\displaystyle X_{t}} be a discrete hidden random variable with N {\displaystyle N} possible values (i.e. We assume there are N {\displaystyle N} states in total). We assume the P ( X t ∣ X t − 1 ) {\displaystyle P(X_{t}\mid X_{t-1})} is independent of time t {\displaystyle t} , which leads to the definition of the time-independent stochastic transition matrix A = { a i j } = P ( X t = j ∣ X t − 1 = i ) . {\displaystyle A=\{a_{ij}\}=P(X_{t}=j\mid X_{t-1}=i).} The initial state distribution (i.e. when t = 1 {\displaystyle t=1} ) is given by π i = P ( X 1 = i ) . {\displaystyle \pi _{i}=P(X_{1}=i).} The observation variables Y t {\displaystyle Y_{t}} can take one of K {\displaystyle K} possible values. We also assume the observation given the "hidden" state is time independent. The probability of a certain observation y i {\displaystyle y_{i}} at time t {\displaystyle t} for state X t = j {\displaystyle X_{t}=j} is given by b j ( y i ) = P ( Y t = y i ∣ X t = j ) . {\displaystyle b_{j}(y_{i})=P(Y_{t}=y_{i}\mid X_{t}=j).} Taking into account all the possible values of Y t {\displaystyle Y_{t}} and X t {\displaystyle X_{t}} , we obtain the N × K {\displaystyle N\times K} matrix B = { b j ( y i ) } {\displaystyle B=\{b_{j}(y_{i})\}} where b j {\displaystyle b_{j}} belongs to all the possible states and y i {\displaystyle y_{i}} belongs to all the observations. An observation sequence is given by Y = ( Y 1 = y 1 , Y 2 = y 2 , … , Y T = y T ) {\displaystyle Y=(Y_{1}=y_{1},Y_{2}=y_{2},\ldots ,Y_{T}=y_{T})} . Thus we can describe a hidden Markov chain by θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} . The Baum–Welch algorithm finds a local maximum for θ ∗ = a r g m a x θ P ( Y ∣ θ ) {\displaystyle \theta ^{}=\operatorname {arg\,max} _{\theta }P(Y\mid \theta )} (i.e. the HMM parameters θ {\displaystyle \theta } that maximize the probability of the observation). === Algorithm === Set θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} with random initial conditions. They can also be set using prior information about the parameters if it is available; this can speed up the algorithm and also steer it toward the desired local maximum. ==== Forward procedure ==== Let α i ( t ) = P ( Y 1 = y 1 , … , Y t = y t , X t = i ∣ θ ) {\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )} , the probability of seeing the observations y 1 , y 2 , … , y t {\displaystyle y_{1},y_{2},\ldots ,y_{t}} and being in state i {\displaystyle i} at time t {\displaystyle t} . This is found recursively: α i ( 1 ) = π i b i ( y 1 ) , {\displaystyle \alpha _{i}(1)=\pi _{i}b_{i}(y_{1}),} α i ( t + 1 ) = b i ( y t + 1 ) ∑ j = 1 N α j ( t ) a j i . {\displaystyle \alpha _{i}(t+1)=b_{i}(y_{t+1})\sum _{j=1}^{N}\alpha _{j}(t)a_{ji}.} Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences. However, this can be avoided in a slightly modified algorithm by scaling α {\displaystyle \alpha } in the forward and β {\displaystyle \beta } in the backward procedure below. ==== Backward procedure ==== Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t {\displaystyle t} . We calculate β i ( t ) {\displaystyle \beta _{i}(t)} as, β i ( T ) = 1 , {\displaystyle \beta _{i}(T)=1,} β i ( t ) = ∑ j = 1 N β j ( t + 1 ) a i j b j ( y t + 1 ) . {\displaystyle \beta _{i}(t)=\sum _{j=1}^{N}\beta _{j}(t+1)a_{ij}b_{j}(y_{t+1}).} ==== Update ==== We can now calculate the temporary variables, according to Bayes' theorem: γ i ( t ) = P ( X t = i ∣ Y , θ ) = P ( X t = i , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) β i ( t ) ∑ j = 1 N α j ( t ) β j ( t ) , {\displaystyle \gamma _{i}(t)=P(X_{t}=i\mid Y,\theta )={\frac {P(X_{t}=i,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)\beta _{i}(t)}{\sum _{j=1}^{N}\alpha _{j}(t)\beta _{j}(t)}},} which is the probability of being in state i {\displaystyle i} at time t {\displaystyle t} given the observed sequence Y {\displaystyle Y} and the parameters θ {\displaystyle \theta } ξ i j ( t ) = P ( X t = i , X t + 1 = j ∣ Y , θ ) = P ( X t = i , X t + 1 = j , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) ∑ k = 1 N ∑ w = 1 N α k ( t ) a k w β w ( t + 1 ) b w ( y t + 1 ) , {\displaystyle \xi _{ij}(t)=P(X_{t}=i,X_{t+1}=j\mid Y,\theta )={\frac {P(X_{t}=i,X_{t+1}=j,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)a_{ij}\beta _{j}(t+1)b_{j}(y_{t+1})}{\sum _{k=1}^{N}\sum _{w=1}^{N}\alpha _{k}(t)a_{kw}\beta _{w}(t+1)b_{w}(y_{t+1})}},} which is the probability of being in state i {\displaystyle i} and j {\displaystyle j} at times t {\displaystyle t} and t + 1 {\displaystyle t+1} respectively given the observed sequence Y {\displaystyle Y} and parameters θ {\displaystyle \theta } . The denominators of γ i ( t ) {\displaystyle \gamma _{i}(t)} and ξ i j ( t ) {\displaystyle \xi _{ij}(t)} are the same ; they represent the probability of making the observation Y {\displaystyle Y} given the parameters θ {\displaystyle \theta } . The parameters of the hidden Markov model θ {\displaystyle \theta } can now be updated: π i ∗ = γ i ( 1 ) , {\displaystyle \pi _{i}^{}=\gamma _{i}(1),} which is the expected frequency spent in state i {\displaystyle i} at time 1 {\displaystyle 1} . a i j ∗ = ∑ t = 1 T − 1 ξ i j ( t ) ∑ t = 1 T − 1 γ i ( t ) , {\displaystyle a_{ij}^{}={\frac {\sum _{t=1}^{T-1}\xi _{ij}(t)}{\sum _{t=1}^{T-1}\gamma _{i}(t)}},} which is the expected number of transitions from state i to state j compared to the expected total number of transitions starting in state i, including from state i to itself. The number of transitions starting in state i is equivalent to the number of times state i is observed in the sequence from t = 1 to t = T − 1. b i ∗ ( v k ) = ∑ t = 1 T 1 y t = v k γ i ( t ) ∑ t = 1 T γ i ( t ) , {\displaystyle b_{i}^{}(v_{k})={\frac {\sum _{t=1}^{T}1_{y_{t}=v_{k}}\gamma _{i}(t)}{\sum _{t=1}^{T}\gamma _{i}(t)}},} where 1 y t = v k = { 1 if y t = v k , 0 otherwise {\displaystyle 1_{y_{t}=v_{k}}={\begin{cases}1&{\text{if }}y_{t}=v_{k},\\0&{\text{otherwise}}\end{cases}}} is an indicator function, and b i ∗ ( v k ) {\displaystyle b_{i}^{}(v_{k})} is the expected number of times the output observations have been equal to v k {\displaystyle v_{k}} while in state i {\displaystyle i} over the expected total number of times in state i {\displaystyle i} . These steps are now repeated iteratively until a desired level of convergence. Note: It is possible to over-fit a particular data set. That is, P ( Y ∣ θ final ) > P ( Y ∣ θ true ) {\displaystyle P(Y\mid \theta _{\text{final}})>P(Y\mid \theta _{\text{true}})} . The algorithm also does not guarantee a global maximum. ==== Multiple sequences ==== The algorithm described thus far assumes a single observed sequence Y = y 1 , … , y T {\displaystyle Y=y_{1},\ldots ,y_{T}} . However, in many situations, there are several sequences observed: Y 1 ,
Universal psychometrics
Universal psychometrics encompasses psychometrics instruments that could measure the psychological properties of any intelligent agent. Up until the early 21st century, psychometrics relied heavily on psychological tests that require the subject to cooperate and answer questions, the most famous example being an intelligence test. Such methods are only applicable to the measurement of human psychological properties. As a result, some researchers have proposed the idea of universal psychometrics - they suggest developing testing methods that allow for the measurement of non-human entities' psychological properties. For example, it has been suggested that the Turing test is a form of universal psychometrics. This test involves having testers (without any foreknowledge) attempt to distinguish a human from a machine by interacting with both (while not being to see either individuals). It is supposed that if the machine is equally intelligent to a human, the testers will not be able to distinguish between the two, i.e., their guesses will not be better than chance. Thus, Turing test could measure the intelligence (a psychological variable) of an AI. Other instruments proposed for universal psychometrics include reinforcement learning and measuring the ability to predict complexity.
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