Allegorical interpretations of Genesis

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https://en.wikipedia.org/wiki/Allegorical_interpretations_of_Genesis 

Allegorical interpretations of Genesis are readings of the biblical Book of Genesis that treat elements of the narrative as symbols or types, rather than viewing them literally as recording historical events. Either way, Judaism and most sects of Christianity treat Genesis as canonical scripture, and believers generally regard it as having spiritual significance.

The opening chapter of Genesis tells a story of God's creation of the universe and of humankind as taking place over the course of six successive days. Some Christian and Jewish schools of thought (such as Christian fundamentalism) read these biblical passages literally, assuming each day of creation as 24 hours in duration. Others (such as Roman Catholic, Eastern Orthodox, and mainline Protestant denominations) read the story allegorically, and hold that the biblical account aims to describe humankind's relationship to creation and the creator, that Genesis 1 does not describe actual historical events, and that the six days of creation simply represents a long period of time.

Genesis 2 records a second account of creation. Chapter 3 introduces a talking serpent, which many Christians believe is Satan in disguise. Many Christians in ancient times regarded the early chapters of Genesis as true both as history and as allegory.

Other Jews and Christians have long regarded the creation account of Genesis as an allegory - even prior to the development of modern science and the scientific accounts (based on the scientific method) of cosmological, biological and human origins. Notable proponents of allegorical interpretation include the Christian theologians Origen, who wrote in the 2nd century that it was inconceivable to consider Genesis literal history, Augustine of Hippo, who in the 4th century, on theological grounds, argued that God created everything in the universe in the same instant, and not in six days as a plain reading of Genesis would require; and the even earlier 1st-century Jewish scholar Philo of Alexandria, who wrote that it would be a mistake to think that creation happened in six days or in any determinate amount of time.

Interpretation

Church historians on allegorical interpretation of Genesis

The literalist reading of some contemporary Christians maligns the allegorical or mythical interpretation of Genesis as a belated attempt to reconcile science with the biblical account. They maintain that the story of origins had always been interpreted literally until modern science (and, specifically, biological evolution) arose and challenged it. This view is not the consensus view, however, as demonstrated below:

According to Rowan Williams: "[For] most of the history of Christianity there's been an awareness that a belief that everything depends on the creative act of God, is quite compatible with a degree of uncertainty or latitude about how precisely that unfolds in creative time."

Some religious historians consider that biblical literalism came about with the rise of Protestantism; before the Reformation, the Bible was not usually interpreted in a completely literal way. Fr. Stanley Jaki, a Benedictine priest and theologian who is also a distinguished physicist, states in his Bible and Science:

Adam and Eve, by Albrecht Dürer (1507).

Insofar as the study of the original languages of the Bible was severed from authoritative ecclesiastical preaching as its matrix, it fueled literalism... Biblical literalism taken for a source of scientific information is making the rounds even nowadays among creationists who would merit Julian Huxley's description of 'bibliolaters.' They merely bring discredit to the Bible as they pile grist upon grist on the mills of latter-day Huxleys, such as Hoyle, Sagan, Gould, and others. The fallacies of creationism go deeper than fallacious reasonings about scientific data. Where creationism is fundamentally at fault is its resting its case on a theological faultline: the biblicism constructed by the [Protestant] Reformers.

However, the Russian Orthodox hieromonk Fr. Seraphim Rose has argued that leading Orthodox saints such as Basil the Great, Gregory the Theologian, John Chrysostom and Ephraim the Syrian believed that Genesis should be treated as a historical account.

Ancient Christian interpretations

Finding allegory in history

Maxine Clarke Beach comments Paul's assertion in Galatians 4:21–31 that the Genesis story of Abraham's sons is an allegory, writing that "This allegorical interpretation has been one of the biblical texts used in the long history of Christian anti-Semitism, which its author could not have imagined or intended".

Other New Testament writers took a similar approach to the Jewish Bible. The Gospel of Matthew reinterprets a number of passages. Where the prophet Hosea has God say of Israel, "Out of Egypt I called my son," (Hosea 11:1), Matthew interprets the phrase as a reference to Jesus. Likewise, Isaiah's promise of a child as a sign to King Ahaz (Isaiah 7:14) is understood by Matthew to refer to Jesus.

Later Christians followed their example. Irenaeus of Lyons, in his work Against Heresies from the middle of the 2nd century, saw the story of Adam, Eve and the serpent pointing to the death of Jesus:

Now in this same day that they did eat, in that also did they die. But according to the cycle and progress of the days, after which one is termed first, another second, and another third, if anybody seeks diligently to learn upon what day out of the seven it was that Adam died, he will find it by examining the dispensation of the Lord. For by summing up in Himself the whole human race from the beginning to the end, He has also summed up its death. From this it is clear that the Lord suffered death, in obedience to His Father, upon that day on which Adam died while he disobeyed God. Now he died on the same day in which he did eat. For God said, 'In that day on which ye shall eat of it, ye shall die by death.' The Lord, therefore, recapitulating in Himself this day, underwent His sufferings upon the day preceding the Sabbath, that is, the sixth day of the creation, on which day man was created; thus granting him a second creation by means of His passion, which is that [creation] out of death.

In the 3rd century, Origen and others of the Alexandrian school claimed that the Bible's true meaning could be found only by reading it allegorically. Origen explained in De Principiis that sometimes spiritual teachings could be gleaned from historical events, and sometimes the lessons could only be taught through stories that, taken literally, would "seem incapable of containing truth."

Days of creation

See also: Genesis creation narrative

Early Christians seem to have been divided over whether to interpret the days of creation in Genesis 1 as literal days, or to understand them allegorically.

For example, St. Basil rejected an allegorical interpretation in his Hexaëmeron, without commenting on the literalism of the days:

I know the laws of allegory, though less by myself than from the works of others. There are those truly, who do not admit the common sense of the Scriptures, for whom water is not water, but some other nature, who see in a plant, in a fish, what their fancy wishes, who change the nature of reptiles and of wild beasts to suit their allegories, like the interpreters of dreams who explain visions in sleep to make them serve their own ends. For me grass is grass; plant, fish, wild beast, domestic animal, I take all in the literal sense. 'For I am not ashamed of the Gospel' [Romans 1:16].

'And there was evening and there was morning: one day.' And the evening and the morning were one day. Why does Scripture say 'one day the first day'? Before speaking to us of the second, the third, and the fourth days, would it not have been more natural to call that one the first which began the series? If it therefore says 'one day,' it is from a wish to determine the measure of day and night, and to combine the time that they contain. Now twenty-four hours fill up the space of one day -- we mean of a day and of a night; and if, at the time of the solstices, they have not both an equal length, the time marked by Scripture does not the less circumscribe their duration. It is as though it said: twenty-four hours measure the space of a day, or that, in reality a day is the time that the heavens starting from one point take to return there. Thus, every time that, in the revolution of the sun, evening and morning occupy the world, their periodical succession never exceeds the space of one day.

Origen of Alexandria, in a passage that was later chosen by Gregory of Nazianzus for inclusion in the Philocalia, an anthology of some of his most important texts, made the following remarks:

For who that has understanding will suppose that the first, and second, and third day, and the evening and the morning, existed without a sun, and moon, and stars? And that the first day was, as it were, also without a sky? And who is so foolish as to suppose that God, after the manner of a husbandman, planted a paradise in Eden, towards the east, and placed in it a tree of life, visible and palpable, so that one tasting of the fruit by the bodily teeth obtained life? And again, that one was a partaker of good and evil by masticating what was taken from the tree? And if God is said to walk in the paradise in the evening, and Adam to hide himself under a tree, I do not suppose that anyone doubts that these things figuratively indicate certain mysteries, the history having taken place in appearance, and not literally.

In Contra Celsum, an apologetic work written in response to the pagan intellectual Celsus, Origen also said:

And with regard to the creation of the light upon the first day, and of the firmament upon the second, and of the gathering together of the waters that are under the heaven into their several reservoirs on the third (the earth thus causing to sprout forth those (fruits) which are under the control of nature alone), and of the (great) lights and stars upon the fourth, and of aquatic animals upon the fifth, and of land animals and man upon the sixth, we have treated to the best of our ability in our notes upon Genesis, as well as in the foregoing pages, when we found fault with those who, taking the words in their apparent signification, said that the time of six days was occupied in the creation of the world.

Saint Augustine, one of the most influential theologians of the Catholic Church, suggested that the Biblical text should not be interpreted literally if it contradicts what we know from science and our God-given reason. From an important passage on his The Literal Interpretation of Genesis (early fifth century, AD), St. Augustine wrote:

St. Augustine of Hippo

It not infrequently happens that something about the earth, about the sky, about other elements of this world, about the motion and rotation or even the magnitude and distances of the stars, about definite eclipses of the sun and moon, about the passage of years and seasons, about the nature of animals, of fruits, of stones, and of other such things, may be known with the greatest certainty by reasoning or by experience, even by one who is not a Christian. It is too disgraceful and ruinous, though, and greatly to be avoided, that he [the non-Christian] should hear a Christian speaking so idiotically on these matters, and as if in accord with Christian writings, that he might say that he could scarcely keep from laughing when he saw how totally in error they are. In view of this and in keeping it in mind constantly while dealing with the book of Genesis, I have, insofar as I was able, explained in detail and set forth for consideration the meanings of obscure passages, taking care not to affirm rashly some one meaning to the prejudice of another and perhaps better explanation.

With the scriptures it is a matter of treating about the faith. For that reason, as I have noted repeatedly, if anyone, not understanding the mode of divine eloquence, should find something about these matters [about the physical universe] in our books, or hear of the same from those books, of such a kind that it seems to be at variance with the perceptions of his own rational faculties, let him believe that these other things are in no way necessary to the admonitions or accounts or predictions of the scriptures. In short, it must be said that our authors knew the truth about the nature of the skies, but it was not the intention of the Spirit of God, who spoke through them, to teach men anything that would not be of use to them for their salvation.

In the book, Augustine took the view that everything in the universe was created simultaneously by God, and not in seven days like a plain account of Genesis would require. He argues that the six-day structure of creation presented in the book of Genesis represents a logical framework, rather than the passage of time in a physical way. Augustine also does not envisage original sin as originating structural changes in the universe, and even suggests that the bodies of Adam and Eve were already created mortal before the Fall. Apart from his specific views, Augustine recognizes that the interpretation of the creation story is difficult, and remarks that we should be willing to change our mind about it as new information comes up.

In The City of God, Augustine rejected both the immortality of the human race proposed by pagans, and contemporary ideas of ages (such as those of certain Greeks and Egyptians) that differed from the Church's sacred writings:

Let us, then, omit the conjectures of men who know not what they say, when they speak of the nature and origin of the human race. For some hold the same opinion regarding men that they hold regarding the world itself, that they have always been... They are deceived, too, by those highly mendacious documents which profess to give the history of many thousand years, though, reckoning by the sacred writings, we find that not 6000 years have yet passed.

However, Augustine is quoting here about the age of human civilization not the age of the Earth based on his use of early Christian histories. Those histories are no longer considered accurate in terms of exact years and therefore either the 6000 years is not an exact number or the years aren't actual literal years.

St. Augustine also comments on the word "day" in the creation week, admitting the interpretation is difficult:

But simultaneously with time the world was made, if in the world's creation change and motion were created, as seems evident from the order of the first six or seven days. For in these days the morning and evening are counted, until, on the sixth day, all things which God then made were finished, and on the seventh the rest of God was mysteriously and sublimely signalized. What kind of days these were it is extremely difficult, or perhaps impossible for us to conceive, and how much more to say!

Contemporary Christian considerations

See also: Framework interpretation (Genesis)

Many modern Christian theologians, Roman Catholic, Eastern Orthodox, and mainline Protestants, have rejected literalistic interpretations of Genesis in favour of allegorical or mythopoietic interpretations such as the literary framework view. Many Christian Fundamentalists have considered such rejection unmerited. Sir Robert Anderson wrote, "Christ and Criticism" in The Fundamentals, which wholly rejected a non-literal interpretation of Genesis by Jesus Christ. In modern times, Answers in Genesis has been a strong advocate of a literal interpretation of Genesis.

Catholic theologian Ludwig Ott in his authoritative Fundamentals of Catholic Dogma, under the section "The Divine Work of Creation," (pages 92–122) covers the "biblical hexahemeron" (the "six days" of creation), the creation of man, Adam/Eve, original sin, the Fall, and the statements of the early Fathers, Saints, Church Councils, and Popes relevant to the matter. Ott makes the following comments on the "science" of Genesis and the Fathers:

as the hagiographers in profane things make use of a popular, that is, a non-scientific form of exposition suitable to the mental perception of their times, a more liberal interpretation, is possible here. The Church gives no positive decisions in regard to purely scientific questions, but limits itself to rejecting errors which endanger faith. Further, in these scientific matters there is no virtue in a consensus of the Fathers since they are not here acting as witnesses of the Faith, but merely as private scientists... Since the findings of reason and the supernatural knowledge of Faith go back to the same source, namely to God, there can never be a real contradiction between the certain discoveries of the profane sciences and the Word of God properly understood.

As the Sacred Writer had not the intention of representing with scientific accuracy the intrinsic constitution of things, and the sequence of the works of creation but of communicating knowledge in a popular way suitable to the idiom and to the pre-scientific development of his time, the account is not to be regarded or measured as if it were couched in language which is strictly scientific... The Biblical account of the duration and order of Creation is merely a literary clothing of the religious truth that the whole world was called into existence by the creative word of God. The Sacred Writer utilized for this purpose the pre-scientific picture of the world existing at the time. The numeral six of the days of Creation is to be understood as an anthropomorphism. God's work of creation represented in schematic form (opus distinctionis — opus ornatus) by the picture of a human working week, the termination of the work by the picture of the Sabbath rest. The purpose of this literary device is to manifest Divine approval of the working week and the Sabbath rest.

Pope John Paul II wrote to the Pontifical Academy of Sciences on the subject of cosmology and how to interpret Genesis:

Cosmogony and cosmology have always aroused great interest among peoples and religions. The Bible itself speaks to us of the origin of the universe and its make-up, not in order to provide us with a scientific treatise, but in order to state the correct relationships of man with God and with the universe. Sacred Scripture wishes simply to declare that the world was created by God, and in order to teach this truth it expresses itself in the terms of the cosmology in use at the time of the writer. The Sacred Book likewise wishes to tell men that the world was not created as the seat of the gods, as was taught by other cosmogonies and cosmologies, but was rather created for the service of man and the glory of God. Any other teaching about the origin and make-up of the universe is alien to the intentions of the Bible, which does not wish to teach how heaven was made but how one goes to heaven.

The "Clergy Letter" Project, drafted in 2004, and signed by thousands of Christian clergy supporting science and faith, states:

We the undersigned, Christian clergy from many different traditions, believe that the timeless truths of the Bible and the discoveries of modern science may comfortably coexist. We believe that the theory of evolution is a foundational scientific truth, one that has stood up to rigorous scrutiny and upon which much of human knowledge and achievement rests. To reject this truth or to treat it as 'one theory among others' is to deliberately embrace scientific ignorance and transmit such ignorance to our children. We believe that among God’s good gifts are human minds capable of critical thought and that the failure to fully employ this gift is a rejection of the will of our Creator.

Prominent evangelical advocates of metaphorical interpretations of Genesis include Meredith G. Kline and Henri Blocher who advocate the literary framework view. In Beyond the Firmament: Understanding Science and the Theology of Creation, evangelical author Gordon J. Glover argues for an ancient near-eastern cosmology interpretation of Genesis, which he labels the theology of creation:

Christians need to understand the first chapter of Genesis for what it is: an 'accurate' rendering of the physical universe by ancient standards that God used as the vehicle to deliver timeless theological truth to His people. We shouldn’t try to make Genesis into something that it’s not by dragging it through 3,500 years of scientific progress. When reading Genesis, Christians today need to transport themselves back to Mt. Sinai and leave our modern minds in the 21st century. If you only remember one thing from this chapter make it this: Genesis is not giving us creation science. It is giving us something much more profound and practical than that. Genesis is giving us a Biblical Theology of Creation.

Rabbinic teachings

Main article: Judaism and evolution

Philo was the first commentator to use allegory on Bible extensively in his writing.

Some medieval philosophical rationalists, such as Maimonides (Mosheh ben Maimon, the "Rambam") held that it was not required to read Genesis literally. In this view, one was obligated to understand Torah in a way that was compatible with the findings of science. Indeed, Maimonides, one of the great rabbis of the Middle Ages, wrote that if science and Torah were misaligned, it was either because science was not understood or the Torah was misinterpreted. Maimonides argued that if science proved a point, then the finding should be accepted and scripture should be interpreted accordingly. Before him Saadia Gaon set rules in the same spirit when allegoric approach can be used, for example, if the plain sense contradicts logic. Solomon ibn Gabirol extensively used allegory in his book "Fountain of Life", cited by Abraham ibn Ezra. In 1305 Shlomo ben Aderet wrote a letter against unrestricted usage of allegory by followers of Maimonides, like Jacob Anatoli in his book "Malmad ha-Talmidim". In spite of this Gersonides copied Maimonides' explanation the story of Adam into his commentary on Genesis, thinly veiled by extensive usage of the word "hint". The main point of Maimonides and Gersonides is that Fall of Man is not a story about one man, but about the human nature. Adam is the pure intellect, Eve is a body, and the Serpent is a fantasy that tries to trap intellect through the body.

Zohar states:

If a man looks upon the Torah as merely a book presenting narratives and everyday matters, alas for him! Such a Torah, one treating with everyday concerns, and indeed a more excellent one, we too, even we, could compile. More than that, in the possession of the rulers of the world there are books of even greater merit, and these we could emulate if we wished to compile some such torah. But the Torah, in all of its words, holds supernal truths and sublime secrets.

Thus the tales related in the Torah are simply her outer garments, and woe to the man who regards that outer garb as the Torah itself, for such a man will be deprived of portion in the next world. Thus David said:" Open Thou mine eyes, that I may behold wondrous things out of Thy law" (Psalms 119:18), that is to say, the things that are underneath. See now. The most visible part of a man are the clothes that he has on, and they who lack understanding, when they look at the man, are apt not to see more in him than these clothes. In reality, however, it is the body of the man that constitutes the pride of his clothes, and his soul constitutes the pride of his body.

Woe to the sinners who look upon the Torah as simply tales pertaining to things of the world, seeing thus only the outer garment. But the righteous whose gaze penetrates to the very Torah, happy are they. Just as wine must be in a jar to keep, so the Torah must also be contained in an outer garment. That garment is made up of the tales and stories; but we, we are bound to penetrate beyond.

Nahmanides, often critical of the rationalist views of Maimonides, pointed out (in his commentary to Genesis) several non-sequiturs stemming from a literal translation of the Bible's account of Creation, and stated that the account actually symbolically refers to spiritual concepts. He quoted the Mishnah in Tractate Chagigah which states that the actual meaning of the Creation account, mystical in nature, was traditionally transmitted from teachers to advanced scholars in a private setting. Many Kabbalistic sources mention Shmitot - cosmic cycles of creation, similar to the Indian concept of yugas.

Adam and Eve in the Baháʼí Faith

The Baháʼí Faith adheres to an allegorical interpretation of the Adam and Eve narrative. In Some Answered Questions, 'Abdu'l-Bahá unequivocally rejects a literal reading, instead holding that the story is a symbolic one containing "divine mysteries and universal meanings"; namely, the fall of Adam symbolizes that humanity became conscious of good and evil.

Alcubierre drive

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Alcubierre_drive 

 

Two-dimensional visualization of an Alcubierre drive, showing the opposing regions of expanding and contracting spacetime that displace the central region

The Alcubierre drive is a speculative warp drive idea according to which a spacecraft could achieve apparent faster-than-light travel by contracting space in front of it and expanding space behind it, under the assumption that a configurable energy-density field lower than that of vacuum (that is, negative mass) could be created. Proposed by theoretical physicist Miguel Alcubierre in 1994, the Alcubierre drive is based on a solution of Einstein's field equations. Since those solutions are metric tensors, the Alcubierre drive is also referred to as Alcubierre metric.

Objects cannot accelerate to the speed of light within normal spacetime; instead, the Alcubierre drive shifts space around an object so that the object would arrive at its destination more quickly than light would in normal space without breaking any physical laws.

Although the metric proposed by Alcubierre is consistent with the Einstein field equations, construction of such a drive is not necessarily possible. The proposed mechanism of the Alcubierre drive implies a negative energy density and therefore requires exotic matter or manipulation of dark energy. If exotic matter with the correct properties cannot exist, then the drive cannot be constructed. At the close of his original article, however, Alcubierre argued (following an argument developed by physicists analyzing traversable wormholes) that the Casimir vacuum between parallel plates could fulfill the negative-energy requirement for the Alcubierre drive. Some research has claimed that such a concept is possible with purely positive energy using 'soliton' waves.

Another possible issue is that, although the Alcubierre metric is consistent with Einstein's equations, general relativity does not incorporate quantum mechanics. Some physicists have presented arguments to suggest that a theory of quantum gravity (which would incorporate both theories) would eliminate those solutions in general relativity that allow for backward time travel (see the chronology protection conjecture) and thus make the Alcubierre drive invalid.

History

In 1994, Miguel Alcubierre proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand. The ship would then ride this wave inside a region of flat space, known as a warp bubble, and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive. The local velocity relative to the deformed space-time would be subluminal, but the speed at which a spacecraft could move would be superluminal, thereby rendering possible interstellar flight, sucn as a visit to Proxima Centauri within a few days.

Alcubierre metric

The Alcubierre metric defines the warp-drive spacetime. It is a Lorentzian manifold that, if interpreted in the context of general relativity, allows a warp bubble to appear in previously flat spacetime and move away at effectively faster-than-light speed. The interior of the bubble is an inertial reference frame and inhabitants experience no proper acceleration. This method of transport does not involve objects in motion at faster-than-light speeds with respect to the contents of the warp bubble; that is, a light beam within the warp bubble would still always move more quickly than the ship. Because objects within the bubble are not moving (locally) more quickly than light, the mathematical formulation of the Alcubierre metric is consistent with the conventional claims of the laws of relativity (namely, that an object with mass cannot attain or exceed the speed of light) and conventional relativistic effects such as time dilation would not apply as they would with conventional motion at near-light speeds.

The Alcubierre drive remains a hypothetical concept with seemingly difficult problems, though the amount of energy required is no longer thought to be unobtainably large. Furthermore, Alexey Bobrick and Gianni Martire claim that, in principle, a class of subluminal, spherically symmetric warp drive spacetimes can be constructed based on physical principles presently known to humanity, such as positive energy.

Mathematics

Using the ADM formalism of general relativity, the spacetime is described by a foliation of space-like hypersurfaces of constant coordinate time t, with the metric taking the following general form:

${\displaystyle ds^{2}=-\left(\alpha ^{2}-\beta _{i}\beta ^{i}\right)\,dt^{2}+2\beta _{i}\,dx^{i}\,dt+\gamma _{ij}\,dx^{i}\,dx^{j},}$

where

• α is the lapse function that gives the interval of proper time between nearby hypersurfaces,
• βⁱ is the shift vector that relates the spatial coordinate systems on different hypersurfaces,
• γ_ij is a positive-definite metric on each of the hypersurfaces.

The particular form that Alcubierre studied is defined by:

${\displaystyle {\begin{aligned}\alpha &=1,\\\beta ^{x}&=-v_{s}(t)f{\big (}r_{s}(t){\big )},\\\beta ^{y}&=\beta ^{z}=0,\\\gamma _{ij}&=\delta _{ij},\end{aligned}}}$

where

${\displaystyle {\begin{aligned}v_{s}(t)&={\frac {dx_{s}(t)}{dt}},\\r_{s}(t)&={\sqrt {{\big (}x-x_{s}(t){\big )}^{2}+y^{2}+z^{2}}},\\f(r_{s})&={\frac {\tanh {\big (}\sigma (r_{s}+R){\big )}-\tanh {\big (}\sigma (r_{s}-R){\big )}}{2\tanh(\sigma R)}},\end{aligned}}}$

with arbitrary parameters R > 0 and σ > 0. Alcubierre's specific form of the metric can thus be written:

${\displaystyle ds^{2}=\left(v_{s}(t)^{2}f{\big (}r_{s}(t){\big )}^{2}-1\right)\,dt^{2}-2v_{s}(t)f{\big (}r_{s}(t){\big )}\,dx\,dt+dx^{2}+dy^{2}+dz^{2}.}$

With this particular form of the metric, it can be shown that the energy density measured by observers whose 4-velocity is normal to the hypersurfaces is given by:

${\displaystyle -{\frac {c^{4}}{8\pi G}}{\frac {v_{s}^{2}\left(y^{2}+z^{2}\right)}{4g^{2}r_{s}^{2}}}\left({\frac {df}{dr_{s}}}\right)^{2},}$

where g is the determinant of the metric tensor.

Thus, because the energy density is negative, one needs exotic matter to travel more quickly than the speed of light. The existence of exotic matter is not theoretically ruled out; however, generating and sustaining enough exotic matter to perform feats such as faster-than-light travel (and to keep open the "throat" of a wormhole) is thought to be impractical. According to writer Robert Low, within the context of general relativity it is impossible to construct a warp drive in the absence of exotic matter.

Connection to dark energy and dark matter

See also: Negative mass

Astrophysicist Jamie Farnes from the University of Oxford has proposed a theory, published in the peer-reviewed scientific journal Astronomy & Astrophysics, that unifies dark energy and dark matter into a single dark fluid, and which is expected to be testable by new scientific instruments capable of being constructed by circa 2030. Farnes found that Albert Einstein had explored the idea of gravitationally repulsive negative masses while developing the equations of general relativity, an idea which leads to a "beautiful" hypothesis where the cosmos has equal amounts of positive and negative qualities. Farnes' theory relies on negative masses that behave identically to the physics of the Alcubierre drive, providing a natural solution for the current "crisis in cosmology" due to a time-variable Hubble parameter.

As Farnes' theory allows a positive mass (i.e. a ship) to reach a speed equal to the speed of light, it has been dubbed "controversial". If the theory is correct, which has been highly debated in the scientific literature, it would explain dark energy, dark matter, allow closed timelike curves (see time travel), and suggest that an Alcubierre drive is physically possible with exotic matter.

Physics

With regard to certain specific effects of special relativity, such as Lorentz contraction and time dilation, the Alcubierre metric has some apparently peculiar aspects. In particular, Alcubierre has shown that a ship using an Alcubierre drive travels on a free-fall geodesic even while the warp bubble is accelerating: its crew would be in free fall while accelerating without experiencing accelerational g-forces. Enormous tidal forces, however, would be present near the edges of the flat-space volume because of the large space curvature there, but a suitable specification of the metric would keep the tidal forces very small within the volume occupied by the ship.

The original warp-drive metric and simple variants of it happen to have the ADM form, which is often used in discussing the initial-value formulation of general relativity. This might explain the widespread misconception that this spacetime is a solution of the field equation of general relativity. Metrics in ADM form are adapted to a certain family of inertial observers, but these observers are not really physically distinguished from other such families. Alcubierre interpreted his "warp bubble" in terms of a contraction of space ahead of the bubble and an expansion behind, but this interpretation could be misleading, since the contraction and expansion actually refer to the relative motion of nearby members of the family of ADM observers.

In general relativity, one often first specifies a plausible distribution of matter and energy, and then finds the geometry of the spacetime associated with it; but it is also possible to run the Einstein field equations in the other direction, first specifying a metric and then finding the energy–momentum tensor associated with it, and this is what Alcubierre did in building his metric. This practice means that the solution can violate various energy conditions and require exotic matter. The need for exotic matter raises questions about whether one can distribute the matter in an initial spacetime that lacks a warp bubble in such a way that the bubble is created at a later time, although some physicists have proposed models of dynamical warp-drive spacetimes in which a warp bubble is formed in a previously flat space. Moreover, according to Serguei Krasnikov, generating a bubble in a previously flat space for a one-way FTL trip requires forcing the exotic matter to move at local faster-than-light speeds, something that would require the existence of tachyons, although Krasnikov also notes that when the spacetime is not flat from the outset, a similar result could be achieved without tachyons by placing in advance some devices along the travel path and programming them to come into operation at preassigned moments and to operate in a preassigned manner. Some suggested methods avoid the problem of tachyonic motion, but would probably generate a naked singularity at the front of the bubble. Allen Everett and Thomas Roman comment on Krasnikov's finding (Krasnikov tube):

[The finding] does not mean that Alcubierre bubbles, if it were possible to create them, could not be used as a means of superluminal travel. It only means that the actions required to change the metric and create the bubble must be taken beforehand by some observer whose forward light cone contains the entire trajectory of the bubble.

For example, if one wanted to travel to Deneb (2,600 light-years away) and arrive less than 2,600 years in the future according to external clocks, it would be required that someone had already begun work on warping the space from Earth to Deneb at least 2,600 years ago:

A spaceship appropriately located with respect to the bubble trajectory could then choose to enter the bubble, rather like a passenger catching a passing trolley car, and thus make the superluminal journey ... as Krasnikov points out, causality considerations do not prevent the crew of a spaceship from arranging, by their own actions, to complete a round trip from Earth to a distant star and back in an arbitrarily short time, as measured by clocks on Earth, by altering the metric along the path of their outbound trip.

Difficulties

The metric of this form has significant difficulties because all known warp-drive spacetime theories violate various energy conditions. Nevertheless, an Alcubierre-type warp drive might be realized by exploiting certain experimentally verified quantum phenomena, such as the Casimir effect, that lead to stress–energy tensors that also violate the energy conditions, such as negative mass–energy, when described in the context of the quantum field theories.

Mass–energy requirement

If certain quantum inequalities conjectured by Ford and Roman hold, the energy requirements for some warp drives may be unfeasibly large as well as negative. For example, the energy equivalent of −10⁶⁴ kg might be required to transport a small spaceship across the Milky Way—an amount orders of magnitude greater than the estimated mass of the observable universe. Counterarguments to these apparent problems have also been offered, although the energy requirements still generally require a Type III civilization on the Kardashev scale.

Chris Van Den Broeck of the Katholieke Universiteit Leuven in Belgium, in 1999, tried to address the potential issues. By contracting the 3+1-dimensional surface area of the bubble being transported by the drive, while at the same time expanding the three-dimensional volume contained inside, Van Den Broeck was able to reduce the total energy needed to transport small atoms to less than three solar masses. Later in 2003, by slightly modifying the Van den Broeck metric, Serguei Krasnikov reduced the necessary total amount of negative mass to a few milligrams. Van Den Broeck detailed this by saying that the total energy can be reduced dramatically by keeping the surface area of the warp bubble itself microscopically small, while at the same time expanding the spatial volume inside the bubble. However, Van Den Broeck concludes that the energy densities required are still unachievable, as are the small size (a few orders of magnitude above the Planck scale) of the spacetime structures needed.

In 2012, physicist Harold White and collaborators announced that modifying the geometry of exotic matter could reduce the mass–energy requirements for a macroscopic space ship from the equivalent of the planet Jupiter to that of the Voyager 1 spacecraft (c. 700 kg) or less, and stated their intent to perform small-scale experiments in constructing warp fields. White proposed to thicken the extremely thin wall of the warp bubble, so the energy is focused in a larger volume, but the overall peak energy density is actually smaller. In a flat 2D representation, the ring of positive and negative energy, initially very thin, becomes a larger, fuzzy torus (donut shape). However, as this less energetic warp bubble also thickens toward the interior region, it leaves less flat space to house the spacecraft, which has to be smaller. Furthermore, if the intensity of the space warp can be oscillated over time, the energy required is reduced even more. According to White, a modified Michelson–Morley interferometer could test the idea: one of the legs of the interferometer would appear to have a slightly different length when the test devices were energised. Alcubierre has expressed skepticism about the experiment, saying: "from my understanding there is no way it can be done, probably not for centuries if at all".

In 2021, physicist Erik Lentz described a way warp drives sourced from known and familiar purely positive energy could exist – warp bubbles based on superluminal self-reinforcing "soliton" waves. He also claimed that he will work on reducing the (positive) energy requirement.

Placement of matter

Krasnikov proposed that if tachyonic matter cannot be found or used, then a solution might be to arrange for masses along the path of the vessel to be set in motion in such a way that the required field was produced. But in this case, the Alcubierre drive vessel can only travel routes that, like a railroad, have first been equipped with the necessary infrastructure. The pilot inside the bubble is causally disconnected from its walls and cannot carry out any action outside the bubble: the bubble cannot be used for the first trip to a distant star because the pilot cannot place infrastructure ahead of the bubble while "in transit". For example, traveling to Vega (which is 25 light-years from Earth) requires arranging everything so that the bubble moving toward Vega with a superluminal velocity would appear; such arrangements will always take more than 25 years.

Coule has argued that schemes, such as the one proposed by Alcubierre, are infeasible because matter placed en route of the intended path of a craft must be placed at superluminal speed—that constructing an Alcubierre drive requires an Alcubierre drive even if the metric that allows it is physically meaningful. Coule further argues that an analogous objection will apply to any proposed method of constructing an Alcubierre drive.

Survivability inside the bubble

An article by José Natário (2002) argues that crew members could not control, steer or stop the ship in its warp bubble because the ship could not send signals to the front of the bubble.

A 2009 article by Carlos Barceló, Stefano Finazzi, and Stefano Liberati uses quantum theory to argue that the Alcubierre drive at faster-than-light velocities is impossible mostly because extremely high temperatures caused by Hawking radiation would destroy anything inside the bubble at superluminal velocities and destabilize the bubble itself; the article also argues that these problems are absent if the bubble velocity is subluminal, although the drive still requires exotic matter.

Damaging effect on destination

Brendan McMonigal, Geraint F. Lewis, and Philip O'Byrne have argued that were an Alcubierre-driven ship to decelerate from superluminal speed, the particles that its bubble had gathered in transit would be released in energetic outbursts akin to the infinitely-blueshifted radiation hypothesized to occur at the inner event horizon of a Kerr black hole; forward-facing particles would thereby be energetic enough to destroy anything at the destination directly in front of the ship.

Wall thickness

The amount of negative energy required for such a propulsion is not yet known. Pfenning and Allen Everett of Tufts hold that a warp bubble traveling at 10-times the speed of light must have a wall thickness of no more than 10⁻³² meters—close to the limiting Planck length, 1.6 × 10⁻³⁵ meters. In Alcubierre's original calculations, a bubble macroscopically large enough to enclose a ship of 200 meters would require a total amount of exotic matter greater than the mass of the observable universe, and straining the exotic matter to an extremely thin band of 10⁻³² meters is considered impractical. Similar constraints apply to Krasnikov's superluminal subway. Chris Van den Broeck constructed a modification of Alcubierre's model that requires much less exotic matter but places the ship in a curved space-time "bottle" whose neck is about 10⁻³² meters.

Causality violation and semiclassical instability

Calculations by physicist Allen Everett show that warp bubbles could be used to create closed timelike curves in general relativity, meaning that the theory predicts that they could be used for backwards time travel. While it is possible that the fundamental laws of physics might allow closed timelike curves, the chronology protection conjecture hypothesizes that in all cases where the classical theory of general relativity allows them, quantum effects would intervene to eliminate the possibility, making these spacetimes impossible to realize. A possible type of effect that would accomplish this is a buildup of vacuum fluctuations on the border of the region of spacetime where time travel would first become possible, causing the energy density to become high enough to destroy the system that would otherwise become a time machine. Some results in semiclassical gravity appear to support the conjecture, including a calculation dealing specifically with quantum effects in warp-drive spacetimes that suggested that warp bubbles would be semiclassically unstable, but ultimately the conjecture can only be decided by a full theory of quantum gravity.

Alcubierre briefly discusses some of these issues in a series of lecture slides posted online, where he writes: "beware: in relativity, any method to travel faster than light can in principle be used to travel back in time (a time machine)". In the next slide, he brings up the chronology protection conjecture and writes: "The conjecture has not been proven (it wouldn't be a conjecture if it had), but there are good arguments in its favor based on quantum field theory. The conjecture does not prohibit faster-than-light travel. It just states that if a method to travel faster than light exists, and one tries to use it to build a time machine, something will go wrong: the energy accumulated will explode, or it will create a black hole."

Relation to Star Trek warp drive

The Star Trek television series and films use the term "warp drive" to describe their method of faster-than-light travel. Neither the Alcubierre theory, nor anything similar, existed when the series was conceived—the term "warp drive" and general concept originated with John W. Campbell's 1931 science fiction novel Islands of Space. Alcubierre stated in an email to William Shatner that his theory was directly inspired by the term used in the show and cites the "'warp drive' of science fiction" in his 1994 article. A USS Alcubierre appears in the Star Trek tabletop RPG Star Trek Adventures.

Expectation–maximization algorithm

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm 

In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step.

EM clustering of Old Faithful eruption data. The random initial model (which, due to the different scales of the axes, appears to be two very flat and wide spheres) is fit to the observed data. In the first iterations, the model changes substantially, but then converges to the two modes of the geyser. Visualized using ELKI.

History

The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele frequencies by Cedric Smith. Another was proposed by H.O. Hartley in 1958, and Hartley and Hocking in 1977, from which many of the ideas in the Dempster-Laird-Rubin paper originated. Hartley’s ideas can be broadened to any grouped discrete distribution. A very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers following his collaboration with Per Martin-Löf and Anders Martin-Löf. The Dempster–Laird–Rubin paper in 1977 generalized the method and sketched a convergence analysis for a wider class of problems. The Dempster–Laird–Rubin paper established the EM method as an important tool of statistical analysis.

The convergence analysis of the Dempster–Laird–Rubin algorithm was flawed and a correct convergence analysis was published by C. F. Jeff Wu in 1983. Wu's proof established the EM method's convergence outside of the exponential family, as claimed by Dempster–Laird–Rubin.

Introduction

The EM algorithm is used to find (local) maximum likelihood parameters of a statistical model in cases where the equations cannot be solved directly. Typically these models involve latent variables in addition to unknown parameters and known data observations. That is, either missing values exist among the data, or the model can be formulated more simply by assuming the existence of further unobserved data points. For example, a mixture model can be described more simply by assuming that each observed data point has a corresponding unobserved data point, or latent variable, specifying the mixture component to which each data point belongs.

Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations. In statistical models with latent variables, this is usually impossible. Instead, the result is typically a set of interlocking equations in which the solution to the parameters requires the values of the latent variables and vice versa, but substituting one set of equations into the other produces an unsolvable equation.

The EM algorithm proceeds from the observation that there is a way to solve these two sets of equations numerically. One can simply pick arbitrary values for one of the two sets of unknowns, use them to estimate the second set, then use these new values to find a better estimate of the first set, and then keep alternating between the two until the resulting values both converge to fixed points. It's not obvious that this will work, but it can be proven in this context. Additionally, it can be proven that the derivative of the likelihood is (arbitrarily close to) zero at that point, which in turn means that the point is either a local maximum or a saddle point. In general, multiple maxima may occur, with no guarantee that the global maximum will be found. Some likelihoods also have singularities in them, i.e., nonsensical maxima. For example, one of the solutions that may be found by EM in a mixture model involves setting one of the components to have zero variance and the mean parameter for the same component to be equal to one of the data points.

Description

Given the statistical model which generates a set ${\displaystyle \mathbf {X} }$ of observed data, a set of unobserved latent data or missing values ${\displaystyle \mathbf {Z} }$, and a vector of unknown parameters ${\displaystyle {\boldsymbol {\theta }}}$, along with a likelihood function ${\displaystyle L({\boldsymbol {\theta }};\mathbf {X} ,\mathbf {Z} )=p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})}$, the maximum likelihood estimate (MLE) of the unknown parameters is determined by maximizing the marginal likelihood of the observed data

${\displaystyle L({\boldsymbol {\theta }};\mathbf {X} )=p(\mathbf {X} \mid {\boldsymbol {\theta }})=\int p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})\,d\mathbf {Z} =\int p(\mathbf {X} \mid \mathbf {Z} ,{\boldsymbol {\theta }})p(\mathbf {Z} \mid {\boldsymbol {\theta }})\,d\mathbf {Z} }$

However, this quantity is often intractable since ${\displaystyle \mathbf {Z} }$ is unobserved and the distribution of ${\displaystyle \mathbf {Z} }$ is unknown before attaining ${\displaystyle {\boldsymbol {\theta }}}$.

The EM algorithm seeks to find the MLE of the marginal likelihood by iteratively applying these two steps:

Expectation step (E step): Define ${\displaystyle Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})}$ as the expected value of the log likelihood function of ${\displaystyle {\boldsymbol {\theta }}}$, with respect to the current conditional distribution of ${\displaystyle \mathbf {Z} }$ given ${\displaystyle \mathbf {X} }$ and the current estimates of the parameters ${\displaystyle {\boldsymbol {\theta }}^{(t)}}$:

${\displaystyle Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})=\operatorname {E} _{\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }}^{(t)}}\left[\log L({\boldsymbol {\theta }};\mathbf {X} ,\mathbf {Z} )\right]\,}$

Maximization step (M step): Find the parameters that maximize this quantity:

${\displaystyle {\boldsymbol {\theta }}^{(t+1)}={\underset {\boldsymbol {\theta }}{\operatorname {arg\,max} }}\ Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})\,}$

The typical models to which EM is applied use ${\displaystyle \mathbf {Z} }$ as a latent variable indicating membership in one of a set of groups:

1. The observed data points ${\displaystyle \mathbf {X} }$ may be discrete (taking values in a finite or countably infinite set) or continuous (taking values in an uncountably infinite set). Associated with each data point may be a vector of observations.
2. The missing values (aka latent variables) ${\displaystyle \mathbf {Z} }$ are discrete, drawn from a fixed number of values, and with one latent variable per observed unit.
3. The parameters are continuous, and are of two kinds: Parameters that are associated with all data points, and those associated with a specific value of a latent variable (i.e., associated with all data points whose corresponding latent variable has that value).

However, it is possible to apply EM to other sorts of models.

The motivation is as follows. If the value of the parameters ${\displaystyle {\boldsymbol {\theta }}}$ is known, usually the value of the latent variables ${\displaystyle \mathbf {Z} }$ can be found by maximizing the log-likelihood over all possible values of ${\displaystyle \mathbf {Z} }$, either simply by iterating over ${\displaystyle \mathbf {Z} }$ or through an algorithm such as the Viterbi algorithm for hidden Markov models. Conversely, if we know the value of the latent variables ${\displaystyle \mathbf {Z} }$, we can find an estimate of the parameters ${\displaystyle {\boldsymbol {\theta }}}$ fairly easily, typically by simply grouping the observed data points according to the value of the associated latent variable and averaging the values, or some function of the values, of the points in each group. This suggests an iterative algorithm, in the case where both ${\displaystyle {\boldsymbol {\theta }}}$ and ${\displaystyle \mathbf {Z} }$ are unknown:

1. First, initialize the parameters ${\displaystyle {\boldsymbol {\theta }}}$ to some random values.
2. Compute the probability of each possible value of ${\displaystyle \mathbf {Z} }$ , given ${\displaystyle {\boldsymbol {\theta }}}$.
3. Then, use the just-computed values of ${\displaystyle \mathbf {Z} }$ to compute a better estimate for the parameters ${\displaystyle {\boldsymbol {\theta }}}$.
4. Iterate steps 2 and 3 until convergence.

The algorithm as just described monotonically approaches a local minimum of the cost function.

Properties

Speaking of an expectation (E) step is a bit of a misnomer. What are calculated in the first step are the fixed, data-dependent parameters of the function ${\displaystyle Q}$. Once the parameters of ${\displaystyle Q}$ are known, it is fully determined and is maximized in the second (M) step of an EM algorithm.

Although an EM iteration does increase the observed data (i.e., marginal) likelihood function, no guarantee exists that the sequence converges to a maximum likelihood estimator. For multimodal distributions, this means that an EM algorithm may converge to a local maximum of the observed data likelihood function, depending on starting values. A variety of heuristic or metaheuristic approaches exist to escape a local maximum, such as random-restart hill climbing (starting with several different random initial estimates ${\displaystyle {\boldsymbol {\theta }}^{(t)}}$), or applying simulated annealing methods.

EM is especially useful when the likelihood is an exponential family: the E step becomes the sum of expectations of sufficient statistics, and the M step involves maximizing a linear function. In such a case, it is usually possible to derive closed-form expression updates for each step, using the Sundberg formula (published by Rolf Sundberg using unpublished results of Per Martin-Löf and Anders Martin-Löf).

The EM method was modified to compute maximum a posteriori (MAP) estimates for Bayesian inference in the original paper by Dempster, Laird, and Rubin.

Other methods exist to find maximum likelihood estimates, such as gradient descent, conjugate gradient, or variants of the Gauss–Newton algorithm. Unlike EM, such methods typically require the evaluation of first and/or second derivatives of the likelihood function.

Proof of correctness

Expectation-Maximization works to improve ${\displaystyle Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})}$ rather than directly improving ${\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})}$. Here it is shown that improvements to the former imply improvements to the latter.

For any ${\displaystyle \mathbf {Z} }$ with non-zero probability ${\displaystyle p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }})}$, we can write

${\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})=\log p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})-\log p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }}).}$

We take the expectation over possible values of the unknown data ${\displaystyle \mathbf {Z} }$ under the current parameter estimate ${\displaystyle \theta ^{(t)}}$ by multiplying both sides by ${\displaystyle p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }}^{(t)})}$ and summing (or integrating) over ${\displaystyle \mathbf {Z} }$. The left-hand side is the expectation of a constant, so we get:

${\displaystyle {\begin{aligned}\log p(\mathbf {X} \mid {\boldsymbol {\theta }})&=\sum _{\mathbf {Z} }p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }}^{(t)})\log p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})-\sum _{\mathbf {Z} }p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }}^{(t)})\log p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }})\\&=Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})+H({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)}),\end{aligned}}}$

where ${\displaystyle H({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})}$ is defined by the negated sum it is replacing. This last equation holds for every value of ${\displaystyle {\boldsymbol {\theta }}}$ including ${\displaystyle {\boldsymbol {\theta }}={\boldsymbol {\theta }}^{(t)}}$,

${\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }}^{(t)})=Q({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})+H({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}),}$

and subtracting this last equation from the previous equation gives

${\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})-\log p(\mathbf {X} \mid {\boldsymbol {\theta }}^{(t)})=Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})-Q({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})+H({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})-H({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}),}$

However, Gibbs' inequality tells us that ${\displaystyle H({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})\geq H({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})}$, so we can conclude that

${\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})-\log p(\mathbf {X} \mid {\boldsymbol {\theta }}^{(t)})\geq Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})-Q({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}).}$

In words, choosing ${\displaystyle {\boldsymbol {\theta }}}$ to improve ${\displaystyle Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})}$ causes ${\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})}$ to improve at least as much.

As a maximization–maximization procedure

The EM algorithm can be viewed as two alternating maximization steps, that is, as an example of coordinate descent. Consider the function:

${\displaystyle F(q,\theta ):=\operatorname {E} _{q}[\log L(\theta ;x,Z)]+H(q),}$

where q is an arbitrary probability distribution over the unobserved data z and H(q) is the entropy of the distribution q. This function can be written as

${\displaystyle F(q,\theta )=-D_{\mathrm {KL} }{\big (}q\parallel p_{Z\mid X}(\cdot \mid x;\theta ){\big )}+\log L(\theta ;x),}$

where ${\displaystyle p_{Z\mid X}(\cdot \mid x;\theta )}$ is the conditional distribution of the unobserved data given the observed data ${\displaystyle x}$ and ${\displaystyle D_{KL}}$ is the Kullback–Leibler divergence.

Then the steps in the EM algorithm may be viewed as:

Expectation step: Choose ${\displaystyle q}$ to maximize ${\displaystyle F}$:

${\displaystyle q^{(t)}=\operatorname {arg\,max} _{q}\ F(q,\theta ^{(t)})}$

Maximization step: Choose ${\displaystyle \theta }$ to maximize ${\displaystyle F}$:

${\displaystyle \theta ^{(t+1)}=\operatorname {arg\,max} _{\theta }\ F(q^{(t)},\theta )}$

Applications

EM is frequently used for parameter estimation of mixed models, notably in quantitative genetics.

In psychometrics, EM is an important tool for estimating item parameters and latent abilities of item response theory models.

With the ability to deal with missing data and observe unidentified variables, EM is becoming a useful tool to price and manage risk of a portfolio.

The EM algorithm (and its faster variant ordered subset expectation maximization) is also widely used in medical image reconstruction, especially in positron emission tomography, single-photon emission computed tomography, and x-ray computed tomography. See below for other faster variants of EM.

In structural engineering, the Structural Identification using Expectation Maximization (STRIDE) algorithm is an output-only method for identifying natural vibration properties of a structural system using sensor data (see Operational Modal Analysis).

EM is also used for data clustering. In natural language processing, two prominent instances of the algorithm are the Baum–Welch algorithm for hidden Markov models, and the inside-outside algorithm for unsupervised induction of probabilistic context-free grammars.

Filtering and smoothing EM algorithms

A Kalman filter is typically used for on-line state estimation and a minimum-variance smoother may be employed for off-line or batch state estimation. However, these minimum-variance solutions require estimates of the state-space model parameters. EM algorithms can be used for solving joint state and parameter estimation problems.

Filtering and smoothing EM algorithms arise by repeating this two-step procedure:

E-step

Operate a Kalman filter or a minimum-variance smoother designed with current parameter estimates to obtain updated state estimates.

M-step

Use the filtered or smoothed state estimates within maximum-likelihood calculations to obtain updated parameter estimates.

Suppose that a Kalman filter or minimum-variance smoother operates on measurements of a single-input-single-output system that possess additive white noise. An updated measurement noise variance estimate can be obtained from the maximum likelihood calculation

${\displaystyle {\widehat {\sigma }}_{v}^{2}={\frac {1}{N}}\sum _{k=1}^{N}{(z_{k}-{\widehat {x}}_{k})}^{2},}$

where ${\displaystyle {\widehat {x}}_{k}}$ are scalar output estimates calculated by a filter or a smoother from N scalar measurements ${\displaystyle z_{k}}$. The above update can also be applied to updating a Poisson measurement noise intensity. Similarly, for a first-order auto-regressive process, an updated process noise variance estimate can be calculated by

${\displaystyle {\widehat {\sigma }}_{w}^{2}={\frac {1}{N}}\sum _{k=1}^{N}{({\widehat {x}}_{k+1}-{\widehat {F}}{\widehat {x}}_{k})}^{2},}$

where ${\displaystyle {\widehat {x}}_{k}}$ and ${\displaystyle {\widehat {x}}_{k+1}}$ are scalar state estimates calculated by a filter or a smoother. The updated model coefficient estimate is obtained via

${\displaystyle {\widehat {F}}={\frac {\sum _{k=1}^{N}{({\widehat {x}}_{k+1}-{\widehat {F}}{\widehat {x}}_{k})}^{2}}{\sum _{k=1}^{N}{\widehat {x}}_{k}^{2}}}.}$

The convergence of parameter estimates such as those above are well studied.

Variants

A number of methods have been proposed to accelerate the sometimes slow convergence of the EM algorithm, such as those using conjugate gradient and modified Newton's methods (Newton–Raphson). Also, EM can be used with constrained estimation methods.

Parameter-expanded expectation maximization (PX-EM) algorithm often provides speed up by "us[ing] a `covariance adjustment' to correct the analysis of the M step, capitalising on extra information captured in the imputed complete data".

Expectation conditional maximization (ECM) replaces each M step with a sequence of conditional maximization (CM) steps in which each parameter θ_i is maximized individually, conditionally on the other parameters remaining fixed. Itself can be extended into the Expectation conditional maximization either (ECME) algorithm.

This idea is further extended in generalized expectation maximization (GEM) algorithm, in which is sought only an increase in the objective function F for both the E step and M step as described in the As a maximization–maximization procedure section. GEM is further developed in a distributed environment and shows promising results.

It is also possible to consider the EM algorithm as a subclass of the MM (Majorize/Minimize or Minorize/Maximize, depending on context) algorithm, and therefore use any machinery developed in the more general case.

α-EM algorithm

The Q-function used in the EM algorithm is based on the log likelihood. Therefore, it is regarded as the log-EM algorithm. The use of the log likelihood can be generalized to that of the α-log likelihood ratio. Then, the α-log likelihood ratio of the observed data can be exactly expressed as equality by using the Q-function of the α-log likelihood ratio and the α-divergence. Obtaining this Q-function is a generalized E step. Its maximization is a generalized M step. This pair is called the α-EM algorithm which contains the log-EM algorithm as its subclass. Thus, the α-EM algorithm by Yasuo Matsuyama is an exact generalization of the log-EM algorithm. No computation of gradient or Hessian matrix is needed. The α-EM shows faster convergence than the log-EM algorithm by choosing an appropriate α. The α-EM algorithm leads to a faster version of the Hidden Markov model estimation algorithm α-HMM. 

Relation to variational Bayes methods

EM is a partially non-Bayesian, maximum likelihood method. Its final result gives a probability distribution over the latent variables (in the Bayesian style) together with a point estimate for θ (either a maximum likelihood estimate or a posterior mode). A fully Bayesian version of this may be wanted, giving a probability distribution over θ and the latent variables. The Bayesian approach to inference is simply to treat θ as another latent variable. In this paradigm, the distinction between the E and M steps disappears. If using the factorized Q approximation as described above (variational Bayes), solving can iterate over each latent variable (now including θ) and optimize them one at a time. Now, k steps per iteration are needed, where k is the number of latent variables. For graphical models this is easy to do as each variable's new Q depends only on its Markov blanket, so local message passing can be used for efficient inference.

Geometric interpretation

Further information: Information geometry

In information geometry, the E step and the M step are interpreted as projections under dual affine connections, called the e-connection and the m-connection; the Kullback–Leibler divergence can also be understood in these terms.

Examples

Gaussian mixture

Comparison of k-means and EM on artificial data visualized with ELKI. Using the variances, the EM algorithm can describe the normal distributions exactly, while k-means splits the data in Voronoi-cells. The cluster center is indicated by the lighter, bigger symbol.

 

An animation demonstrating the EM algorithm fitting a two component Gaussian mixture model to the Old Faithful dataset. The algorithm steps through from a random initialization to convergence.

Let ${\displaystyle \mathbf {x} =(\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n})}$ be a sample of ${\displaystyle n}$ independent observations from a mixture of two multivariate normal distributions of dimension ${\displaystyle d}$, and let ${\displaystyle \mathbf {z} =(z_{1},z_{2},\ldots ,z_{n})}$ be the latent variables that determine the component from which the observation originates.

and ${\displaystyle X_{i}\mid (Z_{i}=2)\sim {\mathcal {N}}_{d}({\boldsymbol {\mu }}_{2},\Sigma _{2}),}$

where

${\displaystyle \operatorname {P} (Z_{i}=1)=\tau _{1}\,}$ and ${\displaystyle \operatorname {P} (Z_{i}=2)=\tau _{2}=1-\tau _{1}.}$

The aim is to estimate the unknown parameters representing the mixing value between the Gaussians and the means and covariances of each:

${\displaystyle \theta ={\big (}{\boldsymbol {\tau }},{\boldsymbol {\mu }}_{1},{\boldsymbol {\mu }}_{2},\Sigma _{1},\Sigma _{2}{\big )},}$

where the incomplete-data likelihood function is

${\displaystyle L(\theta ;\mathbf {x} )=\prod _{i=1}^{n}\sum _{j=1}^{2}\tau _{j}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{j},\Sigma _{j}),}$

and the complete-data likelihood function is

${\displaystyle L(\theta ;\mathbf {x} ,\mathbf {z} )=p(\mathbf {x} ,\mathbf {z} \mid \theta )=\prod _{i=1}^{n}\prod _{j=1}^{2}\ [f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{j},\Sigma _{j})\tau _{j}]^{\mathbb {I} (z_{i}=j)},}$

or

${\displaystyle L(\theta ;\mathbf {x} ,\mathbf {z} )=\exp \left\{\sum _{i=1}^{n}\sum _{j=1}^{2}\mathbb {I} (z_{i}=j){\big [}\log \tau _{j}-{\tfrac {1}{2}}\log |\Sigma _{j}|-{\tfrac {1}{2}}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})^{\top }\Sigma _{j}^{-1}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})-{\tfrac {d}{2}}\log(2\pi ){\big ]}\right\},}$

where ${\displaystyle \mathbb {I} }$ is an indicator function and ${\displaystyle f}$ is the probability density function of a multivariate normal.

In the last equality, for each i, one indicator ${\displaystyle \mathbb {I} (z_{i}=j)}$ is equal to zero, and one indicator is equal to one. The inner sum thus reduces to one term.

E step

Given our current estimate of the parameters θ^(t), the conditional distribution of the Z_i is determined by Bayes theorem to be the proportional height of the normal density weighted by τ:

${\displaystyle T_{j,i}^{(t)}:=\operatorname {P} (Z_{i}=j\mid X_{i}=\mathbf {x} _{i};\theta ^{(t)})={\frac {\tau _{j}^{(t)}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{j}^{(t)},\Sigma _{j}^{(t)})}{\tau _{1}^{(t)}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{1}^{(t)},\Sigma _{1}^{(t)})+\tau _{2}^{(t)}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{2}^{(t)},\Sigma _{2}^{(t)})}}.}$

These are called the "membership probabilities", which are normally considered the output of the E step (although this is not the Q function of below).

This E step corresponds with setting up this function for Q:

${\displaystyle {\begin{aligned}Q(\theta \mid \theta ^{(t)})&=\operatorname {E} _{\mathbf {Z} \mid \mathbf {X} ,\mathbf {\theta } ^{(t)}}[\log L(\theta ;\mathbf {x} ,\mathbf {Z} )]\\&=\operatorname {E} _{\mathbf {Z} \mid \mathbf {X} ,\mathbf {\theta } ^{(t)}}[\log \prod _{i=1}^{n}L(\theta ;\mathbf {x} _{i},Z_{i})]\\&=\operatorname {E} _{\mathbf {Z} \mid \mathbf {X} ,\mathbf {\theta } ^{(t)}}[\sum _{i=1}^{n}\log L(\theta ;\mathbf {x} _{i},Z_{i})]\\&=\sum _{i=1}^{n}\operatorname {E} _{Z_{i}\mid \mathbf {X} ;\mathbf {\theta } ^{(t)}}[\log L(\theta ;\mathbf {x} _{i},Z_{i})]\\&=\sum _{i=1}^{n}\sum _{j=1}^{2}P(Z_{i}=j\mid X_{i}=\mathbf {x} _{i};\theta ^{(t)})\log L(\theta _{j};\mathbf {x} _{i},j)\\&=\sum _{i=1}^{n}\sum _{j=1}^{2}T_{j,i}^{(t)}{\big [}\log \tau _{j}-{\tfrac {1}{2}}\log |\Sigma _{j}|-{\tfrac {1}{2}}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})^{\top }\Sigma _{j}^{-1}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})-{\tfrac {d}{2}}\log(2\pi ){\big ]}.\end{aligned}}}$

The expectation of ${\displaystyle \log L(\theta ;\mathbf {x} _{i},Z_{i})}$ inside the sum is taken with respect to the probability density function ${\displaystyle P(Z_{i}\mid X_{i}=\mathbf {x} _{i};\theta ^{(t)})}$, which might be different for each ${\displaystyle \mathbf {x} _{i}}$ of the training set. Everything in the E step is known before the step is taken except ${\displaystyle T_{j,i}}$, which is computed according to the equation at the beginning of the E step section.

This full conditional expectation does not need to be calculated in one step, because τ and μ/Σ appear in separate linear terms and can thus be maximized independently.

M step

Q(θ | θ^(t)) being quadratic in form means that determining the maximizing values of θ is relatively straightforward. Also, τ, (μ₁,Σ₁) and (μ₂,Σ₂) may all be maximized independently since they all appear in separate linear terms.

To begin, consider τ, which has the constraint τ₁ + τ₂=1:

${\displaystyle {\begin{aligned}{\boldsymbol {\tau }}^{(t+1)}&={\underset {\boldsymbol {\tau }}{\operatorname {arg\,max} }}\ Q(\theta \mid \theta ^{(t)})\\&={\underset {\boldsymbol {\tau }}{\operatorname {arg\,max} }}\ \left\{\left[\sum _{i=1}^{n}T_{1,i}^{(t)}\right]\log \tau _{1}+\left[\sum _{i=1}^{n}T_{2,i}^{(t)}\right]\log \tau _{2}\right\}.\end{aligned}}}$

This has the same form as the MLE for the binomial distribution, so

${\displaystyle \tau _{j}^{(t+1)}={\frac {\sum _{i=1}^{n}T_{j,i}^{(t)}}{\sum _{i=1}^{n}(T_{1,i}^{(t)}+T_{2,i}^{(t)})}}={\frac {1}{n}}\sum _{i=1}^{n}T_{j,i}^{(t)}.}$

For the next estimates of (μ₁,Σ₁):

${\displaystyle {\begin{aligned}({\boldsymbol {\mu }}_{1}^{(t+1)},\Sigma _{1}^{(t+1)})&={\underset {{\boldsymbol {\mu }}_{1},\Sigma _{1}}{\operatorname {arg\,max} }}\ Q(\theta \mid \theta ^{(t)})\\&={\underset {{\boldsymbol {\mu }}_{1},\Sigma _{1}}{\operatorname {arg\,max} }}\ \sum _{i=1}^{n}T_{1,i}^{(t)}\left\{-{\tfrac {1}{2}}\log |\Sigma _{1}|-{\tfrac {1}{2}}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1})^{\top }\Sigma _{1}^{-1}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1})\right\}\end{aligned}}.}$

This has the same form as a weighted MLE for a normal distribution, so

${\displaystyle {\boldsymbol {\mu }}_{1}^{(t+1)}={\frac {\sum _{i=1}^{n}T_{1,i}^{(t)}\mathbf {x} _{i}}{\sum _{i=1}^{n}T_{1,i}^{(t)}}}}$ and ${\displaystyle \Sigma _{1}^{(t+1)}={\frac {\sum _{i=1}^{n}T_{1,i}^{(t)}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1}^{(t+1)})(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1}^{(t+1)})^{\top }}{\sum _{i=1}^{n}T_{1,i}^{(t)}}}}$

and, by symmetry,

${\displaystyle {\boldsymbol {\mu }}_{2}^{(t+1)}={\frac {\sum _{i=1}^{n}T_{2,i}^{(t)}\mathbf {x} _{i}}{\sum _{i=1}^{n}T_{2,i}^{(t)}}}}$ and ${\displaystyle \Sigma _{2}^{(t+1)}={\frac {\sum _{i=1}^{n}T_{2,i}^{(t)}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{2}^{(t+1)})(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{2}^{(t+1)})^{\top }}{\sum _{i=1}^{n}T_{2,i}^{(t)}}}.}$

Termination

Conclude the iterative process if ${\displaystyle E_{Z\mid \theta ^{(t)},\mathbf {x} }[\log L(\theta ^{(t)};\mathbf {x} ,\mathbf {Z} )]\leq E_{Z\mid \theta ^{(t-1)},\mathbf {x} }[\log L(\theta ^{(t-1)};\mathbf {x} ,\mathbf {Z} )]+\varepsilon }$ for ${\displaystyle \varepsilon }$ below some preset threshold.

Generalization

The algorithm illustrated above can be generalized for mixtures of more than two multivariate normal distributions.

Truncated and censored regression

The EM algorithm has been implemented in the case where an underlying linear regression model exists explaining the variation of some quantity, but where the values actually observed are censored or truncated versions of those represented in the model. Special cases of this model include censored or truncated observations from one normal distribution.

Alternatives

EM typically converges to a local optimum, not necessarily the global optimum, with no bound on the convergence rate in general. It is possible that it can be arbitrarily poor in high dimensions and there can be an exponential number of local optima. Hence, a need exists for alternative methods for guaranteed learning, especially in the high-dimensional setting. Alternatives to EM exist with better guarantees for consistency, which are termed moment-based approaches or the so-called spectral techniques. Moment-based approaches to learning the parameters of a probabilistic model are of increasing interest recently since they enjoy guarantees such as global convergence under certain conditions unlike EM which is often plagued by the issue of getting stuck in local optima. Algorithms with guarantees for learning can be derived for a number of important models such as mixture models, HMMs etc. For these spectral methods, no spurious local optima occur, and the true parameters can be consistently estimated under some regularity conditions.