Monthly Archives: May 2016

Statistics with a single history

Only one history is observable to a person – the one that actually happened. Counterfactuals are speculation about what would have happened if choices or some other element of the past history had differed. Only one history is observable to humanity as a whole, to all thinking beings in the universe as a whole, etc. This raises the question of how to do statistics with a single history.

The history is chopped into small pieces, which are assumed similar to each other and to future pieces of history. All conclusions require assumptions. In the case of statistics, the main assumption is “what happened in the past, will continue to happen in the future.” The “what” that is happening can be complicated – a long chaotic pattern can be repeated. It should be specified what the patterns of history consist of before discussing them.

The history observable to a brain consists of the sensory inputs and memory. Nothing else is accessible. This is pointed out by the “brain in a jar” thought experiment. Memory is partly past sensory inputs, but may also depend on spontaneous changes in the brain. Machinery can translate previously unobservable aspects of the world into accessible sensory inputs, for example convert infrared and ultraviolet light into visible wavelengths. Formally, history is a function from time to vectors of sensory inputs.

The brain has a built-in ability to classify sensory inputs by type – visual, auditory, etc. This is why the inputs form a vector. For a given sense, there is a built-in “similarity function” that enables comparing inputs from the same sense at different times.

Inputs distinguished by one person, perhaps with the help of machinery, may look identical to another person. The interpretation is that there are underlying physical quantities that must differ by more than the “just noticeable difference” to be perceived as different. The brain can access physical quantities only through the senses, so whether there is a “real world” cannot be determined, only assumed. If most people’s perceptions agree about something, and machinery also agrees (e.g. measuring tape does not agree with visual illusions), then this “something” is called real and physical. The history accessible to humanity as a whole is a function from time to the concatenation of their sensory input vectors.

The similarity functions of people can also be aggregated, compared to machinery and the result interpreted as a physical quantity taking “similar” values at different times.

A set of finite sequences of vectors of sensory inputs is what I call a pattern of history. For example, a pattern can be a single sequence or everything but a given sequence. Patterns may repeat, due to the indistinguishability of physical quantities close to each other. The finer distinctions one can make, the fewer the instances with the same perception. In the limit of perfect discrimination of all variable values, history is unlikely to ever repeat. In the limit of no perception at all, history is one long repetition of nothing happening. The similarity of patterns is defined based on the similarity function in the brain.

Repeated similar patterns together with assumptions enable learning and prediction. If AB is always followed by C, then learning is easy. Statistics are needed when this is not the case. If half the past instances of AB are followed by C, half by D, then one way to interpret this is by constructing a state space with a probability distribution on it. For example, one may assume the existence of an unperceived variable that can take values c,d and assume that ABc leads deterministically to ABC and ABd to ABD. The past instances of AB can be interpreted as split into equal numbers of ABc and ABd. The prediction after observing AB is equal probabilities of C and D. This is a frequentist setup.

A Bayesian interpretation puts a prior probability distribution on histories and updates it based on the observations. The prior may put probability one on a single future history after each past one. Such a deterministic prediction is easily falsified – one observation contrary to it suffices. Usually, many future histories are assumed to have positive probability. Updating requires conditional probabilities of future histories given the past. The histories that repeat past patterns are usually given higher probability than others. Such a conditional probability system embodies the assumption “what happened in the past, will continue to happen in the future.”

There is a tradeoff between the length of a pattern and the number of times it has repeated. Longer patterns permit prediction further into the future, but fewer repetitions mean more uncertainty. Much research in statistics has gone into finding the optimal pattern length given the data. A long pattern contains many shorter ones, with potentially different predictions. Combining information from different pattern lengths is also a research area. Again, assumptions determine which pattern length and combination is optimal. Assumptions can be tested, but only under other assumptions.

Causality is also a mental construct. It is based on past repetitions of an AB-like pattern, without occurrence of BA or CB-like patterns.

The perception of time is created by sensory inputs and memory, e.g. seeing light and darkness alternate, feeling sleepy or alert due to the circadian rhythm and remembering that this has happened before. History is thus a mental construct. It relies on the assumptions that time exists, there is a past in which things happened and current recall is correlated with what actually happened. The preceding discussion should be restated without assuming time exists.


Milk powder with milk

Dessert recipe: half a bowl of milk, equal volume of powdered milk on top. Eat with a spoon. Tasty, healthy, quick and easy to make, cheap.
The reason not to eat the powdered milk from the pack directly is that a slight breath will make the powder fly. If some of it hits the back of the throat, it causes coughing, which launches the rest of the powder all over the room. Adding liquid milk removes the choking hazard.

On simple answers

Bayes’ rule exercise: is a simple or a complicated answer to a complicated problem more likely to be correct?

Depends on the conditional probabilities: if simple questions are more likely to have simple answers and complex questions complicated, then a complicated answer is more likely to be correct for a complicated problem.

It seems reasonable that the complexity of the answer is correlated with the difficulty of the problem. But this is an empirical question.

If difficult problems are likely to have complex answers, then this is an argument against slogans and ideologies. These seek to give a catchy one-liner as the answer to many problems in society. No need to think – ideology has the solution. Depending on your political leaning, poverty may be due to laziness or exploitation. The foreign policy “solution” is bombing for some, eternal appeasement for others.

The probabilistic preference for complex answers in complicated situations seems to contradict Occam’s razor (among answers equally good at explaining the facts, the simplest answer should be chosen). There is no actual conflict with the above Bayesian exercise. There, the expectation of a complex answer applies to complicated questions, while a symmetric anticipation of a simple answer holds for simple problems. The answers compared are not equally good, because one fits the structure of the question better than the other.

Which ideology is more likely to be wrong?

Exercise in Bayes’ rule: is an ideology more likely to be wrong if it appeals relatively more to poor people than the rich?

More manipulable folks are more likely to lose their money, so less likely to be rich. Stupid people have a lower probability of making money. By Bayes, the rich are on average less manipulable and more intelligent than the poor.

Less manipulable people are less likely to find an ideology built on fallacies appealing. By Bayes, an ideology relatively more appealing to the stupid and credulous is more likely to be wrong. Due to such people being poor with a higher probability, an ideology embraced more by the poor than the rich is more likely to be fallacious.

Another exercise: is an ideology more likely to be wrong if academics like it relatively more than non-academics?

Smarter people are more likely to become academics, so by Bayes’ rule, academics are more likely to be smart. Intelligent people have a relatively higher probability of liking a correct ideology, so by Bayes, an ideology appealing to the intelligent is more likely to be correct. An ideology liked by academics is correct with a higher probability.

Economics to guide materials science

There are too many possible materials to test them all, or even simulate by computer. Materials scientists theorize what combinations of elements are likely to yield the desired properties, but still there are too many possibilities. One way to narrow the choice is to use economics.

If the goal of developing a material is to change the world or make money, the benefit of the invention must exceed the cost. The benefit comes from the improved characteristics of the material relative to existing alternatives. What the market is willing to pay for an improvement depends on its size. There may be a theoretical maximum for a property, or its historical rate of increase may be used to forecast the likely improvement. Once an approximate willingness to pay for a unit of the candidate invented material is known, this can be compared with its estimated cost.

Financial firms dealing in commodity futures forecast the prices of chemical elements over the likely commercialization time horizon. Only materials using a combination of elements that is cheap enough are commercially promising. Cheap enough means that the improved material must cost less per unit than the market is willing to pay for it. An expensive element can be used, but only in appropriately tiny quantity. The requirement that the bundle of elements cost less than some bound cuts down on the number of combinations that are worth testing. Similarly, the manufacturing method must be cheap enough, so some methods may be ignored.

The basic cost-benefit analysis is a simple idea, though the benefit estimation may be complicated in practice. Probably the companies producing various materials are already taking the potential cost and benefit of an innovation into account in their R&D, but academic materials scientists perhaps not. If the goal is to advance fundamental science and satisfy one’s curiosity, then the cost of the material may not be an issue. But for the world to use the material, it must be cheap enough.

A practical recommendation is for an application-oriented lab to put up a periodic table with the prices of the elements added. A spreadsheet with the prices of commodities can be used to calculate the cost of a candidate combination for a new material. Testing the candidates should proceed in the order of decreasing “profit” (benefit minus cost of the material). This profit is not necessarily the same as commercial profit, because the benefit may include its whole contribution to society, not just the revenue to the producer.

On journalistic privilege

Journalists have certain privileges over the average citizen – they get access to inside information, public figures and press conferences. An attack against a journalist creates more outrage, because it is seen as damaging the free press. These advantages are not given so that journalists could make money or satisfy their curiosity. The privileges are provided to help journalists serve the public interest, similarly to the delegation of decision power to politicians. People being people, some journalists abuse the privileges. They do not inform the public, only entertain to make money. This takes the form of sensationalism: covering frivolous topics that sell well, but do not provide useful knowledge. For example celebrity gossip, funny animals, manufactured controversy.

It would be fair to remove journalistic privileges from tabloid reporters and stop calling them journalists. They are just nosy people. The problem is that whoever decides on giving or taking privileges, gets power over the media. Therefore this authority should not be the government, but an independent organization. However, some rights and protections of the media are legislated, so a non-governmental body cannot change them. The legislature would have to delegate its authority in this sphere to the hypothetical independent regulator first. Given how much politicians wish to influence journalists, this decision seems unlikely.