The page says this add-on only works if you disable ‘Google Instant’, which is that feature that tries to guess what you’re typing as you type it.

This really pisses me off. What’s Google trying to do, get lazy people to incorporate Google-specific data in all their links? Take over the internet somehow? Or is it just a stupid bug?

I’m starting to use another web search engine, DuckDuckGo, for PDF file searches. Anyone know other good solutions… or what Google is up to here?

The question, then, is what does the exponential of relative entropy mean? What is it an effective number of? The answer’s not immediately obvious, but I have some half-formed thoughts I should develop a bit before sharing.

The problem in this case arose from this bigger problem…

• Karl Friston, The free-energy principle: a rough guide to the brain?, Trends in Cognitive Science 13 (2009), 293 – 301.

Abstract. This article reviews a free-energy formulation that advances Helmholtz’s agenda to find principles of brain function based on conservation laws and neuronal energy. It rests on advances in statistical physics, theoretical biology and machine learning to explain a remarkable range of facts about brain structure and function. We could have just scratched the surface of what this formulation offers; for example, it is becoming clear that the Bayesian brain is just one facet of the free-energy principle and that perception is an inevitable consequence of active exchange with the environment. Furthermore, one can see easily how constructs like memory, attention, value, reinforcement and salience might disclose their simple relationships within this framework.

As usual, clicking on the paper’s title gives a free version of the paper, while clicking on the journal name gives the official, stable version!

Students in the classroom are said to have learned the probability of a reward occurring at a site in the demonstration, when based on previous experience the student’s guesses for each trial have a frequency per site that matches the probability of the reward occurring at each site established beforehand by the researcher.

Second, schools of fish divide themselves proportionally to the probability of reward coming from a site, displaying a Nash equilibrium.

About the demonstration of probability learning, Gallistel writes “[Students] were greatly surprised to be shown when the demonstration was over that the rat’s behavior [against whom they were competing to get the most reward] was more intelligent than their own. We did not lessen their discomfiture by telling them that if the rat chose under the same conditions they did – under a correction procedure whereby on every trial it ended up knowing which side was the rewarded side – it too would match the relative frequencies of its initial side choices to the relative frequencies of payoffs.”

In Shannon terms, for each site the students experienced an uncertainty -(p)ln(p) that the reward would occur at a particular site. But on occasions when they chose that site – and then saw that the reward occurred at a different site – they also experienced an opposing uncertainty (1-p)ln(1-p) associated with the reward occurring somewhere other than the chosen site. So there is uncertainty on each trial for the reward occurring at a site, and at the same time an uncertainty that it will Not occur at that site. The idea here would be to introduce positive and negative uncertainties, perhaps as Franklin applied positive and negative numbers to electric charge.

But the rat was not given the opposing uncertainty (1-p)ln(1-p) when it chose a site that was not rewarded on the trial, and was simply operating on the uncertainty -(p)ln(p) associated with the reward occurring at the site.

Much has been written about how probability learning is not maximizing utility. (In the rigged demonstration, the rat was set up to maximize utility.) But there must be a way to describe how a group of probability learners do achieve a Nash equilibrium, as do the fish.

By the way, if the two opposing uncertainties are added instead of subtracted, the resulting curve for a two site demonstration is symmetric around 50/50, which seems more intuitive than just the Shannon term -(p)ln(p) alone. The logic seems to be that we are dealing with a corpuscular object, which conforms to the rule that it can only be in one place at a time. This reminds me of the Born rule, where the wave function might represent the possibility of a corpuscular object being at a location, while its complex conjugate might represent the possibility of its NOT being anywhere else. Logically ANDing (numerically multiplying) the two statements, we get “the possibility that it will be here AND the possibility that it will be nowhere else other than here, together, establish the probability of a corpuscular object being here.” Is this where the Born rule comes from?