Pointing Choices

Quote of the day:

"This convention is at variance with that used in many expert systems (e.g. MYCIN), where rules point from evidence to hypothesis (e.g., if symptom, then disease), thus denoting a flow of mental inference. By contrast, the arrows in Bayesian networks point from causes to effects, or from conditions to consequences, thus denoting a flow of constraints attributed to the physical world."

Judea Pearl, Probabilistic Reasoning in Intelligent Systems, p. 151

Bayesian Wasteland?

Despite being overloaded with other concerns, I continue to slog through two books that may (or may not) be helpful with the Machine Understanding project. On the information technology front we have Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference by Judea Pearl. I'm at page 116 as of last night. On the biological front I continue From Neuron to Brain by Stephen W. Kuffler; I'm on page 386.

Regarding Probabilistic Reasoning, so far I have seen a lot of interesting work on the problem of combining probability calculations with logic. I just finished the section on Markov Networks and am about to read up on Bayesian Networks. My problem so far is that I don't see any advantages to marrying anything I've seen of probabilistic reasoning to Jeff Hawkin's theory of predictive memory. Despite having read "Towards a Mathematical Theory of Cortical Micro-circuits," as reported in previous entries. Then again sometimes I have trouble taking up novel ideas. But my impression so far is that neural networks are not operating on a probability basis. The closest I can get, so far, to that kind of model is a signal-mixing basis, where analog functions might represent Pearl's probabilies. Nor do I see how the probability network models can cope with invariants. The word invariant is not found in the index to Probablistic Reasoning. I have higher hopes for a Tensor model, even though my own work on that is very preliminary. [I am reminded of the two seemingly totally different mathematical methods used in early quantum mechanics, which then were proven to be equivalants.]

In From Neuron, I have been reading astonishing details about how synapses and single nerve cells work, including how experiments were conducted, in mind-numbing detail. I am just getting to where how neurons and sets of neurons have been shown to operate, with the first example being neurons that sense the stretching of muscles. Again, I am pointed to tensors, which can be used to represent how multiple muscles representing various degrees of freedom of motion can lead to a coherent knowledge of where a body part is in three-dimensional, euclidean-modeled space.

More oddly, Neuron has basically nothing about Hebbian learning. True, the book is dated 1984. But did no one even try to find a physical basis for Hebbian learning as of that date? If you know of a definitive paper that appears to prove a biochemical mechanism for Hebbian learning, let me and my readers know.

I intended to explain what I was reading, in suitable chunks, in this blog. But it is easier to just keep reading at this point, rather than writing about details that I am not even sure are important yet. I keep finding I have to go back to basics. Today, worrying about tensors, muscles, feedback, and a neuron-level learning model, I am revisiting a two-neuron learning model that, oddly, I first worked on back in the 1980's. If anything comes of it, I'll let you know here.

On Bayesian Belief Propagation

Towards a Mathematical Understanding of Cortical Micro-circuits [Dileep George & Jeff Hawkins, PLoS Computational Biology, October 2009, Volume 5, Issue 10] assumes some knowledge of Bayesian Belief Propagation. I was hoping the article would explain this topic as it went along, but apparently, to AI specialists, this is a well-known area. I found the Wikipedia entry, belief propagation, jumps into mathematical formulism rather quickly. Some words of explanation without math symbols would probably help many people who are interested in machine understanding; that is my goal here. If some of you Bayesian belief propagation experts think I have it wrong, or want to add anything, feel free to add comments. I have ordered a copy of the classic work on the subject, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, and so hope to be more fluent in this area at a later date.

Bayesian Belief Propagation (BBP) was formulated by Judea Pearl, who later authored Probabilistic Reasoning in Intelligent Systems. It appears to be a system for determining unknown probabilities within certain types of complex systems. Let's start with a review of a very simple probability model.

Much of early probability work was formulated in terms of a pair of ordinary gambling dice, six-sided cubes with faces numbered one through six. Thrown many times, they make a nice distribution pattern of numbers two through twelve. It can be shown with a little math and logic that there are percentages that are expected for each of these numbers. If you make a large number of dice throws, say more than 100, and you get results that differ by very much from the expected distribution of results, you may be looking at one of two things. It is possible that you have a distribution skewed purely by chance. On the other hand, you might want to investigate whether your dice and throwing systems are physically constructed to be truly random in the results they give.

Now imagine a black box, say the size of a shoe box. If you shake the box and then stop shaking, it gives you a number. In fact, imagine a room with a variety of such boxes. You shake them, keeping tracks of the resulting numbers for each. If one box consistently gives numbers between 2 and 12 inclusive, and the frequencies for each number match the pattern that we believe two dice would give, then we can conclude that the internal mechanism of the box, whatever it is, has a probability distribution that is created by the same logic that creates the dice probability distribution. There might even be a tiny pair of nano-dice inside.

In computer programming we tend to think in purely causal mechanics. If something does not happen as expected, we have a bug in our program. But in the case of living groups of biological neurons we cannot (so far) look at the inner mechanics of each neuron while watching how the whole group operates. However, if we can control the inputs and read the outputs of a system, then, if we can do the math, we might figure out the probabilities of individual neurons firing given known inputs from the other neurons in the system.

Formally, BBP allows us to calculate the probability characteristics for nodes we can't observe directly, if we can observe enough of the rest of the system. A node does not need to be a neuron. In an AI system, a node is a processing unit with inputs, outputs, or both, that connect it to the system of nodes in question. In our case the nodes are part of a Bayesian network, also known as a belief network. Communication (or messaging) within the network is in terms of probabilities. (See also Bayesian probability)



Think of a network of three people, where each of the people represents a node. Let there be a piece of paper we will call the message (it is blank; the paper is the message, in this case). Let each person have a single die and the rule: if you have the message, throw the die. If it comes up a one, a two, or a three, pass the message to the left. Otherwise, pass it to the right. If we watch the game progress, the message moves around or back and forth in a random pattern. However, it is not a Bayesian network, because it is cyclical. Bayesian networks are not cyclic. Messages more only in one direction.

Now start a series of games in which the message is always handed to person 1 (node 1). Person 1 throws the die and passes the message to person 2 if the result is 3 or less, and to person 3 if the result is 3 or more. Person 2 also throws a die and simply holds the message if the result is 3 or less, but passes if to person 3 if the result is 3 or more. When person 3 gets a message the game is over.

It is a simple game and a tad boring, but if we focus on the propagation time, the number of die throws from the beginning of a game until the end of a game, we can see a set of variations. Over the long run they should form a probability distribution of the times.

Now change the rules slightly. Say person 2 forwards the message if the die comes out 2 or higher. The range of possible results does not change, but the probability distribution will.

Now suppose you know about this type of game, but you can't see the game because it is in a tent. You see a message handed in, and can time how long it is before the message is handed back out by person 3.

By keeping track of the times, you can get the probability distribution. That should tell you the exact rules of the game being played, including the probability that person 1 will give the message directly to person 3, and how long person 2 might keep the message before passing it.

In more complex networks, you need to pin down the probabilities at more of the nodes, if you want to be able to characterize the probability rules for an unknown node.

Which is what the BBP algorithm is supposed to do. It calculates the probabilities for an unobservable node based on the observed nodes.

Markov Chains for Brain Models

I got an email from Numenta today, telling me that Dileep George and Jeff Hawkins had a paper, "Towards a Mathematical Theory of Cortical Micro-circuits", published in the PLoS Computational Biology journal. Part of the abstract states:
“we describe how Bayesian belief propagation in a spatio-temporal hierarchical model, called
Hierarchical Temporal Memory (HTM), can lead to a mathematical model for cortical circuits. An HTM node is abstracted using a coincidence detector and a mixture of Markov chains.”

HTM is the model being used by Numenta and others to crack to machine understanding problem by making an abstract, computable model of the human cortex (brain). I figured the article would explain “Bayesian belief propagation” which parses into three words I understand. I knew I had run into the term “Markov chains” before, in probability theory and elsewhere. I thought I would just refresh my memory about good old Andrey Markov with Wikipedia. But the Markov chains article at Wikipedia was tough going, jumping into the math without clearly explaining it in English. The definition there was comprehensible: “a random process where all information about the future is contained in the present state.” That implies there is a set of rules, expressed as probabilities, so that if you know the current state of a system you can predict its future state probabilities. It neatly combines determinism with indeterminism. For a first impression, I found the Examples of Markov Chains article more helpful.

I thought I might find a better explanation on my book shelf. There was nothing in my Schaum’s Statistics, but my next guess (thank you, cortex) Schaum's Outline of Operations Research
by Richard Bronson and Govindasami Naadimuthu, has an entire chapter, “Finite Markov Chains,” starting on page 369.

Markov chains are a subset of Markov processes. Markov processes consist of a set of objects with a set of states for the objects. At each point in time each object must be in one of the states. As time progresses, “the probability that an object moves from one state to another state in one time period depends only on the two states,” the current state and the probabilities for the future states given the current state.

For example, in the board game Monopoly each “square” or piece of real estate is part of the set of objects. The two dice create a probability structure, the standard one for two dice. The probability that, at any point, a player will transition to the same point, the forward adjacent point, or any point beyond the 12th point forward is zero, because the dice can only roll numbers between 2 and 12. However, more typically there would be a set of states and a probability for the transition to each of the states, including the transition of remaining in the current state.

If there are N possible states, and you know all the probabilities for transitions, you can show this and manipulate it mathematically with a matrix P, the stochastic or transition matrix. There would be N rows and columns in P. As in all probability theory, in any row the probabilities add up to 1.

N=2 is the simplest meaningful number of states. Let’s call them black and white. It is really only interesting if the transitions are different. Say 90% of the time black transitions to white, but only 20% of the time white transitions to black. Then the matrix P would be

.1 .9
.8 .2

There is some ability to predict the future in a statistical way if you have a Markov chain like this. Without doing any math, you would expect over time that if you have only one object, it would probably be white. If it is black, probably the next state of the sequence flips to white, but if it is white, probably the next state will be white. If you have a large number of objects, you can know the likelihood that a given percentage of them will be white or black. Yet you can’t know for sure. Even the least likely state, all objects black, can occur at any time if each black stays black but each white transitions to black.

You can see that Markov chains might be used to represent the states of neurons or of collections of neurons. In my simple examples above, each object influences only its own future state. But in real systems often future state probabilities are determined by the states of a collection of objects. It could be a collection of transistors, or a collection of nerve cells, or HTM nodes.