I'll admit, I looked over https://probmods.org and I don't really get "probabilistic programming". It just looks like you call random() sometimes...? Is there something going on in the language runtime that I'm missing?
The big deal is being able to run the program backwards. I.e., you already have some set of output, obtained from some other source, and you want to figure out how that output could have been produced (i.e., how the source works). So, you write a probabilistic program that encodes your various hypotheses about how the source process works, and then run it backwards to figure out what the forward execution path would have had to have been to produce that output, and thus which hypothesis is correct. That's what "inference" refers to.
Inference is a basic process in scientific investigation- come up with a bunch of hypotheses, do an experiment that generates data through physical processes independently of whatever you hypothesized, and then go backwards to figure out which of your hypotheses would have had to have been true in order to get the experimental output that you did. Which hypothesis is true corresponds to a particular execution path through a probabilistic program, so having efficient implementations of probabilistic languages means we could do much better at accurate analysis of data, which is Good For Science.
What does a compiler-level implementation of probabilistic analysis have to do with accuracy?
As a secondary point, re-implementing inference code for each new model makes it almost certain that there are bugs in said code. So even without changing the model, automatically generated inference code is likely to have fewer bugs and thus give more accurate inferences than hand-written code. (assuming it runs to convergence; naturally there are lots of scenarios in which naively generated code will be slower to converge than something hand-tuned).
Both of those result in the potential for greater confidence in the accuracy of conclusions reached from experimental analysis.
Real problems in the wild are a mix of both types of problem, and intractable part and a part you can do precisely.
In an ideal world you would use the iterative methods ONLY for the parts that don't have an exact solution available. In practice once you have developed the general iterative solution, you might as well just use that for the easy parts too, as you are just burdening yourself with more development work and more scope for bugs by writing two inference implementations.
A language that allowed inference to be switched at no development cost it would be amazing
Really grossly impressively inefficient.
(1) Even very simple linear models in effect encode uncountably many hypotheses (i.e., y = a x + b where a and b are real numbers). You can't just pull (a,b) out of a hat until the agreement looks good, because the hat is too big.
One tool is, obviously, analytic solutions (where possible) like least squares. Beyond that, iterative methods like the EM algorithm. Beyond that, sampling-based methods like MCMC. You can view MCMC as your idea ("generate a bunch of hypotheses and test them"), but with the change that it does not ignore the old hypotheses when generating a new one to test, it uses an old hypothesis to seed a new one, using a probabilistic update rule.
In the presence of a model, you can basically compile (in the CS sense) an inference program that deduces model parameters from input data. The project to do this has been worked on, in fits and starts, since the late 1940s, by a series of geniuses/visionaries.
(2) Once you have a multi-level model, with many parameters, trading off what you gain by tweaking one parameter versus another parameter becomes hard. If the data is
(x1,y1), ..., (xN,yN)
M1: (x1,Z1_1), ..., (xN,Z1_N), i.e. Z1_i = M1(x_i), and
M2: (x1,Z2_1), ..., (xN,Z2_N), i.e. Z2_i = M2(x_i)
So even if you were content to keep generating hypotheses and checking agreement, you wouldn't know how to measure agreement.
Let's say you want to determine if a coin is fair, but you don't know any statistics. What you can do is write a program (model) that represents the fair coin flip process. It might be as simple as:
flips = [random.choice([True, False]) for x in range(0, 10)]
i am speaking from what i gathered reading the article and the other comments. seems pretty interesting to me.
What I'm trying to say is that at the moment it looks like every programming language ever is trivially capable of expressing these sorts of programs using the random number generators that are wildly common. What is different in this programming paradigm?
In particular, this paragraph:
> Another way of thinking about this: unlike a traditional program, which only runs in the forward directions, a probabilistic program is run in both the forward and backward direction. It runs forward to compute the consequences of the assumptions it contains about the world (i.e., the model space it represents), but it also runs backward from the data to constrain the possible explanations.
Are these programs truly running backwards? Is there some backwards reduction relation in their semantics that I could look at?
Presumably, a programming language designed around this kind of task would make it easier to explore and develop larger, more intricate models.
Conceptually, the easiest way to understand things is via rejection sampling. You can "run things backward" but running things forward and just keeping the traces that match the observed output. That's Bayesian inference. Unfortunately, the probability of matching the observed output is zero for most problems so you need to use more advanced methods. Most accounts gloss over these methods since it's a huge research area.
Running things backwards is also not a perfect analogy because if fails to account for the probability distribution you start with.
In one of my lives, I've calculated ballistics. When calculating ballistics, you can probably think of a lot of factors that might go into it.
- Release conditions (velocities, angles, ect...)
- Environmental / trajectory conditions (winds, gusts, different media, ect...)
- Impact conditions and behaviour (spalling, rebounding, fragmentation, yatta....)
For more reading, you can look at something like: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/2007002... (only uses probabilistic methods for transport)
Anyhow, one way to attempt modelling of this is to run a Monte-carlo of the parameter space. Effectively, you run all the permutations of release, all the permutations of those permutations for transport, and then all the permutations^3 of the impact variation.
Now, one of the advances that has come about from efforts like that, is that probabilistic models for some of these steps have been created to approximate the results in a time effective manner.
For example, in the above, they note that instead of running a Monte-carlo of trajectories and releases, a single trajectory is instead run, and then a probabilistic flight cone is created which calculates parameters based on standard deviation from the mean.
A general, and robust version of a language like this would allow you to specify all of your parameters with similar types of effects. IE, you would have mean and then deviation or statistical curve envelopes you would provide around that mean which would be propagated through calculations, and spit out an overall convoluted probability at the end of the pipeline.
In ballistics for example, you might have a mean release condition, with statistical variation on release. That would then be fed into a trajectory calculation, which would do so based on a mean, but with a transport cone that does statistical variation on the mean flight. Finally, impact might be calculated around a mean value, but would then carry the statistical variation through the release and flight. At the end, you would then be able to pull a probability number, as well as impact condition at that probability from any point in the potential impact zone.
At least that's how I see the use of probabilistic languages.
At its simplest, the forward problem is like a functional composition:
y = F( G( H(x) ) )
@gliese1337 nearby, in an excellent comment, addresses the more complex inverse problem: when you have outputs (y's above) from such a process, how do you determine the most likely model (F, G, H) that produced it?
So for example, if you have a bunch of crater ejecta, what can you tell me about the mass of the incoming objects?
- Start with a value + divergence and then propagate that state through your transforms to get a mean + divergent final state.
I should note that in the given example the base divergences need to be found through some method, which is usually running a Monte-carlo around some baseline state, and then making the significant assumption that new calculations (with small deltas) follow roughly the same statistical behavior.
For the backwards case, that can be a significantly more challenging case, as it basically comes down to performing approximate eigenvalue decomposition. However, there are a lot of SOTA tools like pretty much anything from this list:
- http://en.wikipedia.org/wiki/Pattern_recognition#Algorithms
IE, the crater example is fairly close to the setup for a Hidden Markov Model, where we get final data, and assume that it is based on a set of hidden states. Or for lighter reading, almost any of the major tools used in the Netflix recommendation contest, which is effectively an enormous version of this problem.
Hmm, not in general. (I've been working on various problem-specific versions of the backwards case for most of my career.)
Disclaimer: my lab mate has been organizing these workshops for the past couple of years and I'm currently procrastinating from preparing a presentation on this material for a reading group later this week, so I've presently got mixed feelings towards it.
Years ago when I was a grad student, I tried to tackle some NLP problems using similar approaches (ie. unsupervised generative bayesian models organized in hierarchical ways that try to mimic our intuitive concept of causality) but mostly failed at making much headway. I couldn't figure out some of the math and my models usually failed to perform any better than other probabilistic algorithms. No one at my university did anything similar or knew much about bayesian probabilities back then which didn't help.
I always believed in the approach however and until I stumbled on this book, never knew there was significant progress being made. I wish it was available to build upon when I was a student.
In the "Learning as Conditional Inference" chapter the section "Inferring an Arithmetic Expression" show the first example I have seen of a legitimate and flexible approach that could some day allow a computer to program itself. I mean the algorithm is still far from doing general programming tasks but I could imagine extending the approach so that given a programming language grammar and a unit tests as inputs, a computer could write a well chosen and simplified program that pass the tests (simplified with the automatic Occam's razor inherent in the bayesian approaches) . That to me is mind-blowing.
Now not all is perfect in the book, the "Inference about Inference" chapter seems oversimplified to me. The last few chapter seem to get a bit mathematically ad-hoc with an alpha parameter introduced to tie categories and sub-categories together in hierarchical models without much mathematical justification. On the other hand, there is strong evidence for this kind of transfer of information in hierarchies and it's not totally unreasonable to make up a parameter, assign it a bayesian uninformative prior that reflects ignorance of how this parameter should be set and try to induce a value for it from data. It's a perfect example of meta probabilities where probabilities are used to guess parameters that govern other probabilities. It's very bayesian.
Anyways before reading this book I was convinced that general AI was not going to happen in my lifetime. Now I'm not so sure.