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Stream: deprecated: statistics reading group

Topic: McCullagh (2002)

Tomáš Gonda (Jun 27 2020 at 18:22):

Let's discuss anything related to McCullagh's "What is a statistical model?" paper here.

Peter Arndt (Jun 30 2020 at 11:01):

I am totally lost with McCullagh's exercises.

Peter Arndt (Jun 30 2020 at 11:04):

In each case he provides a model in the sense of his first paragraph, i.e. a family of distributions. These are meant to be the possible distributions of what is called the "response variable" or just "response", right?

Peter Arndt (Jun 30 2020 at 11:05):

I am unsure about the terminology here.

Peter Arndt (Jun 30 2020 at 11:14):

I am also not sure what "covariate" means. I guess it is another random variable, and the response variable is meant to depend on the "covariates"? That's what got from googling the term.
But in which way should it depend on the covariates? Through a deterministic function? Through a stochastic function? Is the latter the same as saying that the parameters of the distribution of the response variable depend on the covariates?

Peter Arndt (Jun 30 2020 at 11:18):

Also I don't know what McCullagh means when he writes "inference is required" and "predictions are required". In exercise 1 these two expressions appear side by side, so I guess they mean different things, but what?

Peter Arndt (Jun 30 2020 at 11:20):

I hope you are not too shocked by the level of my ignorance. Maybe part of it is just linguistic - the little statistics that I know I learned in German...

Britt Anderson (Jun 30 2020 at 13:42):

My take on things was more "meta". In one of the examples you have an even N (say 100), fit a logistic function, and then apply a probit function to predict the 101st participant. It seems absurd to use the data to fit one function function and predict the data of a future subject with a different one; but since all the details were specified at the outset you have a well-defined statistical model that maps the \Theta to a probability distribution. Subsequent examples are increasingly subtle demonstrations of the same idea. To me, this seemed to go nice with the Kass article where it emphasizes how non-statisticians often focus on the details and mechanics of a statistical procedure, while the statistician is thinking about the larger question. Of course none of this has an obvious question to category theory so my reading may be way off.

Matias Bossa (Jun 30 2020 at 21:54):

I agree with Britt, however I will try to answer Peter questions because those terms are very common in statistics and will probably show up latter.
"Response" ( aka "outcome"), is a random variable we can observe, is of interest, and is affected/influenced by other variables. We are interested in quantify how those other variables affect the response. For example, a binary response could be "is he infected with the virus?", lets call o_i the observed response for subject i.
"Covariate" are relevant variables, either observations or something that can be manipulated, that could affect the response and, in general, are not affected by noise (i.e. are measured with infinite precision). For example: age, gender, does he receive vaccine or placebo.
"Treatment" is one of the covariates that is of greatest interest and which can be manipulated. For example: vaccine/placebo.
These models are usually linear in the covariates, then we can asume some fixed (unknown) coefficients multiply each covariate: a x_i + b y_i + c z_i, where x, y and z are the covariates, the index i denotes the subject, and a, b and c are the coefficients, i.e. the parameters of the model.
The model is then: o_i ~ Bern( logit^{-1} ( a x_i + b y_i + c z_i ) )
"Inference" (also estimation) is the process of computing an estimate for the unknown model parameters given some observations, in this case the only unknown model parameters are the coefficients associated to each covariate: a, b and c. The inference could be: the computation of an estimator, in a frequentist setting; or the estimations of the posterior probability density, in a Bayesian setting (or some summary of it, e.g. mean, credible interval, etc).
Finally, regarding the last sentence, for a new subject k with known covariates (x_k, y_k, and z_k) but unknown response or outcome (i.e. we don't know if he is infected), we can predict the probability of being infected o_k as p( o_k) = logit^{-1} ( a x_k + b y_k + c z_k ).

The links function is logit or probit depending on the number of subjects, which is absurd. I'm wondering if one can ask the model to behave "well" or "smooth" in order to rule out these models. E.g. a small change in parameters (in this case n, the number of subjects) should produce a small change in posterior probabilities. I think the source of absurdity in this and in the following examples is the dependence of the model on aspect of data (the number of observations could not be part of the model).

Hope it helps

Peter Arndt (Jul 01 2020 at 08:19):

Big thanks to both of you - that was super helplful!

Oliver Shetler (Jul 11 2020 at 19:38):

Hi everybody. Thanks for the great meeing!

We've settled on the following format going forward.

We will have "official" meetings every other week, where we will meet to discuss the assigned readings and exercises. At these meetings, we will also attempt to draw at least one conclusion that is worth posting on the wiki, and we sill decide on our next assignment.

On the weeks where there is no official meeting, we will have an informal study session for people to work on the exercises together, and ask questions. These meetings will just be for people who want extra feedback via video chat. We will not add additional assignments.

Lastly, the assignment for the next meeting is as follows:
(1) Go over sections 3 and 4.
(2) Compare and contrast McCullagh's categorical definition of a statistical model with the standard one given in section 1.
(stretch goal) Explain why the example in Exercise 1 (or an exercise of your choice) fails to satisfy the categorical definition of a statistical model. Diagrams are encouraged.

Andrew Shulaev (Jul 12 2020 at 00:24):

I'm a bit confused: we have already started looking at the Chapter 4, is the plan for the next meeting to go back and go through Chapter 3 then?
Also, the time for the next meeting seems to be different from the last one, is that change intentional?

Peter Arndt (Jul 12 2020 at 01:09):

Chapter 3 is very short and mostly introduces notions and nomenclature for chapter 4, it seems. There will probably not much to discuss, but it will aid in the reading of chapter 4, I think.

Oliver Shetler (Jul 12 2020 at 01:36):

Peter Arndt said:

Chapter 3 is very short and mostly introduces notions and nomenclature for chapter 4, it seems. There will probably not much to discuss, but it will aid in the reading of chapter 4, I think.

Yes. We took a first look at Section 4, but we haven't really fully understood his categorical definition of a statistical model. The goal is to fully process this definition; compare and contrast it with the conventional definition; and if you have time relate it to at least one exercise. Looking at and really reading a section are two different things.

Andrew Shulaev (Jul 12 2020 at 01:45):

Okay thanks for clarification, what about the time change part?

Oliver Shetler (Jul 12 2020 at 05:16):

Andrew Shulaev said:

Okay thanks for clarification, what about the time change part?

Thanks for pointing out the time thing.

That was inadvertent. For some reason the meetup app always reverts to the default time of 7pm EST when I "copy" an event and edit it for the next meeting. Sometimes I forget to change it.

Andrew Shulaev (Jul 12 2020 at 13:22):

Awesome, thanks for clarifying!

Andrew Shulaev (Jul 12 2020 at 18:33):

I've spent a bit more time thinking about the first example in the paper, and want to clarify a couple of things. As far as I understand, logit and probit links are just two different functions $\mathbb R \to [0; 1]$, and the exact difference between them is not particularly important.

Let's denote $X$ to be a covariate space. I interpret the exercise statement to say that the mapping $P : \Theta \to \mathcal P (X^k)$ would depend on parity of $k$ (using one of two different functions). However, per introductory definition a statistical model is one function, but here we have an indexed family of functions, right? Generaly, I think it makes sense for statistical model to be an indexed family so that using different sample size would not technically change the model used, only an index it is instantiated with; and my understanding is that this implicit extension of definition breaks when the dependency on the index is some weird one.

The part I'm not sure about is that the model is indeed a mapping $P : \Theta \to \mathcal P (X^k)$, the other option being $P : \Theta \to \mathcal P (X)$, but if the second option is true I'm not sure I understand how to formulate the problem of inference such that it would be possible to use different link function depending on a number of samples.

Michael Fishman (Jul 25 2020 at 16:46):

Hi Andrew,

I believe you have the wrong definition of a model - a model maps each point in parameter space $\Theta$ to a distribution over sample space $\mathcal{S} = \mathcal{V}^{k}$, not to a distribution over $X^k$
%from page 1236, a statistical model is a mapping from parameter space to distributions over sample space: $\Theta \to \mathcal{P}(\mathcal{V}^{\mathcal{U}})$

About exercise 1:
Consider two designs from exercise 1, $(k \to \Omega)$ and $(k+1 \to \Omega)$. (I'm using $k$ to refer to a set of units with $k$ elements, and $k+1$ to the set with one additional unit)

On page 1236, it says that if the map between covariate spaces is the identity (as it is in exercise 1, since it does not mention changing the covariate space), then the map between distributions over sample spaces is given by marginalization. But since probit and logit are not equal, the map $\mathcal{P}(\mathcal{V}^{k+1} \to \mathcal{P}(\mathcal{V}^{k})$ is not given by marginalizing out the extra unit, so exercise 1 isn't a valid model.

Arthur Parzygnat (Jul 25 2020 at 16:48):

I have to admit I am still struggling quite a lot. I can't come up with a single rigorous example for the definition of a linear or statistical model, and having such an example would be incredibly important to keep track of the abstract definitions in my opinion. A simple example like a coin toss, the height of people in a room, or a linear regression would be fantastic, but I have a hard time figuring out what all the objects are. Here's my attempt to define it for the coin toss since that sounds the simplest (I am using the notation from page 1236 in McCullagh). The response scale will be $V=\{0,1\}$, where $0$ is for tails and $1$ is for heads. Let $\Omega:=\{T,H\}$ (where $H$ stands for heads and $T$ for tails) be the covariate space. Let $U:=\{1,\dots,n\}$, with $n$ a natural number, be the set describing the number of coin tosses. This is one set of units. The sample space for $U$ is $V^{U}=\{0,1\}^{\{1,\dots,n\}}$, and it describes the set of possible outcomes for tosses with numerical value $0$ associated to a toss of tails and $1$ to a toss of heads. Let $\Theta_{\Omega}:=[0,1]$ be the unit interval. A design $\psi:U\to\Omega$ describes a particular experimental outcome? If we have several more coin tosses, we can embed $\varphi:U=\{1,\dots,n\}\hookrightarrow\{1,\dots,n,\dots,n'\}=U'$. The covariate space does not change in this setup since we only allow heads or tails to be observed. As for the statistical model, for each design (experiment) $\psi:U\to\Omega$, we associate a stochastic map $P_{\psi}:\Theta_{\Omega}\rightsquigarrow V^{U}$, which in this case associates to each $p\in[0,1]=\Theta_{\Omega}$ a probability measure on $\{0,1\}^{\{1,\dots,n\}}$. But this is a little strange to me. Why wouldn't we want to instead provide a probability distribution on the covariate space $\Omega$ which, given the experiment $\psi$, describes the probability for the possible outcomes? Why do we instead associate a probability measure on the entire sample space, which is a function space? As one can tell, I'm greatly confused about some things here...

Michael Fishman (Jul 25 2020 at 16:59):

I believe the covariate space is the space of attributes based on which you are trying to predict the response, so $\Omega = \{T,H\}$ is a bit odd

Instead, we could set $\Omega$ to be the kind of coin we are flipping $\Omega = \{nickel, quarter\}$. Then the design would be a set of coins $\{1,...,n\}$ and a map from this set of coins to their type (nickel, quarter)

Oliver Shetler (Jul 25 2020 at 16:59):

Here are some notes and comments on McCullagh's paper courgesy of Paul.

cattheorycomments25July.pdf

Peter Arndt (Jul 25 2020 at 17:01):

I can't find the link anymore - could someone point me there?

Peter Arndt (Jul 25 2020 at 17:01):

The link to the Zoom meeting, I mean

Michael Fishman (Jul 25 2020 at 17:02):

Same here :upside_down:

Arthur Parzygnat (Jul 25 2020 at 17:02):

https://condenast.zoom.us/j/807569857

Peter Arndt (Jul 25 2020 at 17:02):

Thanks!

Arthur Parzygnat (Jul 25 2020 at 18:41):

I think it'll be easier for me to go through Paul's two examples and expand on them rather than trying to do the one I suggested above. If I make any progress, I'll post here. I would encourage anybody else to do similarly if they gained some insight.

Arthur Parzygnat (Jul 27 2020 at 10:12):

I think this was mentioned during our meeting on Saturday, but I have personally found the discussion by Hans Brons beginning on page 1279 to be very helpful from the categorical point of view. For example, at some point McCullagh calls his statistical models functors and at other times natural transformations. Brons clarifies some of those points. In general, I think it's a good idea to read the entire article and the discussions briefly before focusing too heavily on a specific section in case other parts of the article elucidate some points of confusion.

Arthur Parzygnat (Jul 27 2020 at 13:30):

I meant to have many of my previous comments in this stream. If someone has the power, could they move the McCullagh discussion from the #general stream to this one? Anyway, I noticed in Peter Huber's response to McCullagh's paper, he mentioned that he wrote a 10-12 page paper of definitions based on statistical models from a categorical POV back in the early 1960's. He never published this note. Does anyone have access to it? For example, has he made this available online?
Also, Huber mentioned the Blackwell--Sherman--Stein theorem stating that he did not see how category theory could elucidate its proof and in fact ended his discussion with a warning, which we should keep in mind. However, I wonder if @Tomáš Gonda and others have been making progress on exactly this theorem and if they have thoughts on this (I only mentioned Tomas here because I don't think any of the other collaborators are attending our meetings)?

Paolo Perrone (Jul 27 2020 at 14:50):

Arthur Parzygnat said:

I meant to have many of my previous comments in this stream...

Moved!

Also, Huber mentioned the Blackwell--Sherman--Stein theorem stating that he did not see how category theory could elucidate its proof and in fact ended his discussion with a warning, which we should keep in mind. However, I wonder if Tomáš Gonda and others have been making progress on exactly this theorem and if they have thoughts on this (I only mentioned Tomas here because I don't think any of the other collaborators are attending our meetings)?

Where did Huber mention the BSS theorem? I'd love to read that. And yes, we are making progress exactly on that theorem, using category theory :) (The expert there is indeed @Tomáš Gonda.)

Arthur Parzygnat (Jul 27 2020 at 17:26):

@Paolo Perrone , thanks! Also, Huber only mentioned it briefly in his discussion regarding McCullagh's paper on page 1291 (link here: https://projecteuclid.org/euclid.aos/1035844977) at the end of the second last paragraph on that page. I was really surprised to see it mentioned especially since I just heard a talk on it by Tomas a few weeks ago :).

Tomáš Gonda (Jul 27 2020 at 17:57):

It would be indeed nice if someone had those notes! I tried a couple of times to delve into Le Cam's book, but was always left with an impression that to understand what is going on it would help to translate its elements to the language of category theory (and it may indeed be possible, I just haven't done so).

Interestingly, the birth of this project on Blackwell--Sherman--Stein theorem is very much in the same vein as what Huber mentions as his motivations. Namely, I tried to understand the proof of the theorem and in the process of doing so translated it to the language of Markov categories. There are two key features of the framework of Markov categories that allow to elucidate this theorem in my opinion. One is the string diagrammatic language together with its natural interpretation; the other is the synthetic nature of the proof - knowing which axioms are necessary and for what purpose makes the proof much clearer in my mind.

Tomáš Gonda (Jul 27 2020 at 17:57):

This actually touches a bit on @Britt Anderson's question in the last meeting about what the categorical point of view can bring to the table that I didn't get to share my view on. With regards to that, I personally see a lot of value in scientists partaking in "packaging knowledge for future generations" so that those can build on what is known, and this process often need not involve new results in the naive sense. That's where I see a lot of value of category theory. This may of course entail the development of "new knowledge" as a second-order effect. Anyway, this is a bit off-topic, which is also why the discussion didn't go in this direction last Saturday I guess.

Arthur Parzygnat (Jul 27 2020 at 18:03):

So I haven't read your draft yet, but I'm assuming that axioms such as causal, positive, Markov categories, or those with conditionals, etc. are the types of structure you're referring to in formulating the statement/proof of the BSS theorem? Ah yes, I now remember from your talk the existence of conditionals was crucial.

Paolo Perrone (Jul 27 2020 at 20:37):

Tomáš Gonda said:

This actually touches a bit on Britt Anderson's question in the last meeting about what the categorical point of view can bring to the table that I didn't get to share my view on. With regards to that, I personally see a lot of value in scientists partaking in "packaging knowledge for future generations" so that those can build on what is known, and this process often need not involve new results in the naive sense. That's where I see a lot of value of category theory. This may of course entail the development of "new knowledge" as a second-order effect. Anyway, this is a bit off-topic, which is also why the discussion didn't go in this direction last Saturday I guess.

Here's a quote from Sir Michael Atiyah, which I learned from Emily Riehl's book "Category Theory in Context":

The aim of theory really is, to a great extent, that of systematically organizing past experience in such a way that the next generation, our students
and their students and so on, will be able to absorb the essential aspects in as painless a way as possible, and this is the only way in which you can go on cumulatively building up any kind of scientific activity without eventually coming to a dead end.

Britt Anderson (Jul 29 2020 at 20:42):

I find the clarification of goals by @Paolo Perrone and @Tomáš Gonda very helpful for me, someone from outside the community, to understand what an ACT theorist means by the word "applied". Often applied sciences are seen as ones where basic knowledge is given a practical use. From that point of view one might expect category theory applied to statistics (which is kind of applied probability theory) to lead to new statistical tests, a better understanding of what statistical procedures to use when, or practical advice for experimentalists on experimental design choices they might make to enable subsequent statistical procedures - as just a few examples. But I gather now that this would be considered, to poach a phrase I heard at an ACT 2020 presentation, as applied-applied category theory. I think the purpose outlined by Paolo and Tomáš worthwhile and valuable and needing no further justification, but it helps me to manage my expectations to understand what the CT community sees the word "applied" to mean. It also makes it much less surprising that working statisticians and academic statisticians might not be interested in this approach if it is not seen to offer any prospects of changing their day to day statistical work and research, but only to offer a potential future benefit in consolidating existing knowledge into a larger coherent mathematical framework. As someone who follows this discussion to push my own thinking about how human cognition, especially as it relates to probabilistic learning, might be modeled and theorized about I now wonder if applied category theorists see CT merely as one of many potential languages for the "systematic organizing" or "packaging [of] knowledge for future generations" or in some sense the one, true, or "right" one for this project?

Steve Huntsman (Jul 30 2020 at 05:46):

Britt Anderson said:

I find the clarification of goals by Paolo Perrone and Tomáš Gonda very helpful for me, someone from outside the community, to understand what an ACT theorist means by the word "applied". Often applied sciences are seen as ones where basic knowledge is given a practical use. From that point of view one might expect category theory applied to statistics (which is kind of applied probability theory) to lead to new statistical tests, a better understanding of what statistical procedures to use when, or practical advice for experimentalists on experimental design choices they might make to enable subsequent statistical procedures - as just a few examples. But I gather now that this would be considered, to poach a phrase I heard at an ACT 2020 presentation, as applied-applied category theory.

If we take the n-philosophy to heart here we can even imagine infinity-applied category theory, wherein category theory is applied directly in service of real-world problems versus merely other parts of mathematics /s

Arthur Parzygnat (Jul 30 2020 at 08:16):

@Britt Anderson , I see applied category theory (ACT) slightly differently. I don't see ACT as a subject to "consolidate existing knowledge into a larger coherent mathematical framework." It is true that category theory helps me organize things, but that's far from my main goal of using it. This speaks more about the way that I think, how I organize my thoughts, and how I approach problems rather than what problems I hope to solve or what I hope to do with category theory. I would say my main drive is that once you have understood something from a different perspective, it may offer more apparent generalizations than were thought previously. These generalizations may develop into their own largely independent subjects, but such generalizations may also have the interesting benefit of helping us discover or solve problems in our initial domain of interest. This doesn't always happen, but I think it is worth the investigation, and we won't know until we try. Note that this phenomenon happens with many (all?) fields of mathematics and physics (those are two subjects I can vouch for). But category theory seems special to me because it has a way of bringing vastly different subjects together in a way that other fields of mathematics could not.

Britt Anderson (Jul 30 2020 at 11:04):

Tomáš Gonda said:

It would be indeed nice if someone had those notes! ... Interestingly, the birth of this project on Blackwell--Sherman--Stein theorem is very much in the same vein as what Huber mentions as his motivations.

I tracked down an email for Huber and wrote to ask him about the notes. I will let you know if he responds - he is likely in his late 80s. Huber was an amazing scientist. I found this oral history that makes for entertaining reading. https://projecteuclid.org/euclid.ss/1215441288

Morgan Rogers (he/him) (Jul 30 2020 at 14:38):

Britt Anderson said:

I wonder if applied category theorists see CT merely as one of many potential languages for the "systematic organizing" or "packaging [of] knowledge for future generations" or in some sense the one, true, or "right" one for this project?

A resounding yes to this, although I am pretty biased, as might anyone else here be :stuck_out_tongue_wink:

Morgan Rogers (he/him) (Jul 30 2020 at 14:55):

Britt Anderson said:

From that point of view one might expect category theory applied to statistics (which is kind of applied probability theory) to lead to new statistical tests, a better understanding of what statistical procedures to use when... But I gather now that this would be considered, to poach a phrase I heard at an ACT 2020 presentation, as applied-applied category theory.

My instinct was to argue against this, but when I tried to draw analogies I realised that this distinction exists in other scientific domains too! At some point, any theoretical work in physics or chemistry gets which is to be applied gets passed onto someone who considers themselves an engineer. However, the stage that a theory has to be at in order for this transition to occur is usually pretty advanced. So perhaps an applied category theorist is someone who builds up theory in a direction that "categorical engineers" may eventually be able to get a handle on, to develop the concrete advances in statistics that category theory has the potential to deliver?

Britt Anderson (Jul 30 2020 at 18:12):

Dr. Huber was incredibly quick to reply. He said at the time he wrote his commentary he had already lost the notes on Le Cam, and that at this point he feels he has difficulty bringing up the details of this category theory past. So, one of you will just have to do it. I did give him the link to the zulip, but I don't know if he will peak in.

Britt Anderson (Jul 30 2020 at 18:15):

@[Mod] Morgan Rogers wrote:

A resounding yes to this, although I am pretty biased, as might anyone else here be :stuck_out_tongue_wink:

I assumed as much, but am always interested in why? I don't have the expert's perspective, and I would appreciate reading it. Although it may be off topic here and this thread is now "meta" enough that it could be moved elsewhere.

John Baez (Jul 30 2020 at 20:39):

Britt Anderson said:

Often applied sciences are seen as ones where basic knowledge is given a practical use.

Yes. In the case of category theory, basic knowledge can be applied in many different ways.

Often it provides a new way of thinking that can help clarify concepts that were left inexplicit in existing work: "tacit knowledge" that practitioners have, that will need to be formalized if we want to use software to do what they're doing.

Often it provides a high-level view that makes it easier to connect work from one field with another field, "bridging the gaps" or "reducing impedance mismatch" (choose your metaphor).

It can also be used to help design new tools. This tends to take a long time and lot of thought, and this is just getting started. The industry presentations at ACT2020 illustrate how some companies are using category theory in this way.

From that point of view one might expect category theory applied to statistics (which is kind of applied probability theory) to lead to new statistical tests, a better understanding of what statistical procedures to use when, or practical advice for experimentalists on experimental design choices they might make to enable subsequent statistical procedures - as just a few examples.

It might someday do that. It's equally likely to do things that you wouldn't expect: not "more of the same, only better", but something really new.

What's happening right now is a dramatic rethinking of basic concepts in probability and statistics using category theory. We saw that at the conference Categorical Probability and Statistics, which happened online with discussions here. We'll probably see more of it in the reading group here.

It also makes it much less surprising that working statisticians and academic statisticians might not be interested in this approach if it is not seen to offer any prospects of changing their day to day statistical work and research...

There's no need for them to pay attention now unless they want to try something new. They can just wait and see if anything exciting happens.

Bradley Saul (Aug 02 2020 at 14:08):

Would anyone be interested implementing McCullagh's paper in software, perhaps using @Evan Patterson's CatLab.jl? I find that I learn a lot by translating math to code.

Javier Prieto (Aug 02 2020 at 15:40):

@Bradley Saul I'd be interested, but I don't have a good sense of how much work it'd be. Do you have any thoughts on that? For the record, I have no experience with catlab or julia but I do implement a lot of things from scratch just to understand what's going on (mostly in python, lately also haskell)

Britt Anderson (Aug 03 2020 at 12:57):

I was thinking that exact thing. I have not used julia, but would appreciate seeing an implementation as a way to understand the setup concretely. I know that there is also a category theory library for idris, but I don''t have experience with that one either. Is there a ªstandardª library/language around which the community is coalescing?

Bradley Saul (Aug 08 2020 at 17:07):

@Javier Prieto:

I don't have a good sense of how much work it'd be. Do you have any thoughts on that?

It's probably as much work as we let it be. I was thinking of starting small and working out a simple example like example 6.1. And then see where it goes from there.

@Britt Anderson :

Is there a ªstandardª library/language around which the community is coalescing?

I'm not sure. I think catlab.jl is starting to get some traction in the ACT community.

I'll start a repository on github in next couple of weeks and share the link here for those interested.