Probabilistic Graphical Models - Science topic

Hey, I'm using ArangoDB quite some time and it does a pretty good job and scales very well. However, it is a multi-model database, but it provides GraphQL to write your own query on top of the DBMS.

https://www.arangodb.com/

The following link uses multiple datasets:

https://towardsdatascience.com/link-prediction-with-neo4j-part-2-predicting-co-authors-using-scikit-learn-78b42356b44c

go to,paper, that will definitively help you in this regard

Of course it is, its what it is by definition. I probably should have thought more about this, back when I originally posted. If it isnt, then nothing is. Its the "closed" convex hull of its vertices, or all points in [0,1]^3, that can be expressed by convex combinations of its vertices:, (1,0,0), (0,0,1), (0,1,0).

So that any vector (x,y,z), x+y+z=1 , or rather point, (x,y,z), in [0,1]^3, where x+y+z=1,and x >=0,y,>=0z>=0 can be simply expressed by x* (1,0,0)+(0,1,0)+z*(0,0,1)=(x,y,z), as closed under all convex combinations of the vertices convexity (of this form) simply means/ requires that its contains all points in [0,1]^3, that can be expressed by-negative c1, c2,c3 ; c1*(1,0,0)+c2*(0,1,0) +c3*(0,0,1) where c1+c2+c3 =1, it just is the set of 3 all and only non-negative coordinates, that sum to one . And clearly we can set c1=x, c2=y, and c3=y and these are non-negative and sum to one. So its all, and "only" probably triples (x,y,z),x+y+z=1 .

I should have thought this through.

The main questions are (1) and (2):

(1): does the canonical 2 probability simplex, contain the canonical 2 simplex of the sums at each point x+y, x+z, z+y, clearly at each point (x,y,z) in the canonical 2 probability simplex, x+y,in [0,1] x+z\in [0,1], z+yin [0,1] and their sum (x+y+x+z+z+y=2(x+y+z)=2*1=1, but does it contain at "every" point, every such convex combination of (x+y=l, x+z=g, z+y=h), 1>=h>=0,1>=g>=0, 1>=l>=0 & l+g, g+h, h+l in where h+g+l=2 at some point (x,y,z)?

Id say yes, because if there were a (l,g,h), l+g+h=2, 1>=h>=0,1>=g>=0, 1>=l>=0, but with no accompanying (x,y,z) x+y+z=1, (x,y,z)>=0, l= x+y, g=x+z, h=z+y

, g=x+h-y, y=x+h-g, so l=x+x+h-g, l=2x+h-: x=1/2*(l-h+g), which is uniquely determined where l-h+g<=l+g+h=2, as h, g, l>=0, and clearly x+y+z=1/2(2x+2y+2x)1/2(l+g+h)=1, the only way that point could not be in the simplex is if, for example x<0

which in this case of l+g<h, as h<=1, g<=1, l<l, as per the conditions above, that we have a contradiction as (l+g)+h=2 where, l+g<h gives (l+g)+(l+g)<(l+g)+h=2, to 2(l+g)<2, L+g<1 but h<=1, so 1+h<=1+1=2, so, 1+h<=2,

L+g<1 gives (l+g)+h< 1+h

and (l+g)+h<1+h<=2, L+g+h<2, ie l+g+h<1

1/2*(1-g+h)<=1/2*2=1

And moreover, (2):

(3)Does it contain all such h+g+l=1, where the l, g, h, are the sums of the first 2, the first and third, second and third coordinates, respectively of 3 "distinct" cartesian points in the simplex, (x1,y1,z1), (x2,y2,z2), and (x3, z3, y3), l=x1+y1, g=x2+z2, h= y3+z3, h+g+l=1, where none of the x1,x2,x3, x2,y2,z2, x3,z3, y3 are 0?.

Clearly for any m, in [0,1] there is a point in the simplex <x,y,z>, x=m.

So as x is an element of a point, in the simplex, then at those points, where the x coordinate is m, m=x>=0, and clearly x=m<=1.

As, 1>=m>=0 entails that 0<=1-m<=1, then (1-m) is in [0,1],. Then as, for any real in [0,1], there are pts in the simplex, where the x coordinate attains that real, then there is a pt, whose x coordinate is x1, where, (the x coordinate), assumes the value, x1=1-x=1-m.

<x1,y1,z1>, x1+y1+z1=1 so, y1+z1=1-x1=1-(1-m)=m, so x+z assumes the value m,

As the y and z coordinates can assume any value in [0,1] as well, then we have all non negative coordinate of three points, in [0,1]^3:, p1 =(x1,y1,z1), p2=(x2,y2,z2), and p3=(x3, z3, y3), where x1+y2+y3<=3 where x1,y2,y3, are all in [0,1], and thus, the subset, comprising all, sets of three points, p1, p2, p3 in [0,1]^3 where x1+y2+z3=2 , where x1,y2, z3 are non-negative and in [0,1,] are in the simplex.

As, at any pt, in the simplex, and at all pts , x+y+z=1, so: (x1+z1+y1)+)x2+y2+z2)+(x3+y3+z3)=3, and as x1+y2+z3=2 ,

so( x1+y1)+(y2+z2)+(z3+x3)= 3-(x1+y2+z3)= 1,

ie for any real in [0,1] x=m at(x,y,z) if and only there is a point (x1,y1,z1), x1+z1= m

(ie not on the edges of the simplex, where, aeprt from the vertices, only two of the coordinate of a point are non-zero and not 1 ?

At the edges we set y1=0, x1=x1+y1=0+x1= l,

x2=0, so as x2+y2+z2=1, x2+z2 in [0,1], so we set z2=g, z2+x2=z2+0=z2=h so,

z2+x2=g, and at the third point. Let z3=0, h=y3 so y3+z3=0+y3=y3=h

I suppose so, there was a counter-example there wou as if it did not there would have to be no point in the simplex (x,y,z), where 0<=y+z=l=1-g-h<=1 for some, (h,g, l), h+g+l=1 h>=0, g>=0, l>=0, where of course, this gives that 0=1-1<=1-(g+h)=l>=1-0=0 , It also has to express all such combinations because if did not for some l, g, h.

So as long as the the set of all probability triples, 2 canonical probability simplex , contain on its edges, (the 2 positive entries,), the set of call of all probability doubles, that is, the "canonical 1 probability simplex/ the set of 2 non-negative points in R^2 whose sum is 1 (all convex combinations of 2 non-negatives that sum to 1, then the it will contain some such three such edge points, meeting the above sum constraintm(x1,y1,z1), (x2,y2,z2), and (x3, z3, y3), l=x1+y1, g=x2+z2, h= y3+z3, h+g+l=1.

Where along the edges, the double in the canonical 1 probability simplex, which any triple on the edges of canonical 2 probability simplex, identifies, are those doubles, formed by the, 2 elements, of the 3 elements at that point (x,y,z), which happen to be positive (non-zero). SO, setting the first positive coordinate, in said triple (which will be x, or y) to be the x coordinate of the double, and the second positive entry (which will be z or y) to y, ie (x=1/6, y=0, z=5/6) goes to (x=1/6, y=5/6) which will be unique except at the vertices (where one can use both (1,0) or (0,1), and x+y=1 as the other point z will be zero, as the point was taken from the edge of the set of canonical 2 simplex (x,y, z), where precisely one of (x,y,z), is 0, (except at the vertices, themselves, where 2 of them are of course)

Seems to be a very old question, but if someone is still interested there exists a function for exactly this question in the bnlearn package.

It's called arc.strength

For more information see http://www.bnlearn.com/documentation/man/arc.strength.html

1. General remarks:

Research studies are usually carried out on sample of subjects rather than whole populations. The most challenging aspect of fieldwork is drawing a random sample from the target population to which the results of the study would be generalized. The key to a good sample is that it has to be typical of the population from which it is drawn. When the information from a sample is not typical of that in the population in a systematic way, we say that error has occurred. In actual practice, the task is so difficult that several types of errors, i.e. sampling error, non-sampling error, Response error, Processing error,…

In addition, the most important error is the Sampling error, which is statistically defined as the error caused by observing a sample instead of the whole population. The underlying principle that must be followed if we are to have any hope of making inferences from a sample to a population is that the sample be representative of that population. A key way of achieving this is through the use of “randomization”. There several types of random samples, Some of which are: Simple Random Sampling, Stratified Random Sampling, Double-stage Random Sampling... Moreover, the most important sample is the simple random sample which is a sample selected in such a way that every possible sample of the same size is equally likely to be chosen. In order to reduce the sampling error, the simple random sample technique and a large sample size have to be developed.

2. Specific remarks:

The following factors are highly affected the sample size and need to be identified:

· Confidence Level (level of significance) and

Necessary Sample Size = (z-score or t-value)2 * StdDev*(1-StdDev) / (margin of error)2 .

I was able to answer my own question.

I originally had felt that a utility could be develop to auto-populate a CPT from a sufficient number of Monte Carlo runs of a non-BN system that the BN was simulating as an alternative approach. I reasoned this should be a simple set of linear simultaneous equations where the evidence provided to the parents and the inference in the child node are treated as constants and the CPT cell values are the unknowns.

I derived the equations but unfortunately I had a mistake (adding the parent states rather than multiplying them), which made my derivation in error. I since found my error and now all is right with the world.

Thank you P. S. Dharmapala for your comments.

in few words!

fuzzy logic extends probabilistic reasoning ... see Zadeh Works

Dear Professor Mateljevic,

When you say that the unit disk is a real analytic bijection to R2, and has a bijective analytic function rto R^2, does it have one ; is it b-ijective to the simplex; are all pts in (p1, p2, p3)  in IM(F) of the unit disk function.  Do any go missing. I presumed that there were certain issues, with the surface area of the sphere

F: dom (unit disk) to . I was wondering if a quadratic function of the northern surface of the unit sphere, would be bijective as defined by F:[0, 2pi times [0, pi]in R^2\to < [p1=sinpi* cosphi]^2, p2=[sin pi * sin phi]^2, p3=cos^2phi> in R^3where  p1+p2+p3=1, and should be non-negative due to the squares.

Whilst the triangular functions appear to be the other way around and maximize surface area

You are way too dispersed.

Try to undestand well the more basic stuff, one topic at a time.

Then move on.

Do not start from Gleason or weak measurements  for eample.

If you plot is assigned to G, then type FullForm[G]. Pay attention to the head.If you see Graphics[stuff], then you can use the Show command to combine plots, Its you see Graph[stuff], it might still be possible, but is harder. If you code is not too long, I recommend attaching it to your question.

The objective of a regression function is

not to go through all data points but to

get a tradeoff between simplicity of the

function and precision.

Usually a regression function is a polynome

of low degree. In this case it's a nonlinear

function, the combined function of the

neural net.

Regards,

Joachim

Hi Jumoke,

I do not know how it is your basis into Bayesian inference, but you can fit easily Bayesian networks using the open-source software "rstan" (it is a probabilistic programming language implemented in a R package). I used to fit several complex Bayesian networks in this software. There are a lot of probabilities distributions to play with. I could even evaluate multivariate Bayesian networks in this programming environment (assuming multivariate normal priors).

I hope to be helpful,

I like this one: http://szeliski.org/Book/

However, if you are more interested in low-level image processing, I would suggest

Hi buddy,

I  suggest you to take a look at the rstan package.

rstan is a package designed to fit Bayesian models in R. The advantage of this package for fitting hybrid models relies on the flexibility to integrate any form of probability distributions (hybrid models). rstan uses a state-of-the-art variant of the hamiltonian hybrid monte carlo method  known as NUTS  that does not require the user specify any known tuning parameter (tuning parameters are optimized via Metropolis-Hasting variants). Thus, it offers a very flexible inference engine to integrate any form of unknown joint distribution (so called non-conjugate bayesian models), eliminating time consuming steps of explicit mathematical derivation required to derive Gibbs samplers (rstan relies on all properties of Directed Acyclic Graphical models), offering more precision in multiple integration problems than Gibbs sampler due to its property to stationary into leptokurtic regions of the distributions (contrary to Gibbs that have a random walk sampling pattern). rstan is a probabilistic programming language, so you can perform most of the basic operations within a coding block called “transformed parameters”  (its language is a mixture of C+/R, including matrix operation, logical indexing, conditional, sparse operations, loops, buildin-functions and others). In my opinion Stan is the best open-source probabilistic programming software available, it offers a wide-range of probability distributions, does not require model conjugacy, offers programing flexibility, allows parallel computing, and it is fast in its cmdstan version (it have implemented a automatic variational inference algorithm). You can fit easy the model mentioned by Fabrice Clerot in rstan (it supports categorical, and multivariate Gaussian distributions).

I hope to be useful,

Best,

Jhonathan Pedroso

Thanks Dr.Isinkaye. I will read through the files

Dear Mallappa,

Please conduct a simple simulation study of different values of k with accuracy. The value of k which provides you best accuracy should be selected. 

I hope you will understand. If not, drop an email to me at sobhan.sarkar@gmail.com

As I understood are you looking for a "model" that "explains" all the (distributional) features in your data. But I doubt that this is a good idea. A model is a simplication. It is the purpose of a model to focus on some aspect of reality so that we may have a chance to get some understanding about this aspect. The model thereby ignores all apsects deemed to be irrelevant for the aspect under study. 

If a model is (mis?-)used just to describe observed data, then we can not really learn or understand something, especially if the model is flexible enough to follow the data in all its variation. As an example consider a regression model, where a high-order polynomial is fit to model the relationship between x and y. If there are n values, a polynom of order n-1 will give a perfect fit, thus explain the data perfectly, but such a model is (usually) completely worthless. Only if the order is very small (usually only 1-2, rarely 3) we can use the model to understand "gros patterns" in the data that can help us understand some particular aspect of the (in detail all to complex) relationship between x and y.

So I would not say that there will be a "model" for your purpose. It is more a (functional or whatever) description of the distribution of the observed data you are looking for. One way to solve your problem may be to fit a spline to the ecdf of your data. You can then used this fitted spline to generate random values from a distribution that has a very similar shape like the frequency distribution of your observed data. Like polynomials, splines can made flexible enough to follow the ecdf arbitrarily close. The only detail you might need to address is the restriction that the fitted spline must fulfill the conditions of a probability distribution, i.e.

(1)  0 <= F(x) <= 1

(2)  F(x+d) >= F(x) for any d>0

If you have a reasonable amount of data and you might simply use the ecdf directly.

If you want an uninformative prior, you may consider using Jeffrey's Prior.

Kathryn Blackmond Laskey, Bayesian Inference and Decision Theory, Unit 7: Hierarchical Bayesian Models recommends that you transform the hyperparameters of the beta distribution. Dr Laskey proposes this transformation because the two hyperparameters of the beta distribution are so closely correlated that the Gibbs sampling is inefficient. Her transformation, and the only transformation that others consider is U=alpha/(alpha+beta) and V=alpha+beta. U is the mean of the beta distribution and V is a rough measure of the precision of the beta distribution with larger values implying more precision. She suggests a uniform distribution for U and a gamma(1,20) distribution for V. She uses the rat tumour data from Gelman et al.

Dear fa Shin,

I couldn't sleep last night so I thought about your problem a little more. It turns out that it's not hard to solve it using probability theory (no programming).

Given three randomly placed points on a unit circle, we can always choose a coordinate system so that the first point is (1,0) and the second point is (cos(a),sin(a)) for some angle a with 0 <= a <= pi.  Let E be the event that the three points lie on a semicircle. Then, ~E occurs when the third point lies along the arc of the circle from pi to pi+a. Since the third point can lie anywhere on the circle, Pr(~E|a) = a/(2*pi).

We can determine Pr(~E) by "conditioning on a" as follows:

Pr(~E) = Integral_a Pr(~E|a)Pr(a) = Integral_0^{pi} [a/(2*pi)](da/pi) = 1/4.

Therefore, Pr(E) = 1 - Pr(~E) = 1 - 1/4 = 3/4. 

Greg.

Thank you Fabrice, that's very helpful. 

You can use discrete graphical models. each word is a node with value depending on the associated semtiment

I understand that this must be a selective statistical research. For theory-based results 100 objects must have homogeneity. Neural network reminds.

I think the most popular one is deep autoencoders

Useful link

It is usual to find these parameters using Expectation Maximization(EM) algorithm.

these links may help:

Also there are other structures and functions in matlab to estimate GMM parameters.

I think neural network approach is not good way, you should read some paper of Dr. Monteleoni from GWU 

she works on it,

I would suggest to test prtools toolbox for matlab

... and I can also strongly recommend this book:

K.B. Korb and A.E. Nicholson (2010): Bayesian Artificial Intelligence. CRC Press 2nd edition. (http://www.csse.monash.edu.au/bai/book/about.php) ... also here you will find Netica examples

Hi

Load data at the system buses being uncertain, probabilistic load flow aims to obtain set of bus voltage values corresponding to probable values of load (unlike conventional load flow methods) for a certain power system configuration and line parameter. The motivation towards such an approach is generally twofold. The first and foremost one being the conventional method is involved with excessive calculations wherein any load variation will necessitate a renewed solution (in an n bus system, around kn load flows are required, with k different load values at each bus) and secondly with such a vast load flow data, analysis and synthesis of the result is difficult.

Dear Researchers,

I apologize for the late response. Your anwers have been quiet helpful for me.

How about reading the following -

Application oriented papers

(1) http://bioinformatics.oxfordjournals.org/content/23/5/597.full.pdf+html

(2) http://biochem218.stanford.edu/Projects%202007/Emerson.pdf

(3) http://sydney.edu.au/engineering/it/research/tr/tr650.pdf

Tutorial papers

(1) http://homepages.inf.ed.ac.uk/csutton/publications/crftut-fnt.pdf

search for application oriented references. Hope it helps

There it goes:

Sample Spaces and Feature Models: There and Back Again

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4626837&tag=1

I guess it's kinda related to what your thinking into.

Regards!

Depends on your final goal !

For classification purpose let have a look at TANB,

else, for very small dataset (or with high missing data rate), the minimum weighting spanning for tree-shaped-Bayesian (MWST) network could be a starting point if you wanna stay with BN formalism !

I'm working on a system for online control of sensor parameters for the estimation of the interactions between several objects. Currently I'm on an extenstions to infinite elements and more complex sensor model of the attached paper

I would recomend you to read the following PhD thesis and/or contact the author:

I think in clique metods but the metods from syntatic recognition.

Hi, Thanks for your thoughtful suggestions. Actually I am asking about what is the way of implementing these probability calculations in computing s/w s like MATLAB...What should be the data structure to use in these computations to make the program not computationally challenging. Thanks again.