Joint Probability Distribution - Science topic

I want to understand how to configure and apply such a model to identify significant or popular locations in a geographic dataset.

I understand how Gibbs sampling works to allow estimations of joint probability distributions in a simple sense. For example, using a simple two dimensional example (A and B are two different tennis players whose results are not independent, and 0=losing, 1 = winning), you might start with the following conditional probabilities:

P(A=0| B=0) = 0.5,

P(A=0| B=1) = 0.125

P(B=0|A=0) = 0.5

P(B=0| A =1) = 0.125

To get things moving, you would then suppose a starting value for A (let it be 0). We can then use that value of A (A=0) for the next iteration for B, where we therefore look at P(B=0|A=0). As shown above, there is a 0.5 probability that B=0 in that case. Let's run a random number (0-1) generator, which happens to yield 0.67. As this is greater than the probability of 0.5, we take B=1 (using the rule that if the random number is lower than the conditional probability, 0 is yielded, and if the random number is greater than the conditional probability, 1 is yielded). This gives us the first pair of joint values: A=0 and B=1 [which can be written as 0,1]. We now run the next iteration for A using that last value of B (B=1). P(A=0|B=1) is 0.125. The random number generator yields 0.28, so we take A =1. We then look at P(B=0|A=1) [as the last value of A yielded was 1], which is 0.125. The random number generator yields 0.34, which means we take B to be 1. So our second pair of values is: A=1, B=1 [or, 1,1]. We can repeat this process for a very large number of iterations, and if we then count the numbers of paired values that are 0,0; 0,1; 1,0; and 1,1, we should be able to estimate the joint probability distribution. I have attached a simple excel program that carries out such a Gibbs sampling. It can be seen that the estimation of the joint probability distribution is very close to the actual joint probability distribution from which the conditional probabilities were calculated (in practice, of course you wouldn't have access to the true joint probabilities as then you'd have no reason to do the Gibbs sampling). I have largely based my example on an excellent YouTube video by Ben Lambert.

However, this is where I need advice and help. I do not understand how the above example relates to network meta-analyses (NMAs). For example, imagine a network meta-analysis of three treatments A, B and C. How do the data from these studies relate to conditional probabilities? For example, if the odds ratio of outcome X is 0.5 for the comparison of A vs B, the odds ratio of outcome X is 0.2 for the comparison of B vs C, and the odds ratio of outcome X is 0.1 for the comparison of A versus C (clearly no incoherence here!), how do we proceed? I have a vague idea that we could use the odds ratios to calculate conditional probabilities, but can't quite grasp exactly what should happen. I have looked at most of the relevant documents (like the DSU document TSD2) but these don't explain exactly what occurs in the Gibbs sampling itself. Can anyone describe, in simple terms, how the sampling would proceed in an NMA, in relation to the model of Gibbs sampling I have given earlier?

Could any expert try to examine our novel approach for multi-objective optimization?

The brand new approch was entitled "Probability - based multi - objective optimization for material selection", and published by Springer available at https://link.springer.com/book/9789811933509,

DOI: 10.1007/978-981-19-3351-6.

I would like to fit a trivariate joint probability distribution using the nested three-dimensional Copula in MATLAB. To this end, their respective marginal distribution and two dimensional joint distribution have been modelled. I wonder if there is any Copula code or toolbox available to model a multivariate (three or more) joint probability distribution in MATLAB. Thank you in advance!

Suppose we have statistics N(m1, m2), where m1 is the value of the first factor, m2 is the value of the second factor, N(m1, m2) is the number of observations corresponding to the values ​​of factors m1 and m2. In this case, the probability P(m1, m2) = N(m1, m2) /K, where K is the total number of observations. In real situations, detailed statistics N(m1, m2) is often unavailable, and only the normalized marginal values ​​S1(m1) and S2(m2) are known, where S1(m1) is the normalized total number of observations corresponding to the value m1 of the first factor and S2(m2) is the normalized total number of observations corresponding to the value m2 of the second factor. In this case P1(m1) = S1(m1)/K and P2(m2) = S2(m2)/K. It is clear that based on P1(m1) and P2(m2) it is impossible to calculate the exact value of P(m1, m2). But how to do this approximately with the best confidence? Thanks in advance for any advice.

How to find the distance distribution of a random point in a cluster from the origin? I have uniformly distributed cluster heads following the Poisson point process and users are deployed around the cluster head, uniformly, following the Poisson process. I want to compute the distance distribution between a random point in the cluster and the origin. I have attached the image as well, where 'd1', 'd2', and 'theta' are Random Variables. I want to find the distribution of 'r'.

I have several random variables X = [x1, x2,...... xn] represented by columns in a Data matrix and rows are representing random samples. I also have the marginal Probability Density Functions as f(x1), f(x2), ... f(xn) for individual random variables. I would like to calculate their joint PDF as f(x1,x2,....xn).

Im working with Nataf model trying to fit a joint probabilistic model for circular and Linear variables, but I have some difficulties in calculating the correlation matrix because, I could find an equation for calculating the equivalent correlation between two circular variables or between a circular variable and a linear variable.

What is the best way to calculate joint probability distributions from multiple discrete probability distributions?

I need to calculate the combined or joint probability distribution of a number of discrete probability distributions.

Xn={x1;x2;x3...}; P(Xn)={P1;P2;P3...} for n>1.

With regards to the discrete probability distributions: Values must be summed; Probabilities must be multiplied.

Is there a faster way than merely multiplying each discrete probability distribution with each other?

Assume that X, Y, and Z are identical independent Gaussian random variables. I'd like to compute the mean and variance of S=min{P, Q}, where :

Q=(X-Y)²,

P=(X-Z)².

Any help is appreciated.

I have three sets of variables. Two of them are continuous random variable and the other one is discrete in nature. Is there any matlab function which returns the joint probability distribution of these three random variables? Kindly explain with an example.

Hi there,

I'm trying to fit rainfall data to incomplete gamma distribution. So, I don't know how to proceed.

Should I estimate the shape and scale parameters before the Lelleifors test? (i.g. using my data set to fit gamma to it).

Or should I first find out Lilliefors results? I'm confuse.

Kind regards,

Jefferson

is it just possibility versus probability?

Two systems give uncorrerlated or less correlated outputs while their inputs show some correlated behavior. So how to transform and find some complex relationship between inputs-outputs which can possibly give good correlation and copula dependence for outputs.

Hello, I have a question regarding negative binomial (NB) regression. I am not sure if I can include predictor variables that are correlated with the exposure variable (say time)? I'm concerned that the predictor variable (VIF 3.8) is correlated to the exposure variable (VIF 9.8). I have carried out NB regression despite the collinearity and the results are significant. The overall likelihood ratio test is 6.377, df = 1 and sig = 0.012. Can I include the predictor variable in this case? 

What is the difference between joint distribution function and likelihood function?

My research is on coincident flooding using joint probability method but I have limited background knowledge of joint probability. I an expecting a sample of calculations.

Using model f to obtain joint PDF of two parameters.

X and Y are independent, identically distributed log-normal random variables. How can I get the PDF of Z where Z=abs(X-Y)?

Hi

I want to calculate joint probability P(x,y,z) based on P(x,y), P(x,z), P(z,y), P(x), P(y), P(z). How can I do?

x, y, z are three random variables and I compute their probability by estimation.

If anyone has this please give me some idea.

Hello,

Maybe this is a very easy question, maybe not. I have a time-descrete stochastic process X = (X_t). Each X_t has a different pdf, so it is not i.i.d. All X_t have the same sample space and the pdfs are constructed of the same sample size. So now I don't want to have the joint probability function, I want to have the pdf of all realisations of all X_t collected together, as if there was no difference in time. How can I get this "summed up" pdf out of the separate pdfs?

Thanks in advance

I have a random vector whose joint probability distribution is known. However, I would like to sample this vector so that it lies within a convex polytope which can be represented by a set of linear inequalities, such as Ax= b , x>=0. I can naturally use rejection sampling, but the rate of acceptance is very small (<< 1% for practical size applications). I have come across a simple way of uniformly sampling from the unit simplex, and I think this may be adapted to a general convex polytope, but I can't figure out how.

Suppose I have a random vector X = (X_1, X_2, ..., X_n) for which X_i ~ Po(\lambda_i), but the components are not independent, so that the joint distribution is not the product of the marginal distributions, and the covariance matrix of X is not diagonal. It seems the joint distribution does not have an explicit form. I would like to obtain some properties from this distribution, as mean, mode, and variance. How can I sample from it? Is there any good approximation to it other than the multivariate normal?