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rahul_singh asked . 2022-04-21

Calculation of probability using Beta ,lognormal and weibull distribution

Hello,
 
 
I have computed three different properties of sample objects and have stored the discrete data values in three vectors.I have fit weibull, lognormal and beta distribution on these three vectors. Now how can i find the individual probability of each of the data values using their respective fitted distributions? I want to multiply these probabilities for each element to find the joint probability.

weibull , beta distribution ,

Expert Answer

Kshitij Singh answered . 2024-04-21 08:05:39

"if I want to obtain the likelihood of getting a particular combination of values which represent these 3 properties of another cancerous tumor(not from the dataset),i use my fitted distributions for this purpose and i multiply the computed likelihoods to obtain the joint likelihood."
 
Several comments on this.
 
 
First, multiplication is only appropriate here if the 3 properties are independent. If they are at all correlated with one another, the joint likelihood for any given combination of three is not simply the product of the individual ("marginal") likelihoods. You can check this by looking at the pairwise correlations among the three properties across the 200 tumors.
 
 
Second, it sounds like you really do want probabilities rather than likelihoods. For that purpose, you would be best to consider each property as falling within a certain numerical range or "bin". For a first pass, I suggest you classify each tumor as falling above vs below the median with respect to each of the three properties. This will give you 8 categories of tumors: 2x2x2, and you can count how many (out of 200) you have in each category. With a sample of only 200, I doubt that you have enough data to get usable estimates of the probabilities in more than 8 bins, but you might try 3x3x3 if you are a daredevil.
 
 
Third, it still doesn't sound like you quite have appreciated the difference between likelihoods and probabilities. To maybe make that distinction more meaningful to you, I suggest you do the following: Rescore all of your properties into a different unit of measurement--say, for example, that you divide each property value by 10. I hope you agree that logically this changes nothing. Likewise, if you look at the probabilities in the 2x2x2 bins, you will see that the rescoring has also changed nothing. But if you refit your distributions to the rescored property values, you will see that all of the PDF values (i.e., likelihoods) have increased by a factor of 10. The reason is that the PDF values (likelihoods) must integrate to 1 across the whole range of X, so the PDF values--unlike the corresponding probabilities--depend on the units of measurement for X.


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