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I am looking for functional mathematical notation to explain the difference between Predictive Inference and Causal Inference? I list an example model. I also list links further down that give interesting answers, but I think that for each method they use the word input, which is fine/understandable. But I think that what they reference for each type of inference is different for each method even though they call it input (or x).

So I am looking for function composition notation to try to explain what the difference are with say a simple model such as y = a*x, please.

Function notation as in

function( ): {input} -> output := equation(,input_varb ; input_param )

Where the function’s arguments are identified by using comma for variables and semi-colon for parameters. := means an assigned equation. Note equation is not strictly a function.

What is the difference between prediction and inference?

What is the relation between causal inference and prediction?

L92MD14
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  • I don't understand tghe point of your very special function notation. Do you want to identify a mode of inference with a specific function, a specific form of inut to some function, or what? (I doubt that makes much sense). – kjetil b halvorsen Feb 19 '22 at 03:23
  • I hope my notation isn’t too far off from standard notation in mathematics and computer science (I know I can’t cite wiki articles, but those cover the math function notation and variable / parameter / equation assignment). I am not sure what you mean by mode. – L92MD14 Feb 19 '22 at 04:54
  • I also do not understand your aim. You need to make inference about what? Prediction point estimates? Causal point estimates? What are you estimating? You want to understand when the linear model you wrote has causal interpretation? – Plastic Man Feb 19 '22 at 08:44
  • Why not try a textbook or a lecture note on causal inference? (Some references may be found in [these threads](https://stats.stackexchange.com/questions/tagged/causality+references).) They should include a discussion of the difference between causal and noncausal (such as predictive) inference right at the start. – Richard Hardy Feb 19 '22 at 09:47
  • Strongly related: https://stats.stackexchange.com/questions/409301/what-is-the-relationship-between-minimizing-prediction-error-versus-parameter-es/487748#487748 – markowitz Feb 19 '22 at 11:26
  • @PlasticMan in the links I listed the responses that were given tend to use input output & function to try to describe their answer. But one thing they did was use the same input for the both types of methods. I think I question the consistency there. But I may be missing that they are using different inputs for each method so I want to explicitly see that to get clarity. So I think by using the linear (w/ coefficient) model would help me with understanding Predictive-Inference and Causal-Inferences by looking at their functional differences including what inputs they are being feed and why. – L92MD14 Feb 19 '22 at 17:43

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CI is causal inference. PI is predictive inference.

PI( ;a): x -> y:=a*x Where the input: “x” is the variable, and “a” is parameter that describe a subset population. Where the output: is Y.

CI( ): x’-> y:=x’ Where the input: “ x’:=a*x “ is the variable, and therefore “a” and “x” are variable too. So there are no parameters to this function, per se.

But the critical thing is that the “a” can be varied as INTERVENTIONS to represent the total population and so is not just a subset population for this type of analysis.

However if we were to further specify the type of CI analysis by allowing: IF set up “sequentially” it could be uni-variable (single variable) during each computation that is performed as a sequence of slices in the population or in other words given “a”. I might venture to say that this example of CI would just be a series of PIs.

Comma means variable, semi-colon means parameter. := means an assigned equation. Note equation is not strictly a function.

L92MD14
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