আমি প্রস্তাব করব যে ফলাফলগুলি কীভাবে প্রতিবেদন করা হয় তা নিয়ে এটি একটি সমস্যা। "বায়েশিয়ান ড্রামকে পেটাতে" নয় বরং বায়েশিয়ার দৃষ্টিভঙ্গি থেকে মডেল অনিশ্চয়তার কাছে আসা অনুমানের সমস্যা হিসাবে এখানে ব্যাপক সহায়তা করবে। এবং এটি একটি বড় পরিবর্তন হতে হবে না। যদি প্রতিবেদনে এই মডেলটি সত্য বলে সম্ভাবনা থাকে তবে এটি খুব সহায়ক হবে। এটি বিআইসি ব্যবহার করে আনুমানিক এক সহজ পরিমাণ। বিআইসিকে mth মডেল জন্য কল করুন । তারপরে এম মডেলগুলি "সত্য" মডেল হওয়ার সম্ভাবনা যা এম মডেলগুলি ফিট ছিল (এবং যে মডেলগুলির মধ্যে একটি সত্য) তা দ্বারা প্রদত্ত:BICmM
P(model m is true|one of the M models is true)≈wmexp(−12BICm)∑Mj=1wjexp(−12BICj)
=11+∑Mj≠mwjwmexp(−12(BICj−BICm))
Where wj is proportional to the prior probability for the jth model. Note that this includes a "penalty" for trying to many models - and the penalty depends on how well the other models fit the data. Usually you will set wj=1, however, you may have some "theoretical" models within your class that you would expect to be better prior to seeing any data.
Now if somebody else doesn't report all the BIC's from all the models, then I would attempt to infer the above quantity from what you have been given. Suppose you are given the BIC from the model - note that BIC is calculable from the mean square error of the regression model, so you can always get BIC for the reported model. Now if we take the basic premise that the final model was chosen from the smallest BIC then we have BICfinal<BICj. Now, suppose you were told that "forward" or "forward stepwise" model selection was used, starting from the intercept using p potential variables. If the final model is of dimension d, then the procedure must have tried at least
M≥1+p+(p−1)+⋯+(p−d+1)=1+p(p−1)−(p−d)(p−d−1)2
different models (exact for forward selection), If the backwards selection was used, then we know at least
M≥1+p+(p−1)+⋯+(d+1)=1+p(p−1)−d(d−1)2
Models were tried (the +1 comes from the null model or the full model). Now we could try an be more specific, but these are "minimal" parameters which a standard model selection must satisfy. We could specify a probability model for the number of models tried M and the sizes of the BICj - but simply plugging in some values may be useful here anyway. For example suppose that all the BICs were λ bigger than the one of the model chosen so that BICm=BICj−λ, then the probability becomes:
11+(M−1)exp(−λ2)
So what this means is that unless λ is large or M is small, the probability will be small also. From an "over-fitting" perspective, this would occur when the BIC for the bigger model is not much bigger than the BIC for the smaller model - a non-neglible term appears in the denominator. Plugging in the backward selection formula for M we get:
11+p(p−1)−d(d−1)2exp(−λ2)
Now suppose we invert the problem. say p=50 and the backward selection gave d=20 variables, what would λ have to be to make the probability of the model greater than some value P0? we have
λ>−2log(2(1−P0)P0[p(p−1)−d(d−1)])
Setting P0=0.9 we get λ>18.28 - so BIC of the winning model has to win by a lot for the model to be certain.