Malus Law Question

[Xray]

1. Let A and B be correlated events under condition X. Then, per (little-known) Watson's Law*, a law supported by many experiments and (so far) never refuted:

P(AB|X) = P(A|X)P(B|XA) = P(B|X)P(A|XB). (1)

Commutativity thus reciprocity? Assuming a product formula universally applies and is meaningful, (1) seems trivial given all P's are reals - right?

<…SNIP...>

Hi Xray. Regret ever wading into this swampland, having never really studied Bell's theorem beyond a cursory inspection. Seems certain accepted notions of 'correlations' and 'joint probability' are quite foreign to my thinking. Back to the sidelines for me!

Hi Q-reeus,

Many of us here are seeking to understand the swamp; to explain it, not drain it. So, please, no retreat to the sideline just yet; we need all the help that we can get.

Or, please, no retreat until you are clear about the critical importance of equation (1) in all walks of life! For example, via the link to that famous Professor Meadow and his blundering.

Which brings me to this, by you: "Commutativity thus reciprocity? Assuming a product formula universally applies and is meaningful, (1) seems trivial given all P's are reals - right?"

1. I do not understand the relevance of this: "Commutativity thus reciprocity?".

2. To me, there is nothing trivial about (1): eg, see that Prof. Meadow example! Then the next:

3. Why not have a go at deriving (1) from simpler first principles? This might help you to understand why (1) will be very useful for the rest of your days.

4. In my terms: All Ps are in the interval [0,1].

HTH; cheers,

Xray

[Q-reeus]

Q-reeus, you keep going back to post #1, despite the fact that there were serious errors in it which I've since admitted and rectified long before you started posting in this thread, there was no such claimed "unannounced switching":...

Minkwe, your subsequent reposting of #3 does not imo actually clear things up, because nowhere in that #3 do I see you redefine the original actual definitions of P(AB|S) and P(AB|D). What stuck in my mind was that given before in red. I will though admit to failing to notice the import of the follow-on in that same 1st paragraph of post #1:

The reason we use P(AB) = P(A)P(B|A) instead of P(AB) = P(A)P(B) is because for the B filter, the particles under consideration are only the subset which passed through A. P(B|A) reads "for those particles which went through A, what proportion of them will now go through B."

Having absorbed the original definition - net count past both filters, that which may be called a 'refinement' in blue text above was so much blah blah blah to me by that stage. So in a way we are both right and wrong but for different reasons. I accept that according to your subsequently clarified definition of joint probability, one gets 0.25 for the given D case example earlier. For me it is a post-processed value of no direct physical significance - the raw count is 0.5 and that's what one measures. Evidently re Bell etc., the real interest in correlations arises where the extra refinement of selecting for spin-up vs spin-down, V-polarized vs H-polarized etc. comes in.
Anyway, trust there is no hard feelings now that this ongoing misunderstanding has hopefully been sorted out.

[Q-reeus]

Hi Q-reeus,

Many of us here are seeking to understand the swamp; to explain it, not drain it. So, please, no retreat to the sideline just yet; we need all the help that we can get.

Or, please, no retreat until you are clear about the critical importance of equation (1) in all walks of life! For example, via the link to that famous Professor Meadow and his blundering.

Which brings me to this, by you: "Commutativity thus reciprocity? Assuming a product formula universally applies and is meaningful, (1) seems trivial given all P's are reals - right?"

1. I do not understand the relevance of this: "Commutativity thus reciprocity?".

Hi Xray. Well P(AB|X) = P(A|X)P(B|XA) = P(B|X)P(A|XB). (1), applied to say minkwe's S case, is equivalent to saying one can always 'run the experiment backwards' i.e. reverse the filter ordering, and result will not change. That's a reciprocal system (non-reciprocity applies for some physical systems). And commutative in practice - even though the exact form of (1) is not strictly commutative. I believe JC uses non-commutative algebra extensively in his proofs which is why I wondered if those P's were always simply real number values - as you say in (4), in the range [0,1]. Evidently so.

2. To me, there is nothing trivial about (1): eg, see that Prof. Meadow example! Then the next:

3. Why not have a go at deriving (1) from simpler first principles? This might help you to understand why (1) will be very useful for the rest of your days.

4. In my terms: All Ps are in the interval [0,1].

HTH; cheers,

Xray

Will give your above suggestion some thought. With any luck will never be in a situation needing to remind a courthouse of the correct use of stats!