George Rebane
Here’s an exercise in critical thinking that’s been circling the backwaters of my brainbone. Most of the nation has debated Biden’s mental capacity and come down on this side or that. And during this time a lot of new evidence keeps coming in on the policies he’s intitiated and how he’s handling his office. So, if we are to surmise that one of four mutually independent characterizations (i.e. hypotheses) of his administration is true, how do we incorporate the newly arriving evidence to update our belief across the four Biden cases. Let me define these as follows.
H1 – Biden is mentally fit, politically strong, and in full charge/command of his office.
H2 – Biden is mentally fit, politically weak, and is an instrument of his party’s radical leftwing.
H3 – Biden is not mentally fit for the office, is politically weak, and is an instrument of his party’s radical leftwing.
H4 – Biden is not mentally fit for office and is being prepared for an early resignation.
Assigning beliefs (probability values summing to one) to these cases, we can now examine arriving evidence – e.g. his avoiding a press conference and spontaneous encounters with the press – and update our beliefs. Once you determine how likely it is that the evidence supports each case/hypothesis, it’s real easy to update your beliefs. With these four cases in mind, a piece of evidence E will let you assign four likelihood ratios L1 through L4, one for each case.
From a previous discussion on the use of likelihood ratios, I draw your attention to ‘Bayes sans equations and tears’. Each likelihood ratio is formally the belief/probability that E occurs when H is true, divided by the belief/probability that E occurs when H is false (not true). But to use the belief updating method with likelihoods, we don’t have actually know or compute with those probabilities. We can do a direct subjective assessment by answering the question ‘how many more times is it likely the E supports H, than not?’ Let’s take H3 as an example, and let E be the evidence of Biden’s having avoided press conferences and encounters with reporters in which he has to provide spontaneous answers. One could easily say that L3 in this case is at least 10, that it’s ten times as likely for him to avoid the press if he were mentally unfit and his leftwing was controlling him. However, for H1 the value of L1 from E might be 1/20, which means that as being mentally competent and in control, the likelihood of avoiding the press would be 1/20 = 0.05. Another way, that says that it’s twenty times as likely that one of the other cases – H2, H3, H4 – would be true than H1 being true. Anyway, you get the idea that if L is greater than one for some H, then the evidence E supports H, and if less than one, then the evidence supports some other case/hypothesis than H. In this way, the update formula increases and decreases your post-evidence beliefs as the next new evidence comes in.
OK, so let’s update the beliefs in the above cases by considering two pieces of evidence.
E1 = Biden and his handlers avoid press potentially embarrassing press encounters that require spontaneous and/or extemporaneous answers to questions.
E2 = Biden has opened borders to invite and permit illegal aliens, some infected with C19, to be turned loose into our countryside.
For me L1 is about 1/20, meaning that a mentally competent and in control president would want to address the press more frequently than not, and get the correct word out on all the policy changes he has made. L2 may be a bit lower since his leftwing would probably like to get him out there in front of the press to allay all the rumors that he’s a bumblebrain – so I’ll pick L2 = 1/30. L3 is the likelihood that Biden’s leftwing judges him to be mentally compromised and therefor has not wanted him and the Democratic machine to be embarrassed in front of the press – therefore I’ll set L3 = 15, meaning that E1 strongly supports H3. Finally, if the Dems were preparing Bumblebrain for an early out, they most certainly don’t want to either burnish his presidency or have him embarrass the Democrat leadership – therefore I’ll set L4 = 30, twice as likely for as if H3 were true.
To update our beliefs, we have to have a pre-evidence set of beliefs or prior probabilities for each H. Let’s make the safe assumption that we start out by being ignorant of which case is actually true. To express this quantitatively we set all the prior probabilities to be equal. This means that P(H1) = P(H2) = P(H3) = P(H4) = 0.25 since they have to sum to one (certainty). Here each post-evidence belief or posterior probability P(H|E) is computed by the simple formula
P(Hi|E1) = Li*P(Hi)/[sum of the products for i = 1,2,3,4]
In our case for H1 we have
P(H1|E1) = L1*P(H1)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)]
= 0.05*0.25/[0.05*0.25+0.033*0.25+15*0.25+30*0.25] = 0.001.
Similarly computing the remaining post-evidence beliefs, we get
P(H2|E1) = L2*P(H2)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.007,
P(H3|E1) = L3*P(H3)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.331,
P(H4|E1) = L4*P(H4)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.661.
This tells us that incorporating E1 our belief in H1 and H2, that Biden is mentally competent, plummets.
Now we go on to consider E2, the open borders evidence. In my opinion, were Biden mentally competent and in control, H1, then he most likely would not invite an unending tsunami of illegals and turn them loose in the country. In other words, I’d like to believe that his campaign rhetoric had some pro-American truth to it in preserving the cultural and economic make-up of our nation. Therefore, I will set L1 = ½, saying it is twice as likely that he then would not create a crisis and flood the country in a chaotic manner. However, for H2 I think it at least five times as likely that mentally competent or not, his radical and anti-American leftwing handlers would force the presence chaos on the border, therefore L2 = 5. For H3, a mentally unfit Biden, the likelihood of the Democratic leftwing holding sway on the current insane border policy is much higher, say, L3 = 25. And finally for H4, I assess that E2 would be similar here as it was for H3, therefore L4 = 25. Doing these calculations as above, we now use the above calculated beliefs for E1 as our priors for doing the E2 calculations. We then get –
P(H1|E2) = L1*P(H1)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.000,
P(H2|E2) = L2*P(H2)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.001,
P(H3|E2) = L3*P(H3)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.333,
P(H4|E2) = L4*P(H4)/[L1*P(H1) + L2*P(H2) + L3*P(H3) + L4*P(H4)] = 0.666.
Here we that the border chaos evidence E2 did not materially change the beliefs in Biden’s mental status that we computed with E1, the press avoidance evidence. The probability that Biden is mentally competent and in full control of his office, H1, is zero to three decimal places. And the belief in H2 has sunk to one seventh of what it was with E1. Finally, the beliefs in either H3 or H4 being true have increased marginally as expected because for an incompetent Biden, the Dems’ leftwing would have created the border crisis E2 in any case.
From the above presentation, it should be clear that different people with different assessments of the evidence would reasonably come to a set of different beliefs. The important notion here is ‘reasonably’ because that is how we have proceeded through this exercise of updating beliefs from considering two separate, but not equally impactive, pieces of evidence in assessing Biden’s mental capacity.
For those who are concerned about the validity/accuracy of such a sequence of updating beliefs that may be affected by the selected initial or prior values, not to worry. You can always start from a position of total ignorance – a tabula rasa if you will – and assign equal probabilities to all cases. As evidence starts coming in, no matter the sequence, the posterior belief values will quickly converge on the 'correct' values that represent your subjective view of reality.
Exercises in the reasonable updating of subjective beliefs when a collection of evidence is considered should not present a problem for the layman. The whole thing can be set up in a spreadsheet for handy calculation as new evidence arrives that impact on a set of mutually exclusive hypotheses or representations of truth.
rebanesruminations.com 
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