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The last two slides nudged a weight and watched the price move. That worked, but it is the wrong direction. The machine does not get to try a weight and see what happens. It only ever learns one thing: the price came out $16.25 too low.
So the calculation has to run the other way. Start at the error and walk left.
Press start at the error. The guess is $23.75 against a real price of $40. That gap is the only fact the bottle gives us.
Before starting, be clear about what this walk produces, because it is easy to expect the wrong thing.
It does not correct anything. No weight moves. Nothing is fixed. At the end of the walk the machine still guesses $23.75.
What it produces is one number per weight, and every one of those numbers answers the same question:
If I nudged this weight up by 1, how much would the final price move?
That is a measurement, not a repair. It tells us how much each weight matters, which is what we need before we can sensibly change any of them. The changing comes later.
One honest note about the $16.25. These effects do not depend on it. The weight after hidden 1 would still be worth $.50 per unit if the bottle had cost $30 instead of $40. What the error decides is how far to actually move, and that is a later slide's job. Here the error tells us only which way is too low, so we know a bigger price is the improvement we are looking for.
One warning about reading the picture. The machine's own numbers and these new effect numbers sit close together, and some of them coincide. The plain numbers inside the boxes and discs are the machine's existing values, and they never change. Anything in orange is an effect, an answer to the question above.
The first crossing: the weights that touch the price.
Press cross the output weights. The output weight after hidden 1 sits right beside the price. Slide 22 already measured it: take that weight from 25 to 26 and the guess would go from $23.75 to $24.25, a rise of $.50, because hidden 1 was sending .50.
That is the weight's effect, and it turns out to equal the signal its unit sent. The reason is worth a second. Adding 1 to the weight adds one extra copy of whatever the unit sent, and the unit sent .50, so the price gains $.50.
This produces a coincidence on screen that can trip you up. Hidden 1 displays 0.50 and its weight's effect is also +$.50. They are not the same thing. One is a signal the unit sent during the guess. The other is what would happen to the price if we nudged the weight. They match because of the rule just given, not because they are one number.
The four hidden units send .50, .20, .15, .10, so the four output weights have effects of +$.50, +$.20, +$.15, +$.10. The unit that spoke loudest has the most effect on the price, exactly as in Chapter 1.
All four are worked out here, in this one crossing. Nothing is queued for later.
That was the easy layer. These weights touch the price directly, so their effects were close at hand. The error now has to get past them, into a unit that never touches the price at all, and that crossing changes the numbers dramatically. It gets the next slide to itself.
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