What is the physical basis behind burnt calories estimates? we see on treadmills, bikes, fitness trackers, etc., estimations of calories burnt. These are clearly based on correlations to weight, age, hear rate. How were these values correlated?
Is there a physical way to measure the amount of calories being burnt in the body directly? Energy balance seems too primitive since calories can be expelled unburnt, can be stored, and metabolism is different between individuals. I tried to look it up but anything related to fitness on the Internet seems to be loaded with armchair experts' guesses. 
 A: I think this is off topic here and I'm not sure of a better site for it (maybe fitness.SE)... 
I'll try to put as much physics as I can to keep it on topic here though. It's all about an energy balance and understanding where the energy can go when an animal exerts effort.
You can measure the basal metabolic rate (BMR) of an animal using respirometry. In particular, the volume of carbon dioxide produced in each breath can be measured directly with proper equipment and this can be related to the basal metabolic rate. This will tell you the amount of energy the animal expends just to keep itself warm, awake, thinking, etc.. The relationship between carbon dioxide production and energy required is based on the chemical reactions that occur within the body, of which carbon dioxide is a product. So you measure how much was produced and you know how much of the reactions took place and now you can determine the energy required for that to happen.
You can measure this on a whole bunch of different people of different age, weight, fitness, height, etc. and generate approximations based on the data. For BMR this can give a few different approximations.
Next you need to measure how energy output changes with whatever other metric you can measure. Ideally you would use power output (either on a treadmill, bike, etc.) along with the amount of oxygen consumed in each breath and quite possibly the change in body temperature. 
At given power output levels (or given force levels depending on the testing), you can add up the heat generated by the body, the heat lost due to radiation from the skin, the heat lost due to sweat evaporation and the heat produced by using up the oxygen in the air to produce power, all of which when you sum them together will give you a number for the energy expended. Not all of these variables can be measured easily and may need to be estimated either from approximations generated by other experiments or from looking at simplified models. Based on collecting a whole bunch of data that way, you could go ahead and compute a set of approximations for height, weight, age based on speed (to replace power since most units don't measure power). Some base it on heart rate also.
However, if you are on a device that does measure power directly and also provides you with an energy estimate, it is most likely doing it a different way. By integrating the power it measures vs time, it is providing you with an energy output. This is technically not the energy used by you but the energy that made it into the exercise equipment! In other words, it doesn't count all the energy lost to body heat and only counts "useful energy." Some devices may apply a correction and add to this number an estimate for energy output based on estimates for BMR, heating etc.. 
You could dig through physiological journals and probably pull out expressions for these curve fits. That doesn't mean that what you see on your treadmill is using them though. Manufacturers like to hide what they use for their computations and like to only report accuracy and precision of their measurements. And usually only one one variable, not on the rest. So for example, if you purchase a power meter for a bike and it says it has accurate power measurements to within X%+/-Y%, that only tells you how good it is at measuring power. If they don't also provide bounds on their energy estimate, or equations for how they determine it, there is no telling how accurate it really is. 
