Recursive Bayesian estimation, update

Recursive Bayesian estimation, update. You have observations like for example, it has a beak, it has feathers, it swims, etc. You want to figure the probability of it being a duck. You know about the definition of a duck with information like p(has_features | it's_a_duck) but you want to figure out probability it's_a_duck given all observations up until now, making one observation at a time.

Recursive Bayesian estimation turns information like p(has_features | it's_a_duck) into what you want in this case. If you already knew probability it's_a_duck given any sequence of observations, you wouldn't need this.

From Wikipedia: 

(1) 

p(x_k | z_k) = c p(z_k | x_k) p(x_k | z_k-1)

 We just ignore the constant c. 

p(x_k | z_k) = p(z_k | x_k) p(x_k | z_k-1)

This is recursive. You could write it in words as:
current_estimate = p(z_k | x_k) estimate_based_on_last_observation

Now an example:

p(x₃ | z₃) = p(z₃ | x₃) p(x₃ | z₂)

Need to find p(x₃ | z₂) 

p(x₃ | z₂) = p(z₂ | x₃) p(x₃ | z₁)

Need to find p(x₃ | z₁) 

Equation (1) no longer applies, but you could say:

p(x₃ | z₁) = p(z₁ | x₃) p(x₃)

p(x_k | z_k)

in terms of p(z_i | x₃)

And we assume these are known. If the hypothesis is that it's a duck. z1 and z2 are pieces of evidence picked up along the way that it's duck or not. Suppose z_{1 is beak observation.} This p(z₁ | x₃) would be probability of beak given it's a duck, like almost 1.