Q:

Bokeh dynamically changing BoxAnnotation

Is there possible to update bokeh figure's renderes in IPython's interact function. I have code which looks like:
x = [0, 1, 2, 3, 4]
y = [0, 1, 2, 3, 4]
source = ColumnDataSource(data=dict(x=x, y=y)
f = figure()
f.line(x, y, source=source)
show(f)

def update_func(selected_data):
   source.data['y'] = ...
   source.push_notebook()
   <here I would like to add BoxAnnotation to figure f, and rerender it>

interactive(update_func, selected_data=[0,1,2])

A:

You could use CustomJS to insert some JavaScript code that will be used to change the bottom and top values of the BoxAnnotation. I'm using the Slider from Bokeh in this example:
from bokeh.io import vform
from bokeh.models import CustomJS, Slider
from bokeh.plotting import figure, show
from bokeh.models import BoxAnnotation

plot = figure(plot_width=300, plot_height=300)
plot.line([0,1],[0,1], line_width=3, line_alpha=0.6)

box_l = BoxAnnotation(plot=plot, top=0.4, 
                      fill_alpha=0.1, fill_color='red')
box_m = BoxAnnotation(plot=plot, bottom = 0.4,top=0.6, 
                      fill_alpha=0.1, fill_color='green')
box_h = BoxAnnotation(plot=plot, bottom=0.6, 
                      fill_alpha=0.1, fill_color='red')
plot.renderers.extend([box_l, box_m, box_h])

callb_low = CustomJS(args=dict(box_l=box_l,box_m=box_m,plot=plot),
    code="""
        var level = cb_obj.get('value')
        box_l.set({"top":level})
        box_m.set({"bottom":level})
        plot.trigger('change');
    """)

callb_high = CustomJS(args=dict(box_m=box_m,box_h=box_h,plot=plot),
    code="""
        var level = cb_obj.get('value')
        box_m.set({"top":level})
        box_h.set({"bottom":level})
        plot.trigger('change');
    """)

slider1 = Slider(start=0.1, end=1, value=0.4, step=.01, title="low",
                 callback=callb_low)
slider2 = Slider(start=0.1, end=1, value=0.6, step=.01, title="high",
                 callback=callb_high)
layout = vform(slider1,slider2, plot)
show(layout)

The output will look like:

Based on the suggestions by bigreddot, you can have the following done in ipython notebooks:
from bokeh.io import push_notebook
from bokeh.plotting import figure, show
from bokeh.models import BoxAnnotation
from ipywidgets import interact

p = figure(x_range=(0,1), y_range=(0,1),plot_width=300, plot_height=300)
box_L = BoxAnnotation(plot=p, top=0.4, 
                      fill_alpha=0.1, fill_color='red')
box_M = BoxAnnotation(plot=p, bottom = 0.4,top=0.6, 
                      fill_alpha=0.1, fill_color='green')
box_H = BoxAnnotation(plot=p, bottom=0.6, 
                      fill_alpha=0.1, fill_color='red')
p.renderers.extend([box_L, box_M, box_H])

def update_func(thresh_L=0.4, thresh_H=0.6):
    box_L.top = box_M.bottom = thresh_L;
    box_M.top = box_H.bottom=thresh_H;
    p.renderers.extend([box_L, box_M, box_H])
    push_notebook()  # note, just a function, not a method on "source"

show(p)

Then in a separate cell you start your sliders like:
interact(update_func, thresh_L=(0, 1, 0.1),thresh_H=(0, 1, 0.1))

A:

As of Bokeh 0.11 the push_notebook function can now update arbitrary Bokeh model objects. So just set/update the properties on your box annotation and call push_notebook. Your code would look something like:
x = [0, 1, 2, 3, 4]
y = [0, 1, 2, 3, 4]
source = ColumnDataSource(data=dict(x=x, y=y)
f = figure()
f.line(x, y, source=source)
show(f)

def update_func(selected_data):
    source.data['y'] = ...
    # update box annotation (or any other properties) here
    push_notebook()  # note, just a function, not a method on "source"

interactive(update_func, selected_data=[0,1,2]) 

You can see some howto notebooks here:
https://github.com/bokeh/bokeh/blob/master/examples/howto/notebook_comms/Basic%20Usage.ipynb
https://github.com/bokeh/bokeh/blob/master/examples/howto/notebook_comms/Continuous%20Updating.ipynb
https://github.com/bokeh/bokeh/blob/master/examples/howto/notebook_comms/Jupyter%20Interactors.ipynb

