help me attach the bayesian inference!!

[nguyenhoaiThanhbk2811]

help me attach the bayesian inference!!

[aloctavodia]

Hi @nguyenhoaiThanhbk2811

• You assign prior distributions to parameters, not problems. Thus when you say "My problem has a prior value that is uniform distribution [0,1]", you should identify which parameter in your model has such a prior.

• You say the likelihood is a normal distribution, that's fine. But you also say is a normal with mean 1.35 and standard deviation 0.05. If that's true then you don't need to perform any inference. Because you already know the values of all the parameters in your model.

Thus to solve your problem you must first identify the unknown parameters in your model and then assign priors to those unknown parameters. For example you may have a normal likelihood with know mean, let say 0 and unknown standard deviation. In that case your PyMC3 model will be something like this:

with pm.Model() as model:
    std = pm.HalfNormal('std', 25)
    y = pm.Normal('y', 0, std, observed=your_data)

or maybe you have something like this

with pm.Model() as model:
    mean = pm.Normal('mean', 1.35, 0.05)
    std = pm.Uniform('std', 0, 1)
    y = pm.Normal('y', mean, std, observed=your_data)

[nguyenhoaiThanhbk2811]

Thanks you. you replied your message.
My problem means:
I has a coefficient C1 with [0,1] uniform distribution (P (C1): priors ) and the likelihood of observations P(d/C1) with mean 1.3 and standard deviation 0.5.
My aim is how to find the posterior distribution, P(C1/d) via Bayesian inference with MCMC by pymc3.
Could you give me the tutorial is same as this problem.
Thanks you very much

[aloctavodia]

If you say your data is distributed as Normal with unknown mean and unknown standard deviation, then you put priors over the mean and standard deviation and after that you can perform Bayesian inference. As a result you will get the posterior distribution of the mean and standard deviation.

If instead you say your data is distributed as a normal with a mean 1.35 and standard deviation 0.05, then you already have the answer to your problem. The mean is 1.35 and the standard deviation is 0.05 (without any uncertainty).

A few questions:

Are you saying that the mean is 1.35 and the standard deviation 0.05, because you compute that directly from the data?

How is C1 related to your data?

Could you try to write your model down?

[nguyenhoaiThanhbk2811]

Hi you,
Now, I say detail my problem, I hope that you could help me.
my problem: I have a velocity of an airplane. I don't know exactly the velocity of the airplane. So I want to experiment to have the value of this velocity. I assume that I have a range value of the velocity is [0,1] m/s. Getting 200 random values in the range of the velocity I have corresponding to the 200 values of drag force. I saw the values of drag force which is normal distribution with mean = 1.35 and standard deviation 0.05.
I want to apply the Bayesian inference to find the posterior distribution p(velocity/drag).
maybe my thoughts about the distribution of drag force (likelihood function p(drag/velocity)) is wrong.
Do you know how to apply the Bayesian inference to find the posterior distribution p(velocity/drag).
thanks you.

[aloctavodia]

Hi,

The speed of an object is related to the drag force by a deterministic physical model, something like

Fd = S * k

where:
Fd = drag force
S = speed of the object
k = some constant that should be tabulated somewhere.

Thus, if you know the speed you can directly compute the drag force (or vice versa). And if you have 200 samples of the speed you can directly compute 200 samples of the drag force.

I am not sure from your description what is the observed quantity (the speed or the drag force) but given that you can compute one from the other this should not be a problem, so let assume you have samples of the drag force and you know the value of k from some table, one possible model is to find the mean and standard deviation of the drag force under a Gaussian model, then you will have something like

with pm.Model() as model_0:
    mean_fd = pm.HalfNormal('mean_fd', 25)  # I am assuming the Fd is always positive, the value "25" is just arbitrary.
    sd_fd = pm.HalfNormal('sd_fd', 25)  #  The value "25" is just arbitrary.

    obs_fd = pm.Normal('obs_fd', mu=mean_fd, sigma=sd_fd, observed=data)
    trace = pm.sample()

You can also have a model that put a prior over the mean of the speed and transform the speed into drag force.

with pm.Model() as model_1:
    S = pm.HalfNorma('S', 0, 1)  # I am using a HalfNormal prior instead of the uniform prior you suggested, this generally works better, unless you really know speed over 1 are impossible.
    sd = pm.HalfNormal('sd', 25)  # just some default prior, you should check if this makes sense. 

    Fd = S * k

    obs_fd = pm.Normal('obs_fd', mu=Fd, sigma=sd_fd, observed=data)
    trace = pm.sample()

I hope this helps.

[nguyenhoaiThanhbk2811]

http://www.fz-juelich.de/SharedDocs/Downloads/CONFERENCES/IRW12/EN/MondayAfternoon_08.pdf?__blob=publicationFile

this is the link of paper which I am referring. At the page 7 It is same as the step which I have used Stochastic collocation method based on gPC to find the histogram of the drag force (in this page is power). Similar with the paper, If I can find the "uncertainties affecting the signals" how to apply Bayesian to that?. in my problem is " velocity and drag force" and paper's is "h,tc,Lc with power".

[aloctavodia]

The model in you link is more or less the same as model_1 in my previous comment. In the link F is a function of h,tc,Lc, in model_1, Fd is a function of S and k. In the link the unknown standard deviation is σ, in model_1 sd_fd. In your link the observations are labelled as d and sd_fd in model_1.

[nguyenhoaiThanhbk2811]

Dear you,
If I find the function between input and out. how to consider the posterior distribution.
I have a input Uniform distribution (0.9,1.1). the relative between input and output is a linear regression.

"
import pymc3 as pm
import numpy as np
import scipy.stats as stats
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt

np.random.seed(123)

observed_values = np.array([0.0204232,0.0205054,0.0204971,0.0204463,0.0206686,0.0206678,0.0206883,0.0204627,0.020426,0.0206532,0.0206322,0.020677,0.0204431,0.0205319,0.0204508,0.0204115,0.020721,0.0206988,0.0206179,0.0206418])

basic_model = pm.Model()

with pm.Model () as model:
ce1 = pm.Uniform ('ce1',lower = 0.9, upper =1.1) # prior distribution

# expected value of outcome

linear regression

# likelihood  of observation
y = pm.Normal ('y',mu=mu,sd=ce1,observed=observed_values)

map_estimate = pm.find_MAP(model = basic_model)

print (map_estimate)
"

how to find the posterior distribution of input.

[aloctavodia]

with pm.Model() as model:
    ce1 = pm.Uniform ('ce1',lower = 0.9, upper =1.1)  # prior
    a = pm.Normal('a', 0, 100)  # prior
    b = pm.Normal('b', 0, 100)  # prior
    mu = a + b * x_obs_values   # linear model
    y = pm.Normal ('y', mu=mu, sd=ce1,observed=y_obs_values)
    trace = pm.sample()

Please do not use pm.find_MAP() is not generally a good idea.

The trace object will contain samples from the posterior distribution.

[aloctavodia]

Hi, @nguyenhoaiThanhbk2811 I am closing this. Feel free to reopen if you think this issue has not been solved.

[nguyenhoaiThanhbk2811]

Hi You,
Now If I estimated the surrogate model between input and output.
y = 2.6642 + 2.3829x^4 - 9.8086x^3+15.111x^2-10.334x
how to find the posterior distribution of x.?
with pior distribution of X is normal distribution (mu=1, sd=0.05).

[aloctavodia]

Please explain your problem in detail, what is your data and what do you want to know, how do you get your priors? also how this question is related with the previous ones, is this still the same problem? Are you reading the book? To which model in the book do you think your problem is closer to?