Return to Answer

Assume that $X_1,..., X_n$ arrive sequentially.

How can we update our inference on $\theta$, given that we start with prior $p(\theta)$?

At time step $n$ we know the likelihood $$ p(x_1,...,x_n|\theta) = p(x_1| θ)p(x_2 | x_1, \theta),..., p(x_n | x_{1:n−1}, θ) $$ After observing $n$ points we can compute the posterior distribution $$ p(\theta|x_{1:n}) \propto p(\theta) p(x_{1:n}|\theta). $$

Now, assume that $x_{n+1}$ is available. We can compute the new posterior distribution in the same way as before, starting with the prior and including all $n+1$ points: $$ p(\theta|x_{1:n+1}) \propto p(\theta)p(x_{1:n+1}|\theta). $$

Another way would be to consider $p(\theta|x_{1:n})$ as the current prior on $\theta$ and update this prior with the new observation:

$$ p(\theta|x_{1:n+1}) \propto p(\theta) p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n},\theta) =p(\theta)p(x_{1:n+1}|\theta). $$

This approach provides a simple way to update the posterior distribution, also called the current "state of knowledge", in an online fashion, after arrival of each single data point.

The two approaches yield the same posterior distribution.

Assume that $X_1,..., X_n$ arrive sequentially.

How can we update our inference on $\theta$, given that we start with prior $p(\theta)$?

At time step $n$ we know the likelihood $$ p(x_1,...,x_n|\theta) = p(x_1| \theta)p(x_2 | x_1, \theta),..., p(x_n | x_{1:n−1}, θ) $$ After observing $n$ points we can compute the posterior distribution $$ p(\theta|x_{1:n}) \propto p(\theta) p(x_{1:n}|\theta). $$

Now, assume that $x_{n+1}$ is available. We can compute the new posterior distribution in the same way as before, starting with the prior and including all $n+1$ points: $$ p(\theta|x_{1:n+1}) \propto p(\theta)p(x_{1:n+1}|\theta). $$

Another way would be to consider $p(\theta|x_{1:n})$ as the current prior on $\theta$ and update this prior with the new observation:

$$ p(\theta|x_{1:n+1}) \propto p(\theta) p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n},\theta) =p(\theta)p(x_{1:n+1}|\theta). $$

This approach provides a simple way to update the posterior distribution, also called the current "state of knowledge", in an online fashion, after arrival of each single data point.

The two approaches yield the same posterior distribution.

Assume that $X_1,..., X_n$ arrive sequentially.

How can we update our inference on $\theta$, given that we start with prior $p(\theta)$?

At time step $n$ we know the likelihood $$ p(x_1,...,x_n|\theta) = p(x_1| θ)p(x_2 | x_1, \theta),..., p(x_n | x_{1:n−1}, θ) $$ After observing $n$ points we can compute the posterior distribution $$ p(\theta|x_{1:n}) \propto p(\theta) p(x_{1:n}|\theta). $$

Now, assume that $x_{n+1}$ is available. We can compute the new posterior distribution in the same way as before, starting with the prior and including all $n+1$ points: $$ p(\theta|x_{1:n+1}) \propto p(\theta)p(x_{1:n+1}|\theta). $$

Another way would be to consider $p(\theta|x_{1:n})$ as the current prior on $\theta$ and update this prior with the new observation:

$$ p(\theta|x_{1:n+1}) \propto p(\theta) p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n},\theta) =p(\theta)p(x_{1:n+1}|\theta) $$

The two approaches yield the same posterior distribution.

Assume that $X_1,..., X_n$ arrive sequentially.

How can we update our inference on $\theta$, given that we start with prior $p(\theta)$?

At time step $n$ we know the likelihood $$ p(x_1,...,x_n|\theta) = p(x_1| θ)p(x_2 | x_1, \theta),..., p(x_n | x_{1:n−1}, θ) $$ After observing $n$ points we can compute the posterior distribution $$ p(\theta|x_{1:n}) \propto p(\theta) p(x_{1:n}|\theta). $$

Now, assume that $x_{n+1}$ is available. We can compute the new posterior distribution in the same way as before, starting with the prior and including all $n+1$ points: $$ p(\theta|x_{1:n+1}) \propto p(\theta)p(x_{1:n+1}|\theta). $$

Another way would be to consider $p(\theta|x_{1:n})$ as the current prior on $\theta$ and update this prior with the new observation:

$$ p(\theta|x_{1:n+1}) \propto p(\theta) p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n},\theta) =p(\theta)p(x_{1:n+1}|\theta). $$

This approach provides a simple way to update the posterior distribution, also called the current "state of knowledge", in an online fashion, after arrival of each single data point.

The two approaches yield the same posterior distribution.

Assume that $X_1,..., X_n$ arrive sequentially.

How can we update our inference on $\theta$, given that we start with prior $p(\theta)$?

At time step $n$ we know the likelihood $$ p(x_1,...,x_n|\theta) = p(x_1| θ)p(x_2 | x_1, \theta),..., p(x_n | x_{1:n−1}, θ) $$ After observing $n$ points we can compute the posterior distribution $$ p(\theta|x_{1:n}) \propto p(\theta) p(x_{1:n}|\theta). $$

Now, assume that $x_{n+1}$ is available. We can compute the new posterior distribution as $$ p(\theta|x_{1:n+1}) \propto p(\theta)p(x_{1:n+1}|\theta). $$

Another way would be to consider $p(\theta|x_{1:n})$ as the current prior on $\theta$ and update this prior with the new observation:

$$ p(\theta|x_{1:n+1}) \propto p(\theta) p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n},\theta) =p(\theta)p(x_{1:n+1}|\theta) $$

The two approaches yield the same posterior distribution.

Assume that $X_1,..., X_n$ arrive sequentially.

How can we update our inference on $\theta$, given that we start with prior $p(\theta)$?

At time step $n$ we know the likelihood $$ p(x_1,...,x_n|\theta) = p(x_1| θ)p(x_2 | x_1, \theta),..., p(x_n | x_{1:n−1}, θ) $$ After observing $n$ points we can compute the posterior distribution $$ p(\theta|x_{1:n}) \propto p(\theta) p(x_{1:n}|\theta). $$

Now, assume that $x_{n+1}$ is available. We can compute the new posterior distribution in the same way as before, starting with the prior and including all $n+1$ points: $$ p(\theta|x_{1:n+1}) \propto p(\theta)p(x_{1:n+1}|\theta). $$

Another way would be to consider $p(\theta|x_{1:n})$ as the current prior on $\theta$ and update this prior with the new observation:

$$ p(\theta|x_{1:n+1}) \propto p(\theta) p(x_{1:n}|\theta)p(x_{n+1}|x_{1:n},\theta) =p(\theta)p(x_{1:n+1}|\theta) $$

The two approaches yield the same posterior distribution.