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bayes_algorithm [2016/10/26 22:44] nikolaj |
bayes_algorithm [2016/10/29 16:33] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $W_z:X\to {\mathbb R}$ | | | @#55CCEE: context | @#55CCEE: $W_z:X\to {\mathbb R}$ | | ||
| | @#FF9944: definition | @#FF9944: $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ | | | @#FF9944: definition | @#FF9944: $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ | | ||
| - | | @#FF9944: definition | @#FF9944: $bel_{\mathrm out}[bel_{\mathrm in}](x) := N^*W_z(x)\int_A K_u(x,x')\,bel_{\mathrm in}(x'){\mathrm d}$ | | + | | @#FF9944: definition | @#FF9944: $bel_{\mathrm out}[bel_{\mathrm in}](x) := N^*W_z(x)\int_A K_u(x,x')\,bel_{\mathrm in}(x'){\mathrm d}x'$ | |
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| + | >this is the algorithm for the case where all the ingredient have these types. In practice, Coming up with an initial $bel$ is a also part of the task. | ||
| + | >$N^*$ is supposed to be the normalization of the whole term on the right of it - a normalization to the sum/integral of $bel_{\mathrm in}$. In practice, the latter should normalize to $1$. | ||
| ----- | ----- | ||
| === Discussion === | === Discussion === | ||
| - | $N^*$ is supposed to be the normalization of the whole term on the right of it - a normalization to the sum/integral of $bel_{\mathrm in}$. In practice, the latter should normalize to $1$. | ||
| $K_u(x,x')$ ought to capture the propagation, possibly determined by actions $u$. | $K_u(x,x')$ ought to capture the propagation, possibly determined by actions $u$. | ||
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| and you apply the Bayes Filter with | and you apply the Bayes Filter with | ||
| - | $ P(x_k, x_{k-1}) = p( x_k \mid x_{k-1}) = \mathcal{N} ( {F}_k x_{k-1} + B_k u_k, {Q}_k) $ | + | $ P_u(x_k, x_{k-1}) = p( x_k \mid x_{k-1}, u_k) = \mathcal{N} ( {F}_k x_{k-1} + B_k u_k, {Q}_k) $ |
| - | $ O(x_k) = p( {z}_k\mid x_k) = \mathcal{N}( {H}_{k} x_k, {R}_k) $ | + | $ O_z(x_k) = p( {z}_k\mid x_k) = \mathcal{N}( {H}_{k} x_k, {R}_k) $ |
| where $ \mathcal{N}( x, \sigma^2) $ is the normal distribution, except of course with multivariate arguments. | where $ \mathcal{N}( x, \sigma^2) $ is the normal distribution, except of course with multivariate arguments. | ||
